Finite Group Databasebeta

← #29686962

Subgroups — #29686962

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#21454721C211yesmaximal, characteristic, complemented
3#21454718C211yesmaximal, characteristic, complemented
4C2×#16839881C211yesmaximal, characteristic, complemented
5#21454248C211yesmaximal, characteristic, complemented
6#21446716C211yesmaximal, characteristic, complemented
7#21446606C211yesmaximal, characteristic, complemented
8#21446488C211yesmaximal, characteristic, complemented
9#21446323C211yesmaximal, characteristic, complemented
10#21446319C211yesmaximal, characteristic, complemented
11#21445872C211yesmaximal, characteristic, complemented
12#21445702C211yesmaximal, characteristic, complemented
13#21445622C211yesmaximal, characteristic, complemented
14#21445618C211yesmaximal, characteristic, complemented
15C61×#1932C211yesmaximal, characteristic, complemented
16#16840562C2211yescharacteristic
17#16840560C2211yescharacteristic
18C2×#14519797C2211yescharacteristic
19C2×#14519785C2211yescharacteristic
20C2×#14519785C2211yescharacteristic
21C2×#14519785C2211yescharacteristic
22C2×#14519784C2211yescharacteristic
23C4×#11758222C2211yescharacteristic
24#16840036C412yescomplemented
25#16840031C412yescomplemented
26#16840028C412yescomplemented
27#16840024C412yescomplemented
28#16839883—22no—
29#16839881—22no—
30#16839881—21no—
31#16839881—21no—
32#16839881—21no—
33#16839881—21no—
34#16839878—22no—
35#16839876—22no—
36#16839810C2211yescharacteristic
37C2×#14519712C2211yescharacteristic
38C2×#14519712C2211yescharacteristic
39#16839798C2211yescharacteristic
40#16839780C2211yescharacteristic
41#16839778C2211yescharacteristic
42#16839778C2211yescharacteristic
43C2×#14519705C2211yescharacteristic
44#16839744C2211yescharacteristic
45#16839736C2211yescharacteristic
46#16839735C2211yescharacteristic
47#16839720C2211yescharacteristic
48#16839720C2211yescharacteristic
49#16839720C2211yescharacteristic
50#16839718C2211yescharacteristic
51#16839716—21no—
52#16839716—21no—
53#16839714—21no—
54C2×#14519688C2211yescharacteristic
55#97×D61—21no—
56#16839534C2211yescharacteristic
57#16839534C2211yescharacteristic
58#16839533C2211yescharacteristic
59#16839522C2211yescharacteristic
60C61×#430C2211yescharacteristic
61C61×#421C2211yescharacteristic
62C2×#104×C61C2211yescharacteristic
63C61×#413C2211yescharacteristic
64C4×M16×C61C2211yescharacteristic
65C61×#344C2211yescharacteristic
66C61×#339C2211yescharacteristic
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#14519785—21no—
90#14519785—21no—
91#14519785—21no—
92#14519785—21no—
93#14519785—21no—
94#14519785—21no—
95#14519785C2×C412yes—
96#14519785C2×C412yes—
97#14519785D412yes—
98#14519785D412yes—
99#14519784—21no—
100#14519784—21no—
101#14519784—21no—
102#14519784—21no—
103C2×#11758194—21no—
104C2×#11758194C2311yescharacteristic
105C2×#11758194C2311yescharacteristic
106C2×#11758189—21no—
107C2×#11758189—21no—
108C2×#11758189—21no—
109C22×#92418—21no—
110C22×#92418C2311yescharacteristic
111C22×#92418C2311yescharacteristic
112C2×C8×D61—21no—
113C2×C8×D61—21no—
114C2×C8×D61—21no—
115C4×Dic122—21no—
116C4×D244—21no—
117C4×C22⋊D61—22no—
118#14519736C2311yescharacteristic
119#14519736C2311yescharacteristic
