Finite Group Databasebeta

← #29686961

Subgroups — #29686961

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455538C211yesmaximal, characteristic
2#21454723C211yesmaximal, characteristic, complemented
3#21454720C211yesmaximal, characteristic, complemented
4C2×#16839882C211yesmaximal, characteristic, complemented
5#21454246C211yesmaximal, characteristic, complemented
6#21446714C211yesmaximal, characteristic, complemented
7#21446608C211yesmaximal, characteristic, complemented
8#21446490C211yesmaximal, characteristic, complemented
9#21446486C211yesmaximal, characteristic, complemented
10#21446321C211yesmaximal, characteristic, complemented
11#21446317C211yesmaximal, characteristic, complemented
12#21445872C211yesmaximal, characteristic, complemented
13#21445704C211yesmaximal, characteristic, complemented
14#21445620C211yesmaximal, characteristic, complemented
15C61×#1757C211yesmaximal, characteristic, complemented
16#16840562C2211yescharacteristic
17#16840560C2211yescharacteristic
18C2×#14519797C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20C2×#14519783C2211yescharacteristic
21C2×#14519783C2211yescharacteristic
22#16840230C2211yescharacteristic
23#16840036C412yescomplemented
24#16840031C412yescomplemented
25#16840028C412yescomplemented
26#16840024C412yescomplemented
27#16839885—22no—
28#16839882—22no—
29#16839882—21no—
30#16839882—21no—
31#16839882—21no—
32#16839882—21no—
33#16839880—22no—
34#16839877—22no—
35#16839811C2211yescharacteristic
36#16839810C2211yescharacteristic
37C2×#14519712C2211yescharacteristic
38C2×#14519712C2211yescharacteristic
39#16839798C2211yescharacteristic
40#16839780C2211yescharacteristic
41#16839780C2211yescharacteristic
42#16839779C2211yescharacteristic
43#16839778C2211yescharacteristic
44C2×#14519704C2211yescharacteristic
45C2×#14519703C2211yescharacteristic
46#16839744C2211yescharacteristic
47#16839735C2211yescharacteristic
48#16839719—21no—
49#16839718C2211yescharacteristic
50#16839717C2211yescharacteristic
51#16839716—21no—
52M16×Dic61C2211yescharacteristic
53C4×#11758189—21no—
54C2×#14519687C2211yescharacteristic
55#97×D61—21no—
56#16839534C2211yescharacteristic
57#16839532C2211yescharacteristic
58#16839523C2211yescharacteristic
59#16839521C2211yescharacteristic
60C2×#131×C61C2211yescharacteristic
61C61×#413C2211yescharacteristic
62C61×#410C2211yescharacteristic
63C61×#405C2211yescharacteristic
64C61×#405C2211yescharacteristic
65C61×#344C2211yescharacteristic
66C61×#344C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68C2×#11758222C2311yescharacteristic
69#14519810—21no—
70#14519810C2×C412yes—
71#14519808C2×C412yes—
72#14519807—21no—
73#14519805—21no—
74#14519805C2×C412yes—
75#14519803C2×C412yes—
76#14519802—21no—
77#14519799—22no—
78#14519797—22no—
79#14519797—21no—
80#14519797—21no—
81#14519797—21no—
82#14519797—21no—
