Finite Group Databasebeta

← #29686949

Subgroups — #29686949

Order 15616.

876 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#21455536C211yesmaximal, characteristic
2#21454722C211yesmaximal, characteristic, complemented
3#21454718C211yesmaximal, characteristic, complemented
4C2×#16839876C211yesmaximal, characteristic, complemented
5C4×#14519784C211yesmaximal, characteristic, complemented
6#21446714C211yesmaximal, characteristic, complemented
7#21446608C211yesmaximal, characteristic, complemented
8#21446489C211yesmaximal, characteristic, complemented
9#21446485C211yesmaximal, characteristic, complemented
10#21446320C211yesmaximal, characteristic, complemented
11#21445871C211yesmaximal, characteristic, complemented
12#21445697C211yesmaximal, characteristic, complemented
13#21445621C211yesmaximal, characteristic, complemented
14#21445618C211yesmaximal, characteristic, complemented
15C61×#1906C211yesmaximal, characteristic, complemented
16#16840560C2211yescharacteristic
17C2×#14519797C2211yescharacteristic
18C2×#14519785C2211yescharacteristic
19C2×#14519784C2211yescharacteristic
20C2×#14519784C2211yescharacteristic
21C2×#14519784C2211yescharacteristic
22#16840035C412yescomplemented
23#16840030C412yescomplemented
24#16840027C412yescomplemented
25#16840023C412yescomplemented
26#16839884—22no—
27#16839881—22no—
28#16839879—22no—
29#16839876—22no—
30#16839876—21no—
31#16839876—21no—
32#16839876—21no—
33#16839876—21no—
34#16839811C2211yescharacteristic
35#16839810C2211yescharacteristic
36C2×#14519712C2211yescharacteristic
37C2×#14519712C2211yescharacteristic
38#16839798C2211yescharacteristic
39#16839780C2211yescharacteristic
40#16839779C2211yescharacteristic
41#16839779C2211yescharacteristic
42C2×#14519705C2211yescharacteristic
43#16839746C2211yescharacteristic
44#16839744C2211yescharacteristic
45#16839735C2211yescharacteristic
46#16839720C2211yescharacteristic
47#16839718—21no—
48#16839718—21no—
49#16839718C2211yescharacteristic
50#16839718C2211yescharacteristic
51#16839718C2211yescharacteristic
52C4×#11758194—21no—
53C4×#11758194C2211yescharacteristic
54C2×C4×#92418—21no—
55C2×C4×#92418C2211yescharacteristic
56#16839533C2211yescharacteristic
57#16839532C2211yescharacteristic
58#16839532C2211yescharacteristic
59#16839522C2211yescharacteristic
60C61×#575C2211yescharacteristic
61C4×#41×C61C2211yescharacteristic
62C61×#421C2211yescharacteristic
63C2×#104×C61C2211yescharacteristic
64C61×#410C2211yescharacteristic
65C61×#344C2211yescharacteristic
66C61×#339C2211yescharacteristic
67C2×#11758222C2311yescharacteristic
68#14519808—21no—
69#14519807C2×C412yes—
70#14519803—21no—
71#14519802C2×C412yes—
72#14519797—22no—
73#14519797—21no—
74#14519797—21no—
75#14519797—21no—
76#14519797—21no—
77C2×#11758202C2×C411yescharacteristic, complemented
78C2×#11758200C2×C411yescharacteristic, complemented
79C2×#11758201C2×C411yescharacteristic, complemented
80C2×#11758199C2×C411yescharacteristic, complemented
81#14519785—21no—
82#14519785—21no—
83#14519785—21no—
84#14519785—21no—
85#14519784—21no—
86#14519784—21no—
87#14519784—21no—
88#14519784—21no—
89#14519784—21no—
90#14519784—21no—
91#14519784—21no—
92#14519784—21no—
93#14519784C2×C412yes—
94#14519784C2×C412yes—
95#14519784D412yes—
96#14519784D412yes—
97C2×#11758194—21no—
98C2×#11758194—21no—
99C2×#11758194—21no—
100C2×#11758194C2311yescharacteristic
101C2×#11758194C2311yescharacteristic
102C2×#11758194C2311yescharacteristic
103C2×#11758194C2311yescharacteristic
104C2×#11758189—21no—
105C22×#92418—21no—
106C22×#92418—21no—
107C22×#92418—21no—
108C22×#92418C2311yescharacteristic