120#14519734—21no—
121C42×D61—21no—
122#14519712—22no—
123#14519712—21no—
124#14519712—21no—
125#14519712C2×C412yescomplemented
126#14519712C2×C412yescomplemented
127#14519710—21no—
128#14519708—21no—
129#14519705—22no—
130#14519705—21no—
131#14519705—21no—
132#14519703—22no—
133#14519703—22no—
134#14519695C2×C412yescomplemented
135#14519693C2×C412yescomplemented
136#14519692—22no—
137#14519691—22no—
138#14519688—21no—
139#14519688D411yescharacteristic
140#14519688D411yescharacteristic
141#14519688D411yescharacteristic
142#14519688D411yescharacteristic
143#14519687—21no—
144#14519687—21no—
145#14519687—21no—
146C4×#92418—21no—
147C8×Dic61—21no—
148C8×Dic61—21no—
149C8×Dic61—21no—
150C2×#41×C61C2311yescharacteristic
151C2×M16×C61C2311yescharacteristic
152C2×M16×C61C2311yescharacteristic
153C2×M16×C61C2311yescharacteristic
154C2×M16×C61C2311yescharacteristic
155#117×C61C2311yescharacteristic
156C2×C42×C61C2311yescharacteristic
157#108×C61—22no—
158#104×C61—22no—
159#104×C61—21no—
160#104×C61—21no—
161#101×C61C2×C412yescomplemented
162#100×C61C2×C412yescomplemented
163#97×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×C61—21no—
174C22×C4×C61C2411yescharacteristic
175C2×C22⋊D61—21no—
176C2×C22⋊D61—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—21no—
197M16×C61C22×C412yes—
198M16×C61C22×C412yes—
199M16×C61C2×D412yes—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C2×C8×C61—21no—
203C2×C8×C61—21no—
204C4⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C42×C61—21no—
207C42×C61—21no—
208C42×C61C2×D411yescharacteristic
209C42×C61C2×D411yescharacteristic
210C22⋊Dic61—21no—
211C22⋊Dic61#4112yes—
212#11758202—22no—
213#11758202—21no—
214#11758202—21no—
215#11758201—22no—
216#11758201—21no—
217#11758201—21no—
218#11758200—22no—
219#11758200—21no—
220#11758200—21no—
221#11758199—22no—
222#11758199—21no—
223#11758199—21no—
224#11758198—22no—
225C4⋊Dic61—21no—
226C4⋊Dic61—21no—
227#11758196—22no—
228C4×Dic61—22no—
229C4×Dic61—21no—
230C4×Dic61—21no—
231#11758194—21no—
232#11758194—21no—
233#11758194—21no—
234#11758194—41no—
235#11758194—41no—
236#11758194C22×C412yes—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—41no—
240C2×#92418—41no—
241C2×#92418C22×C411yescharacteristic
242C2×#92418C22×C411yescharacteristic
243C2×#92418C2×D411yescharacteristic
244C2×#92418C2×D411yescharacteristic
245C2×#92418C2×D411yescharacteristic
246C2×#92418C2×D411yescharacteristic
247C2×#92418C2×D411yescharacteristic
248C2×#92418C2×D411yescharacteristic
249C2×#92418#4111yescharacteristic
250C2×#92418#4111yescharacteristic
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—21no—
281C2×C4×C61—21no—
282C2×C4×C61—41no—
283C2×C4×C61—41no—
284C2×C4×C61—41no—
285C2×C4×C61—81no—
286C2×C4×C61C23×C411yescharacteristic
287C2×C4×C61C22×D411yescharacteristic
288C2×C4×C61C22×D411yescharacteristic
289C2×C4×C61C2×#4111yescharacteristic
290C2×C4×C61C2×#4111yescharacteristic
291C2×C4×C61C2×#4111yescharacteristic
292C22⋊D61—22no—
293C22⋊D61—22no—
294C22⋊D61—41no—
295C22⋊D61—41no—
296C2×Dic61—21no—
297C2×Dic61—21no—
298C2×Dic61—22no—
299C2×Dic61—41no—
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—22no—
316C8×C61—41no—
317#92418—22no—
318#92418—41no—