83C2×#11758202C2×C411yescharacteristic, complemented
84C2×#11758200C2×C411yescharacteristic, complemented
85C2×#11758201C2×C411yescharacteristic, complemented
86C2×#11758199C2×C411yescharacteristic, complemented
87#14519785—21no—
88#14519785—21no—
89#14519785C2×C412yes—
90#14519785C2×C412yes—
91#14519783—21no—
92#14519783—21no—
93#14519783—21no—
94#14519783—21no—
95#14519783—21no—
96#14519783—21no—
97#14519783D412yes—
98#14519783D412yes—
99C2×#11758194C2311yescharacteristic
100C2×#11758194C2311yescharacteristic
101C2×#11758194C2311yescharacteristic
102C2×#11758189—21no—
103C2×#11758189—21no—
104C2×#11758189—21no—
105C22×#92418C2311yescharacteristic
106C2×C8×D61—21no—
107C2×C8×D61—21no—
108C2×C8×D61—21no—
109#14519742—21no—
110#14519739—21no—
111#14519737—22no—
112#14519736C2311yescharacteristic
113#14519734—21no—
114C2×C4×Dic61C2311yescharacteristic
115C42×D61—21no—
116#14519713—22no—
117#14519712—22no—
118#14519712—21no—
119#14519712—21no—
120#14519712C2×C412yescomplemented
121#14519712C2×C412yescomplemented
122#14519710—21no—
123#14519708—21no—
124#14519705—22no—
125#14519705—22no—
126#14519704—21no—
127#14519704—21no—
128#14519704—22no—
129#14519703—22no—
130#14519703—21no—
131#14519703—21no—
132#14519695C2×C412yescomplemented
133#14519693C2×C412yescomplemented
134#14519691—22no—
135#14519688—21no—
136#14519687—21no—
137#14519687—21no—
138#14519687D411yescharacteristic
139#14519687D411yescharacteristic
140#14519687D411yescharacteristic
141#14519687D411yescharacteristic
142C4×#92418—21no—
143C8×Dic61—21no—
144C8×Dic61—21no—
145C2×#41×C61C2311yescharacteristic
146#131×C61—21no—
147#131×C61—21no—
148#131×C61—21no—
149#131×C61—21no—
150C2×M16×C61—21no—
151C2×M16×C61C2311yescharacteristic
152C2×M16×C61C2311yescharacteristic
153C2×M16×C61C2311yescharacteristic
154C22×C8×C61—21no—
155C22×C8×C61C2311yescharacteristic
156#117×C61C2311yescharacteristic
157#117×C61C2311yescharacteristic
158#101×C61C2×C412yescomplemented
159#100×C61C2×C412yescomplemented
160#99×C61—22no—
161#97×C61—21no—
162#97×C61—21no—
163C4×C8×C61—21no—
164C4×C8×C61—21no—
165#41×C61—21no—
166#41×C61—21no—
167#41×C61—21no—
168#41×C61—21no—
169C2×Q8×C61C22×C411yescharacteristic
170C2×D4×C61—21no—
171C2×D4×C61C22×C411yescharacteristic
172C22×C4×C61—21no—
173C22×C4×C61C2411yescharacteristic
174C2×C22⋊D61—21no—
175C2×C22⋊D61—21no—
176C22×Dic61—21no—
177#11758222—21no—
178#11758222—21no—
179#11758222—21no—
180#11758222—21no—
181#11758222—21no—
182#11758222—21no—
183#11758222C22×C412yes—
184#11758222C2×D412yes—
185C2×D244C22×C411yescharacteristic
186C2×D244#4111yescharacteristic
187C2×C4×D61—21no—
188C2×C4×D61—21no—
189C2×C4×D61—22no—
190C2×Dic122C22×C411yescharacteristic
191C2×Dic122#4111yescharacteristic
192M16×C61—21no—
193M16×C61—21no—
194M16×C61—21no—
195M16×C61—21no—
196M16×C61—41no—
197M16×C61—41no—
