Finite Group Databasebeta

← #29658455

Subgroups — #29658455

Order 15616.

852 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic, complemented
2#21456791C211yesmaximal, characteristic
3#21456538C211yesmaximal, characteristic, complemented
4#21445075C211yesmaximal, characteristic, complemented
5#21445061C211yesmaximal, characteristic, complemented
6#21445040C211yesmaximal, characteristic, complemented
7#21445026C211yesmaximal, characteristic, complemented
8#21444929C211yesmaximal, characteristic, complemented
9#21444915C211yesmaximal, characteristic, complemented
10#21444897C211yesmaximal, characteristic, complemented
11#21444891C211yesmaximal, characteristic, complemented
12#21444476C211yesmaximal, characteristic, complemented
13#21444240C211yesmaximal, characteristic, complemented
14#21444232C211yesmaximal, characteristic, complemented
15C61×#3555C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840144—21no—
19#16840136C2211yescharacteristic
20#16840119—21no—
21#16840112C2211yescharacteristic
22#16840110C2211yescharacteristic
23#16840049—21no—
24#16840047C2211yescharacteristic
25#16840047C2211yescharacteristic
26#16840010—22no—
27#16839993—21no—
28#16839993—21no—
29#16839972C2211yescharacteristic
30#16839967C2211yescharacteristic
31#16839964—21no—
32#16839962C2211yescharacteristic
33#16839962C2211yescharacteristic
34#16839962C2211yescharacteristic
35#16839957C2211yescharacteristic
36#16839922—21no—
37#16839905—21no—
38#16839860—22no—
39#16839815C2211yescharacteristic
40C2×C4⋊#92418C2211yescharacteristic
41C4×#11758194C2211yescharacteristic
42#16839511C2211yescharacteristic
43#16839511C2211yescharacteristic
44#16839508C2211yescharacteristic
45#16839508C2211yescharacteristic
46Q8⋊#92418C2211yescharacteristic
47Q8⋊#92418C2211yescharacteristic
48Q8⋊#92418C2211yescharacteristic
49Q8⋊#92418C2211yescharacteristic
50Q8⋊#92418C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54C61×#563—22no—
55Q82×C61C2211yescharacteristic
56C61×#557—21no—
57C61×#557C2211yescharacteristic
58C61×#556—21no—
59C61×#556—21no—
60C61×#556C2211yescharacteristic
61C61×#554C2211yescharacteristic
62C61×#552—21no—
63C61×#544—21no—
64C61×#542—21no—
65C61×#541—21no—
66C61×#541—21no—
67C61×#541—22no—
68C61×#533C2211yescharacteristic
69C61×#533C2211yescharacteristic
70C61×#529C2211yescharacteristic
71C2×#11758222—21no—
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758201—22no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201—21no—
78C2×#11758199—21no—
79C2×#11758194—21no—
80C22×#92418—21no—
81#14519777C2311yescharacteristic
82#14519774C2311yescharacteristic
83#14519765—21no—
84#14519751—21no—
85#14519751—21no—
86C4×D244—21no—
87C4×D244—21no—
88#14519710—22no—
89#14519708—21no—
90#14519708—21no—
91C4⋊#92418—21no—
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95C4⋊#92418C2311yescharacteristic
96#14519701—21no—
97#14519701—21no—
98#14519701—22no—
99#14519700—21no—
100#14519700—21no—