109C22×#92418C2311yescharacteristic
110C2×C8×D61—21no—
111#14519736C2311yescharacteristic
112#14519713—22no—
113#14519712—22no—
114#14519712—21no—
115#14519712—21no—
116#14519712C2×C412yescomplemented
117#14519712C2×C412yescomplemented
118#14519710—21no—
119#14519708—21no—
120#14519705—21no—
121#14519705—21no—
122#14519705—22no—
123#14519704—22no—
124#14519704—22no—
125#14519696C2×C412yescomplemented
126#14519695C2×C412yescomplemented
127#14519694C2×C412yescomplemented
128#14519693C2×C412yescomplemented
129#14519691—22no—
130#14519688—21no—
131#14519688—21no—
132#14519688—21no—
133#14519688—21no—
134#14519687—21no—
135C4×#92418—21no—
136C4×#92418—21no—
137C4×#92418—21no—
138C4×#92418—21no—
139C4×#92418D411yescharacteristic
140C4×#92418D411yescharacteristic
141C4×#92418D411yescharacteristic
142C4×#92418D411yescharacteristic
143C8×Dic61—21no—
144C2×#41×C61C2311yescharacteristic
145C2×#41×C61C2311yescharacteristic
146#137×C61—21no—
147#137×C61C2×C412yes—
148AGL(1,ℤ/8)×C61—21no—
149AGL(1,ℤ/8)×C61C2×C412yes—
150#135×C61—22no—
151C2×M16×C61C2311yescharacteristic
152C2×M16×C61C2311yescharacteristic
153C4×Q8×C61—21no—
154C4×D4×C61—21no—
155C4×D4×C61—22no—
156#117×C61—21no—
157#117×C61C2311yescharacteristic
158#117×C61C2311yescharacteristic
159C2×C42×C61—21no—
160C2×C42×C61C2311yescharacteristic
161#104×C61—22no—
162#104×C61—21no—
163#104×C61—21no—
164#99×C61—22no—
165#41×C61—21no—
166#41×C61—21no—
167#41×C61—21no—
168#41×C61—21no—
169#41×C61—21no—
170#41×C61—21no—
171#41×C61C22×C412yes—
172#41×C61C2×D412yes—
173C2×Q8×C61C22×C411yescharacteristic
174C2×Q8×C61#4111yescharacteristic
175C2×D4×C61—21no—
176C2×D4×C61—21no—
177C2×D4×C61C22×C411yescharacteristic
178C2×D4×C61#4111yescharacteristic
179C22×C4×C61—21no—
180C22×C4×C61—22no—
181C22×C4×C61—21no—
182C22×C4×C61—21no—
183C22×C4×C61C2411yescharacteristic
184C2×C22⋊D61—21no—
185#11758222—21no—
186#11758222—21no—
187#11758222—21no—
188#11758222—21no—
189C2×D244C22×C411yescharacteristic
190C2×C4×D61—21no—
191C2×Dic122C22×C411yescharacteristic
192Q16×C61—22no—
193SD16×C61—22no—
194SD16×C61—22no—
195D8×C61—22no—
196M16×C61—21no—
197M16×C61—21no—
198M16×C61—21no—
199M16×C61C22×C412yes—
200C2×C8×C61—21no—
201C2×C8×C61—21no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—21no—
207C22⋊C4×C61#4112yes—
208C42×C61—22no—
209C42×C61—21no—
210C42×C61—21no—
211C42×C61—21no—
212C42×C61C2×D411yescharacteristic
213C42×C61C2×D411yescharacteristic
214C22⋊Dic61—21no—
215#11758202—22no—
216#11758202—21no—
217#11758202—21no—
218#11758201—22no—
219#11758201—21no—
220#11758201—21no—
221#11758200—22no—
222#11758200—21no—
223#11758200—21no—
224#11758199—22no—
225#11758199—21no—
226#11758199—21no—
227C4⋊Dic61—21no—
228C4×Dic61—21no—
229#11758194—21no—
230#11758194—21no—
231#11758194—21no—
232#11758194—22no—
233#11758194—22no—
234#11758194—22no—
235#11758194—22no—
236#11758194—21no—
237#11758194—21no—
238#11758194—41no—
239#11758194—41no—
240#11758194C22×C412yes—
241#11758194C22×C412yes—
242#11758194C2×D412yes—
243C2×#92418—21no—
244C2×#92418—21no—
245C2×#92418—21no—
246C2×#92418—22no—
247C2×#92418—22no—
248C2×#92418—22no—
249C2×#92418—22no—
250C2×#92418—21no—
251C2×#92418—41no—
252C2×#92418—41no—
253C2×#92418C22×C411yescharacteristic
254C2×#92418C22×C411yescharacteristic
255C2×#92418C2×D411yescharacteristic
256C2×#92418C2×D411yescharacteristic
257C2×#92418C2×D411yescharacteristic
258C2×#92418C2×D411yescharacteristic
259C2×#92418C2×D411yescharacteristic
260C2×#92418C2×D411yescharacteristic
261C2×#92418#4111yescharacteristic
262C2×#92418#4111yescharacteristic