198M16×C61C22×C412yes—
199M16×C61C22×C412yes—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61—41no—
204C2×C8×C61—41no—
205C2×C8×C61C2×D411yescharacteristic
206C2×C8×C61C2×D411yescharacteristic
207C2×C8×C61C2×D411yescharacteristic
208C2×C8×C61C2×D411yescharacteristic
209C2×C8×C61#4111yescharacteristic
210C2×C8×C61#4111yescharacteristic
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C22⋊C4×C61—21no—
214C22⋊C4×C61#4112yes—
215C42×C61—21no—
216C42×C61—21no—
217C22⋊Dic61—21no—
218#11758202—22no—
219#11758202—21no—
220#11758202—21no—
221#11758201—22no—
222#11758201—21no—
223#11758201—21no—
224#11758200—22no—
225#11758200—21no—
226#11758200—21no—
227#11758199—22no—
228#11758199—21no—
229#11758199—21no—
230#11758198—22no—
231C4⋊Dic61—21no—
232#11758196—22no—
233C4×Dic61—22no—
234C4×Dic61—21no—
235C4×Dic61—21no—
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238#11758194—21no—
239#11758194—21no—
240#11758194—21no—
241#11758194—21no—
242#11758194C22×C412yes—
243#11758194C2×D412yes—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—21no—
247C2×#92418C22×C411yescharacteristic
248C2×#92418C22×C411yescharacteristic
249C2×#92418C2×D411yescharacteristic
250C2×#92418C2×D411yescharacteristic
251C2×#92418#4111yescharacteristic
252C2×#92418#4111yescharacteristic
253Dic244—22no—
254D488—22no—
255#11758190—22no—
256#11758190—22no—
257#11758189—22no—
258#11758189—22no—
259#11758189—22no—
260#11758189—22no—
261#11758189—41no—
262#11758189—41no—
263C8×D61—22no—
264C8×D61—22no—
265C8×D61—22no—
266C8×D61—22no—
267C8×D61—41no—
268C8×D61—41no—
269C23×C61—21no—
270C23×C61C2×#4111yescharacteristic
271C22×D61—21no—
272C22×D61—21no—
273Q8×C61—21no—
274Q8×C61—21no—
275D4×C61—21no—
276D4×C61—21no—
277D4×C61—41no—
278D4×C61—41no—
279C2×C4×C61—21no—
280C2×C4×C61—22no—
281C2×C4×C61—41no—
282C2×C4×C61—41no—
283C2×C4×C61—41no—
284C2×C4×C61C23×C411yescharacteristic
285C2×C4×C61C22×D411yescharacteristic
286C2×C4×C61C22×D411yescharacteristic
287C2×C4×C61C2×#4111yescharacteristic
288C2×C4×C61C2×#4111yescharacteristic
289C2×C4×C61C2×#4111yescharacteristic
290C22⋊D61—22no—
291C22⋊D61—22no—
292C22⋊D61—41no—
293C22⋊D61—41no—
294C2×Dic61—21no—
295C2×Dic61—21no—
296C2×Dic61—21no—
297C2×Dic61—21no—
298C2×Dic61—41no—
299C2×Dic61—81no—
300D244—21no—
301D244—21no—
302D244—21no—
303D244C4×D412yes—
304C4×D61—22no—
305C4×D61—22no—
306C4×D61—41no—
307C4×D61—41no—
308C4×D61—82no—
309Dic122—21no—
310Dic122—21no—
311Dic122—21no—
312Dic122C4×D412yes—
313C8×C61—22no—
314C8×C61—22no—
315C8×C61—41no—
316C8×C61#12111yescharacteristic
317C8×C61#12111yescharacteristic
318C8×C61#12111yescharacteristic
319C8×C61#12111yescharacteristic
320#92418—22no—
321#92418—22no—
322#92418—41no—
323#92418C4×D411yescharacteristic
324#92418C4×D411yescharacteristic
325#92418C4×D411yescharacteristic
326#92418C4×D411yescharacteristic