319#92418C4×D411yescharacteristic
320#92418C4×D411yescharacteristic
321#92418C4×D411yescharacteristic
322#92418C4×D411yescharacteristic
323#92418#12111yescharacteristic
324#92418#12111yescharacteristic
325#92418#12111yescharacteristic
326#92418#12111yescharacteristic
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—41no—
339C4×C61—41no—
340C4×C61—41no—
341C4×C61—81no—
342C4×C61C2×C4×D411yescharacteristic
343C4×C61C2×C4×D411yescharacteristic
344C4×C61C2×#12111yescharacteristic
345C4×C61#52911yescharacteristic
346Dic61—22no—
347Dic61—42no—
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#1932D6111yescharacteristic, complemented
359#1906—611no—
360#1906—611no—
361#1814—611no—
362#1773—611no—
363#1757—611no—
364#1757—611no—
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×#131—611no—
379C4×#41—611no—
380#473—612no—
381#472—612no—
382#471—612no—
383#471—612no—
384#444—1221no—
385#444—1221no—
386#444—1221no—
387#444—1221no—
388#444—1222no—
389#444—1222no—
390#443—1222no—
391#443—1222no—
392#430—611no—
393#430C2×D6111yescharacteristic
394C2×#108—611no—
395C2×#108—611no—
396#427—611no—
397#421—611no—
398#421—611no—
399#421—611no—
400#421C2×D6111yescharacteristic
401C2×#104—611no—
402C2×#104C2×D6111yescharacteristic
403#413—611no—
404#413C2×D6111yescharacteristic
405#410—611no—
406#410—611no—
407#405—611no—
408#405—611no—
409#405—611no—
410#405—611no—
411#405—1221no—
412#405—1221no—
413C4×M16—1221no—
414C4×M16C2×D6111yescharacteristic
415C2×#97—611no—
416C2×#97—1221no—
417#344—611no—
418#344—611no—
419#344—611no—
420#344C2×D6111yescharacteristic
421#339—611no—
422#339C2×D6111yescharacteristic
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—612no—
446#131—612no—
447#131—612no—
448#131—612no—
449#131—1221no—
450#131—1221no—
451#131—1221no—
452#131—1221no—
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—611no—
462C2×M16—611no—
463C2×M16—1221no—
464C2×M16—1221no—
465C2×M16—1221no—
466C2×M16—1221no—
467C2×M16C22×D6111yescharacteristic
468C2×M16C22×D6111yescharacteristic
469C2×M16C22×D6111yescharacteristic
470C2×M16C22×D6111yescharacteristic
471C22×C8—611no—
472C22×C8—611no—
473C22×C8—1221no—
474C22×C8—1221no—
475C22×C8—1221no—
476C22×C8—1221no—
477C4×Q8—1221no—
478C4×D4—1221no—
479C4×D4—1222no—
480#117—611no—
481#117—611no—
482#117—1221no—
483#117C22×D6111yescharacteristic
484C2×C42—1221no—
485C2×C42C22×D6111yescharacteristic
486#108—22no—
487#108—612no—
488#108—612no—
489#108—1221no—
490#108—1221no—
491#108—1222no—
492#107—1221no—
493#106—1221no—
494#104—22no—
495#104—21no—
496#104—21no—
497#104—1222no—
498#104—1222no—
499#104—1222no—
500#104—1221no—
501#104—1221no—
502#101—612no—
503#101C4×D6112yescomplemented
504#100—612no—
505#100C4×D6112yescomplemented
506#99—1222no—
507#99—1222no—
508#97—21no—
509#97—611no—
510#97—611no—
511#97—611no—
512#97—611no—
513#97—1221no—
514#97—1221no—
515#97—1221no—
516#97—1221no—
517C4×C8—21no—
518C4×C8—1221no—
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—21no—
545C22×C4—1222no—
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—21no—
570M16—612no—
571M16—1222no—
572M16—1222no—
573M16—1222no—
574M16—1222no—
575M16—1221no—
576M16—1221no—
577M16—1221no—
578M16—2441no—
579M16—2441no—
580M16—2441no—