263C2×#92418#4111yescharacteristic
264C2×#92418#4111yescharacteristic
265#11758189—41no—
266#11758189—41no—
267C8×D61—41no—
268C8×D61—41no—
269C23×C61—21no—
270C23×C61—21no—
271C23×C61C2×#4111yescharacteristic
272C22×D61—21no—
273Q8×C61—21no—
274Q8×C61—21no—
275Q8×C61—21no—
276Q8×C61C4×D412yes—
277D4×C61—22no—
278D4×C61—22no—
279D4×C61—21no—
280D4×C61—21no—
281D4×C61—21no—
282D4×C61—41no—
283D4×C61—41no—
284D4×C61C4×D412yes—
285C2×C4×C61—21no—
286C2×C4×C61—21no—
287C2×C4×C61—22no—
288C2×C4×C61—22no—
289C2×C4×C61—21no—
290C2×C4×C61—22no—
291C2×C4×C61—21no—
292C2×C4×C61—42no—
293C2×C4×C61—44no—
294C2×C4×C61—41no—
295C2×C4×C61—41no—
296C2×C4×C61—41no—
297C2×C4×C61C23×C411yescharacteristic
298C2×C4×C61C22×D411yescharacteristic
299C2×C4×C61C22×D411yescharacteristic
300C2×C4×C61C2×#4111yescharacteristic
301C2×C4×C61C2×#4111yescharacteristic
302C2×C4×C61C2×#4111yescharacteristic
303C22⋊D61—41no—
304C22⋊D61—41no—
305C2×Dic61—21no—
306C2×Dic61—41no—
307D244—21no—
308D244—21no—
309C4×D61—41no—
310C4×D61—41no—
311Dic122—21no—
312Dic122—21no—
313C8×C61—22no—
314C8×C61—41no—
315#92418—22no—
316#92418—22no—
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#21558—611nomaximal, Sylow
328C22×C61—22no—
329C22×C61—41no—
330C22×C61—42no—
331C22×C61—41no—
332C22×C61—81no—
333C22×C61C4×#4111yescharacteristic
334C22×C61#52511yescharacteristic
335C22×C61#53211yescharacteristic
336C2×D61—41no—
337C2×D61—81no—
338C4×C61—22no—
339C4×C61—22no—
340C4×C61—42no—
341C4×C61—41no—
342C4×C61—41no—
343C4×C61—81no—
344C4×C61C2×C4×D411yescharacteristic
345C4×C61C2×C4×D411yescharacteristic
346C4×C61C2×#12111yescharacteristic
347C4×C61#52911yescharacteristic
348Dic61—41no—
349Dic61—81no—
350#3067—611no—
351#2955—611no—
352#2955—611no—
353C2×#443—611no—
354C4×#131—611no—
355#1976—611no—
356#1953—611no—
357#1932—611no—
358#1932—611no—
359#1906—611no—
360#1906D6111yescharacteristic, complemented
361#1814—611no—
362#1772—611no—
363#1757—611no—
364#1757—611no—
365C2×C61—21no—
366C2×C61—21no—
367C2×C61—21no—
368C2×C61—42no—
369C2×C61—81no—
370C2×C61C4×#12111yescharacteristic
371D61—81no—
372#575—611no—
373#575C2×D6111yescharacteristic
374C2×#135—611no—
375C2×#131—611no—
376C2×#131—611no—
377C2×#131—611no—
378C2×#131—611no—
379C4×#41C2×D6111yescharacteristic
380#473—612no—
381#472—612no—
382#471—612no—
383#471—612no—
384#444—1222no—
385#444—1222no—
386#443—1221no—
387#443—1221no—
388#443—1221no—
389#443—1221no—
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#421C2×D6111yescharacteristic
401C2×#104—611no—
402C2×#104C2×D6111yescharacteristic
403#413—611no—
404#413—611no—
405#410—611no—
406#410C2×D6111yescharacteristic
407#405—611no—
408#405—611no—
409#405—611no—
410#405—611no—
411#405—1221no—
412#405—1221no—
413C4×M16—611no—
414C4×M16—1221no—
415C2×C4×C8—611no—
416C2×C4×C8—1221no—
417#344—611no—
418#344—611no—
419#344—611no—
420#344C2×D6111yescharacteristic
421#339—611no—
422#339C2×D6111yescharacteristic
423C61#2155811yescharacteristic, complemented, Sylow
424C2×#41—611no—
425C2×#41C22×D6111yescharacteristic
426C2×#41C22×D6111yescharacteristic
427#137—21no—
428#137—612no—
429#137—1221no—
430#137C4×D6112yes—
431AGL(1,ℤ/8)—21no—
432AGL(1,ℤ/8)—612no—
433AGL(1,ℤ/8)—1221no—
434AGL(1,ℤ/8)C4×D6112yes—
435#135—22no—
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—611no—
464C2×M16—611no—
465C2×M16—1221no—
466C2×M16—1221no—
467C2×M16—1221no—
468C2×M16—1221no—
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—21no—
478C4×D4—21no—
479C4×D4—22no—
480#117—21no—
481#117—611no—
482#117C22×D6111yescharacteristic
483#117C22×D6111yescharacteristic