327#21559—611nomaximal, Sylow
328C22×C61—41no—
329C22×C61—41no—
330C22×C61—81no—
331C22×C61C4×#4111yescharacteristic
332C22×C61#52511yescharacteristic
333C22×C61#53211yescharacteristic
334C2×D61—22no—
335C2×D61—42no—
336C2×D61—41no—
337C2×D61—81no—
338C4×C61—42no—
339C4×C61—41no—
340C4×C61—81no—
341C4×C61C2×C4×D411yescharacteristic
342C4×C61C2×C4×D411yescharacteristic
343C4×C61C2×#12111yescharacteristic
344C4×C61#52911yescharacteristic
345Dic61—22no—
346Dic61—41no—
347Dic61—41no—
348Dic61—41no—
349Dic61—81no—
350#3067—611no—
351#2955—611no—
352#2955—611no—
353C2×#444—611no—
354#2875—611no—
355#1976—611no—
356#1953—611no—
357#1932—611no—
358#1932—611no—
359#1906—611no—
360#1906—611no—
361#1814—611no—
362#1773—611no—
363#1757—611no—
364#1757D6111yescharacteristic, complemented
365C2×C61—21no—
366C2×C61—21no—
367C2×C61—21no—
368C2×C61—81no—
369C2×C61C4×#12111yescharacteristic
370D61—42no—
371D61—81no—
372#575—611no—
373#575—611no—
374C2×#135—611no—
375C2×#131—611no—
376C2×#131—611no—
377C2×#131—611no—
378C2×#131C2×D6111yescharacteristic
379C4×#41—611no—
380#473—612no—
381#472—612no—
382#471—612no—
383#471—612no—
384#444—1222no—
385#444—1221no—
386#444—1221no—
387#444—1221no—
388#444—1221no—
389#444—1222no—
390#443—1222no—
391#443—1222no—
392#430—611no—
393#430—611no—
394C2×#108—611no—
395C2×#108—611no—
396#427—611no—
397#421—611no—
398#421—611no—
399#421—611no—
400#421—611no—
401C2×#104—611no—
402C2×#104—611no—
403#413—611no—
404#413C2×D6111yescharacteristic
405#410—611no—
406#410C2×D6111yescharacteristic
407#405—611no—
408#405—611no—
409#405—1221no—
410#405—1221no—
411#405C2×D6111yescharacteristic
412#405C2×D6111yescharacteristic
413C4×M16—611no—
414C4×M16—1221no—
415C2×#97—611no—
416C2×#97—1221no—
417#344—611no—
418#344—611no—
419#344C2×D6111yescharacteristic
420#344C2×D6111yescharacteristic
421#339—611no—
422#339—611no—
423C61#2155911yescharacteristic, complemented, Sylow
424C2×#41—611no—
425C2×#41—611no—
426C2×#41C22×D6111yescharacteristic
427#137—612no—
428#137—612no—
429#137—1221no—
430#137—1221no—
431AGL(1,ℤ/8)—612no—
432AGL(1,ℤ/8)—612no—
433AGL(1,ℤ/8)—1221no—
434AGL(1,ℤ/8)—1221no—
435#135—1222no—
436#135—1221no—
437#135—1221no—
438#135—1221no—
439#135—1221no—
440#135—1222no—
441C2×Q16—611no—
442C2×SD16—611no—
443C2×SD16—611no—
444C2×D8—611no—
445#131—21no—
446#131—21no—
447#131—21no—
448#131—21no—
449#131—612no—
450#131—612no—
451#131—612no—
452#131—612no—
453#131—1221no—
454#131—1221no—
455#131—1221no—
456#131—1221no—
457#131—1221no—
458#131—1221no—
459#131—1221no—
460#131—1221no—
461C2×M16—21no—
462C2×M16—611no—
463C2×M16—611no—
464C2×M16—611no—
465C2×M16—1221no—
466C2×M16—1221no—
467C2×M16—1221no—
468C2×M16C22×D6111yescharacteristic
469C2×M16C22×D6111yescharacteristic
470C2×M16C22×D6111yescharacteristic
471C22×C8—21no—
472C22×C8—611no—
473C22×C8—1221no—