581M16—2441no—
582M16C2×C4×D6112yes—
583M16C2×C4×D6112yes—
584M16D4×D6112yes—
585C2×C8—21no—
586C2×C8—21no—
587C2×C8—21no—
588C2×C8—21no—
589C2×C8—611no—
590C2×C8—611no—
591C2×C8—611no—
592C2×C8—611no—
593C2×C8—611no—
594C2×C8—611no—
595C2×C8—611no—
596C2×C8—611no—
597C2×C8—611no—
598C2×C8—611no—
599C2×C8—611no—
600C2×C8—611no—
601C2×C8—1222no—
602C2×C8—1222no—
603C2×C8—1222no—
604C2×C8—1222no—
605C2×C8—1221no—
606C2×C8—1221no—
607C2×C8—2441no—
608C2×C8—2441no—
609C2×C8—2441no—
610C2×C8—2441no—
611C4⋊C4—21no—
612C4⋊C4—1222no—
613C4⋊C4—1221no—
614C4⋊C4—1221no—
615C22⋊C4—21no—
616C22⋊C4—612no—
617C22⋊C4—1222no—
618C22⋊C4—1221no—
619C42—21no—
620C42—21no—
621C42—1222no—
622C42—1221no—
623C42—1221no—
624C42D4×D6111yescharacteristic
625C42D4×D6111yescharacteristic
626C23—21no—
627C23—1221no—
628C23—1221no—
629C23#41×D6111yescharacteristic
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—21no—
650C2×C4—21no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—81no—
655C2×C4—1221no—
656C2×C4—1222no—
657C2×C4—1222no—
658C2×C4—1221no—
659C2×C4—1222no—
660C2×C4—4882no—
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×C4#41×D6111yescharacteristic
670C8—22no—
671C8—22no—
672C8—22no—
673C8—41no—
674C8—611no—
675C8—611no—
676C8—611no—
677C8—611no—
678C8—611no—
679C8—611no—
680C8—611no—
681C8—611no—
682C8—1222no—
683C8—2441no—
684C22—41no—
685C22—41no—
686C22—81no—
687C22—1222no—
688C22—4881no—
689C22—2442no—
690C22—2441no—
691C22#1684023011yescharacteristic
692C22#1684027011yescharacteristic
693C22#1684031111yescharacteristic
694C4—41no—
695C4—41no—
696C4—41no—
697C4—81no—
698C4—1222no—
699C4—4881no—
700C4—2442no—
701C4—2441no—
702C4C4×D4×D6111yescharacteristic
703C4C4×D4×D6111yescharacteristic
704C4#121×D6111yescharacteristic
705C4#1684029511yescharacteristic
706C2—21no—
707C2—21no—
708C2—21no—
709C2—81no—
710C2—4881no—
711C2—2442no—
712C2#2144833111yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1932C2
socleC2×C61C4×#121
prime radical#21455538C2
cosocleC22×C4×C61C24
hypercenter#1932D61
hypocenterC61#21559
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1932

Complements

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

NH ≅ G/Ntype
2. #21454721C2semidirect
3. #21454718C2semidirect
4. C2×#16839881C2semidirect
5. #21454248C2semidirect
6. #21446716C2semidirect
7. #21446606C2semidirect
8. #21446488C2semidirect
9. #21446323C2semidirect
10. #21446319C2semidirect
11. #21445872C2semidirect
12. #21445702C2semidirect
13. #21445622C2semidirect
14. #21445618C2semidirect
15. C61×#1932C2semidirect
24. #16840036C4semidirect
25. #16840031C4semidirect
26. #16840028C4semidirect
27. #16840024C4semidirect
83. C2×#11758202C2×C4semidirect
84. C2×#11758200C2×C4semidirect
85. C2×#11758201C2×C4semidirect
86. C2×#11758199C2×C4semidirect
125. #14519712C2×C4semidirect
126. #14519712C2×C4semidirect
134. #14519695C2×C4semidirect
135. #14519693C2×C4semidirect
161. #101×C61C2×C4semidirect
162. #100×C61C2×C4semidirect
358. #1932D61semidirect
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
#214547212
#214547182
C2×#168398812
#214542482
#214467162
#214466062
#214464882
#214463232
#214463192
#214458722
#214457022
#214456222
#214456182
C61×#19322
#2155961