484C2×C42—21no—
485C2×C42C22×D6111yescharacteristic
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—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#101—612no—
504#100—612no—
505#100—612no—
506#99—22no—
507#99—1222no—
508#97—1221no—
509#97—1221no—
510#97—1221no—
511#97—1221no—
512#97—1221no—
513C4×C8—611no—
514C4×C8—611no—
515C4×C8—611no—
516C4×C8—611no—
517C4×C8—1221no—
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—21no—
527#41—21no—
528#41—1221no—
529#41—1221no—
530#41—1221no—
531#41—1221no—
532#41C2×C4×D6112yes—
533#41D4×D6112yes—
534C2×Q8—611no—
535C2×Q8C2×C4×D6111yescharacteristic
536C2×Q8#1175822411yescharacteristic
537C2×D4—21no—
538C2×D4—21no—
539C2×D4—611no—
540C2×D4—1221no—
541C2×D4C2×C4×D6111yescharacteristic
542C2×D4#1175822411yescharacteristic
543C22×C4—21no—
544C22×C4—22no—
545C22×C4—21no—
546C22×C4—21no—
547C22×C4—1221no—
548C22×C4C23×D6111yescharacteristic
549Q16—22no—
550Q16—1221no—
551Q16—1221no—
552Q16—1222no—
553SD16—22no—
554SD16—22no—
555SD16—1221no—
556SD16—1221no—
557SD16—1221no—
558SD16—1221no—
559SD16—1222no—
560SD16—1222no—
561D8—22no—
562D8—1221no—
563D8—1221no—
564D8—1222no—
565M16—21no—
566M16—21no—
567M16—21no—
568M16—612no—
569M16—612no—
570M16—612no—
571M16—1222no—
572M16—1222no—
573M16—1222no—
574M16—1222no—
575M16—1221no—
576M16—1221no—
577M16—1221no—
578M16—1221no—
579M16—1221no—
580M16—2441no—
581M16—2441no—
582M16—2441no—
583M16—2441no—
584M16C2×C4×D6112yes—
585C2×C8—21no—
586C2×C8—21no—
587C2×C8—611no—
588C2×C8—611no—
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—1222no—
600C2×C8—1222no—
601C2×C8—1222no—
602C2×C8—1222no—
603C2×C8—1221no—
604C2×C8—1221no—
605C2×C8—1221no—
606C2×C8—1221no—
607C2×C8—2441no—
608C2×C8—2441no—
609C2×C8—2441no—
610C2×C8—2441no—
611C4⋊C4—22no—
612C4⋊C4—21no—
613C4⋊C4—21no—
614C4⋊C4—1221no—
615C22⋊C4—22no—
616C22⋊C4—21no—
617C22⋊C4—1221no—
618C22⋊C4#1175822412yes—
619C42—22no—
620C42—21no—
621C42—21no—
622C42—21no—
623C42—1221no—
624C42D4×D6111yescharacteristic
625C42D4×D6111yescharacteristic
626C23—21no—
627C23—21no—
628C23—1221no—
629C23C2×#1175822411yescharacteristic
630Q8—21no—
631Q8—21no—
632Q8—21no—
633Q8—1221no—
634Q8—1221no—
635Q8#1451973712yes—
636D4—22no—
637D4—22no—
638D4—21no—
639D4—21no—
640D4—21no—
641D4—41no—
642D4—41no—
643D4—1221no—
644D4—1221no—
645D4—2441no—
646D4—2441no—
647D4#1451973712yes—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—21no—
653C2×C4—22no—
654C2×C4—21no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—1221no—
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×#1175822411yescharacteristic
670C8—22no—
671C8—41no—
672C8—611no—
673C8—611no—
674C8—611no—
675C8—611no—
676C8—611no—
677C8—611no—
678C8—611no—
679C8—611no—
680C8—1222no—
681C8—1222no—
682C8—1222no—
683C8—2441no—
684C22—22no—
685C22—41no—
686C22—42no—
687C22—41no—
688C22—81no—
689C22—4881no—
690C22—2441no—
691C22C4×#1175822411yescharacteristic
692C22#1684026811yescharacteristic
693C22#1684030711yescharacteristic
694C4—22no—
695C4—22no—
696C4—42no—
697C4—41no—
698C4—41no—
699C4—81no—
700C4—4881no—
701C4—2441no—
702C4C4×D4×D6111yescharacteristic
703C4C2×#1451973711yescharacteristic
704C4#121×D6111yescharacteristic
705C4#1684028911yescharacteristic
706C2—21no—
707C2—21no—
708C2—21no—
709C2—42no—
710C2—81no—
711C2—4881no—
712C2#2144832911yescharacteristic
subgroupG/H
centerC4#121×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1906C2
socleC2×C61C4×#121
prime radical#21455536C2
cosocleC22×C4×C61C24
hypercenter#1906D61
hypocenterC61#21558
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1906