474C22×C8—1221no—
475C22×C8—1221no—
476C22×C8C22×D6111yescharacteristic
477C4×Q8—1221no—
478C4×D4—1221no—
479C4×D4—1222no—
480#117—611no—
481#117—1221no—
482#117C22×D6111yescharacteristic
483#117C22×D6111yescharacteristic
484C2×C42—611no—
485C2×C42—1221no—
486#108—612no—
487#108—612no—
488#108—1222no—
489#108—1221no—
490#108—1221no—
491#108—1222no—
492#107—1221no—
493#106—1221no—
494#104—1222no—
495#104—1222no—
496#104—1221no—
497#104—1221no—
498#104—1221no—
499#104—1221no—
500#104—1222no—
501#104—1222no—
502#101—612no—
503#101C4×D6112yescomplemented
504#100—612no—
505#100C4×D6112yescomplemented
506#99—22no—
507#99—1222no—
508#97—21no—
509#97—21no—
510#97—611no—
511#97—611no—
512#97—611no—
513#97—611no—
514#97—1221no—
515#97—1221no—
516#97—1221no—
517C4×C8—21no—
518C4×C8—21no—
519C4×C8—1221no—
520C4×C8—1221no—
521C4×C8—1221no—
522#41—21no—
523#41—21no—
524#41—21no—
525#41—21no—
526#41—612no—
527#41—612no—
528#41—1221no—
529#41—1221no—
530#41—1221no—
531#41—1221no—
532#41—1221no—
533#41—1221no—
534C2×Q8—611no—
535C2×Q8—611no—
536C2×Q8C2×C4×D6111yescharacteristic
537C2×D4—21no—
538C2×D4—611no—
539C2×D4—611no—
540C2×D4—1221no—
541C2×D4—1221no—
542C2×D4C2×C4×D6111yescharacteristic
543C22×C4—21no—
544C22×C4—1222no—
545C22×C4—1221no—
546C22×C4—1221no—
547C22×C4—1221no—
548C22×C4C23×D6111yescharacteristic
549Q16—1222no—
550Q16—1221no—
551Q16—1221no—
552Q16—1222no—
553SD16—1222no—
554SD16—1222no—
555SD16—1221no—
556SD16—1221no—
557SD16—1221no—
558SD16—1221no—
559SD16—1222no—
560SD16—1222no—
561D8—1222no—
562D8—1221no—
563D8—1221no—
564D8—1222no—
565M16—21no—
566M16—21no—
567M16—21no—
568M16—21no—
569M16—41no—
570M16—41no—
571M16—612no—
572M16—612no—
573M16—1222no—
574M16—1222no—
575M16—1221no—
576M16—1221no—
577M16—1222no—
578M16—1222no—
579M16—1221no—
580M16—1221no—
581M16—2441no—
582M16—2441no—
583M16C2×C4×D6112yes—
584M16C2×C4×D6112yes—
585C2×C8—21no—
586C2×C8—21no—
587C2×C8—21no—
588C2×C8—41no—
589C2×C8—41no—
590C2×C8—611no—
591C2×C8—611no—
592C2×C8—611no—
593C2×C8—611no—
594C2×C8—611no—
595C2×C8—611no—
596C2×C8—1222no—
597C2×C8—1222no—
598C2×C8—1221no—
599C2×C8—1221no—
600C2×C8—1222no—
601C2×C8—1222no—
602C2×C8—1221no—
603C2×C8—2441no—
604C2×C8—2441no—
605C2×C8#1175822211yescharacteristic
606C2×C8#1175822211yescharacteristic
607C2×C8D4×D6111yescharacteristic
608C2×C8D4×D6111yescharacteristic
609C2×C8D4×D6111yescharacteristic
610C2×C8D4×D6111yescharacteristic
611C4⋊C4—21no—
612C4⋊C4—21no—
613C4⋊C4—1222no—
614C4⋊C4—1221no—
615C22⋊C4—21no—
616C22⋊C4—1222no—
617C22⋊C4—1221no—
618C22⋊C4#1175822212yes—
619C42—21no—
620C42—21no—
621C42—611no—
622C42—611no—
623C42—1222no—
624C42—1221no—
625C42—1221no—
626C23—21no—
627C23—1221no—
628C23—1221no—
629C23C2×#1175822211yescharacteristic
630Q8—21no—
631Q8—21no—
632Q8—612no—
633Q8—1221no—
634Q8—1221no—
635Q8—1221no—
636D4—21no—
637D4—21no—