Complements

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

NH ≅ G/Ntype
2. #21454722C2semidirect
3. #21454718C2semidirect
4. C2×#16839876C2semidirect
5. C4×#14519784C2semidirect
6. #21446714C2semidirect
7. #21446608C2semidirect
8. #21446489C2semidirect
9. #21446485C2semidirect
10. #21446320C2semidirect
11. #21445871C2semidirect
12. #21445697C2semidirect
13. #21445621C2semidirect
14. #21445618C2semidirect
15. C61×#1906C2semidirect
22. #16840035C4semidirect
23. #16840030C4semidirect
24. #16840027C4semidirect
25. #16840023C4semidirect
77. C2×#11758202C2×C4semidirect
78. C2×#11758200C2×C4semidirect
79. C2×#11758201C2×C4semidirect
80. C2×#11758199C2×C4semidirect
116. #14519712C2×C4semidirect
117. #14519712C2×C4semidirect
125. #14519696C2×C4semidirect
126. #14519695C2×C4semidirect
127. #14519694C2×C4semidirect
128. #14519693C2×C4semidirect
360. #1906D61semidirect
423. C61#21558semidirect

Maximal subgroups

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

subgroupindex
#214555362
#214547222
#214547182
C2×#168398762
C4×#145197842
#214467142
#214466082
#214464892
#214464852
#214463202
#214458712
#214456972
#214456212
#214456182
C61×#19062
#2155861