638D4—41no—
639D4—41no—
640D4—612no—
641D4—1222no—
642D4—1222no—
643D4—1221no—
644D4—1221no—
645D4—1221no—
646D4—2441no—
647D4—2441no—
648C2×C4—21no—
649C2×C4—22no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—1221no—
654C2×C4—1222no—
655C2×C4—1222no—
656C2×C4—1221no—
657C2×C4—1221no—
658C2×C4—1221no—
659C2×C4—4882no—
660C2×C4—4881no—
661C2×C4—2441no—
662C2×C4—2441no—
663C2×C4—2441no—
664C2×C4C22×C4×D6111yescharacteristic
665C2×C4C2×D4×D6111yescharacteristic
666C2×C4C2×D4×D6111yescharacteristic
667C2×C4#41×D6111yescharacteristic
668C2×C4#41×D6111yescharacteristic
669C2×C4C2×#1175822211yescharacteristic
670C8—22no—
671C8—22no—
672C8—41no—
673C8—611no—
674C8—611no—
675C8—611no—
676C8—611no—
677C8—1222no—
678C8—1222no—
679C8—2441no—
680C8#1451975011yescharacteristic
681C8#1451975011yescharacteristic
682C8#1451975011yescharacteristic
683C8#1451975011yescharacteristic
684C22—41no—
685C22—41no—
686C22—81no—
687C22—1222no—
688C22—4881no—
689C22—2442no—
690C22—2441no—
691C22C4×#1175822211yescharacteristic
692C22#1684027111yescharacteristic
693C22#1684030911yescharacteristic
694C4—42no—
695C4—41no—
696C4—81no—
697C4—1222no—
698C4—4881no—
699C4—2441no—
700C4—2441no—
701C4—2441no—
702C4C4×D4×D6111yescharacteristic
703C4C4×D4×D6111yescharacteristic
704C4C2×#1451975011yescharacteristic
705C4#1684029011yescharacteristic
706C2—21no—
707C2—21no—
708C2—21no—
709C2—81no—
710C2—4881no—
711C2—2442no—
712C2C4×#1451975011yescharacteristic
subgroupG/H
centerC4C2×#14519750
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1757C2
socleC2×C61C4×#121
prime radical#21455538C2
cosocleC22×C4×C61C24
hypercenter#1757D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1757

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21454723C2semidirect
3. #21454720C2semidirect
4. C2×#16839882C2semidirect
5. #21454246C2semidirect
6. #21446714C2semidirect
7. #21446608C2semidirect
8. #21446490C2semidirect
9. #21446486C2semidirect
10. #21446321C2semidirect
11. #21446317C2semidirect
12. #21445872C2semidirect
13. #21445704C2semidirect
14. #21445620C2semidirect
15. C61×#1757C2semidirect
23. #16840036C4semidirect
24. #16840031C4semidirect
25. #16840028C4semidirect
26. #16840024C4semidirect
83. C2×#11758202C2×C4semidirect
84. C2×#11758200C2×C4semidirect
85. C2×#11758201C2×C4semidirect
86. C2×#11758199C2×C4semidirect
120. #14519712C2×C4semidirect
121. #14519712C2×C4semidirect
132. #14519695C2×C4semidirect
133. #14519693C2×C4semidirect
158. #101×C61C2×C4semidirect
159. #100×C61C2×C4semidirect
364. #1757D61semidirect
423. C61#21559semidirect
503. #101C4×D61semidirect
505. #100C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214555382
#214547232
#214547202
C2×#168398822
#214542462
#214467142
#214466082
#214464902
#214464862
#214463212
#214463172
#214458722
#214457042
#214456202
C61×#17572
#2155961