101#14519697—21no—
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105#14519697—22no—
106C22⋊#92418—21no—
107#14519688—21no—
108C4×#92418—21no—
109C4×#92418C2311yescharacteristic
110C4×#92418C2311yescharacteristic
111C4⋊Q8×C61—22no—
112C4⋊Q8×C61—21no—
113C4⋊Q8×C61—22no—
114C4⋊Q8×C61—21no—
115C4⋊Q8×C61—21no—
116C4⋊Q8×C61D411yescharacteristic
117C4⋊Q8×C61D411yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120#126×C61—22no—
121#126×C61—22no—
122#126×C61—22no—
123#126×C61—22no—
124#126×C61—22no—
125#126×C61—22no—
126#126×C61—21no—
127#125×C61—22no—
128#125×C61—22no—
129#125×C61—22no—
130#125×C61—22no—
131#125×C61—21no—
132#125×C61—22no—
133#125×C61—21no—
134#125×C61—21no—
135#125×C61—21no—
136#125×C61D411yescharacteristic
137#124×C61—22no—
138#124×C61—21no—
139#124×C61D411yescharacteristic
140#123×C61—22no—
141#123×C61—21no—
142C22⋊Q8×C61—22no—
143C22⋊Q8×C61—22no—
144C22⋊Q8×C61—21no—
145C22⋊Q8×C61—21no—
146C22⋊Q8×C61—22no—
147C22⋊Q8×C61—21no—
148C22⋊Q8×C61—21no—
149C22⋊Q8×C61D411yescharacteristic
150C22⋊Q8×C61D411yescharacteristic
151C22⋊Q8×C61D412yes—
152C22⋊Q8×C61D411yescharacteristic
153#121×C61D411yescharacteristic
154C4×Q8×C61—22no—
155C4×Q8×C61—21no—
156C4×Q8×C61—21no—
157C4×Q8×C61—21no—
158C4×Q8×C61—21no—
159C4×Q8×C61—21no—
160C4×Q8×C61—21no—
161C4×Q8×C61—21no—
162C4×Q8×C61D412yes—
163C4×Q8×C61C2311yescharacteristic
164C4×Q8×C61C2311yescharacteristic
165C4×Q8×C61C2311yescharacteristic
166C4×D4×C61—21no—
167C4×D4×C61—21no—
168C4×D4×C61C2311yescharacteristic
169#117×C61—22no—
170#117×C61—21no—
171#117×C61—21no—
172#117×C61—21no—
173#117×C61—21no—
174C2×C4⋊C4×C61—21no—
175C2×C42×C61C2311yescharacteristic
176C2×Q8×C61—22no—
177C2×Q8×C61—21no—
178C2×Q8×C61—21no—
179C2×Q8×C61C2×D411yescharacteristic
180C2×Q8×C61C2×D411yescharacteristic
181C2×Q8×C61C2×D411yescharacteristic
182C2×Q8×C61C2×D412yes—
183C2×D4×C61—21no—
184C2×D4×C61C2×D411yescharacteristic
185C22×C4×C61—22no—
186C22×C4×C61—21no—
187C22×C4×C61C2×D411yescharacteristic
188C22×C4×C61C2×D411yescharacteristic
189C22×C4×C61C2×D411yescharacteristic
190C2×C22⋊D61—41no—
191#11758222—81no—
192#11758222—81no—
193C2×D244—21no—
194C2×D244—21no—
195C2×D244—21no—
196C2×C4×D61—41no—
197C2×Dic122—21no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—22no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—22no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61—21no—
208C4⋊C4×C61—21no—
209C4⋊C4×C61—21no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61—22no—
214C4⋊C4×C61—21no—
215C4⋊C4×C61—22no—
216C4⋊C4×C61—21no—
217C4⋊C4×C61—21no—
218C4⋊C4×C61—21no—
219C4⋊C4×C61—21no—
220C4⋊C4×C61—21no—
221C4⋊C4×C61—21no—
222C4⋊C4×C61C2×D411yescharacteristic
223C4⋊C4×C61C2×D411yescharacteristic
224C4⋊C4×C61C2×D411yescharacteristic
225C4⋊C4×C61C2×D411yescharacteristic
226C4⋊C4×C61C2×D412yes—
227C22⋊C4×C61—22no—
228C22⋊C4×C61—22no—
229C22⋊C4×C61—22no—
230C22⋊C4×C61—21no—
231C22⋊C4×C61—22no—
232C22⋊C4×C61—21no—
233C22⋊C4×C61—21no—
234C22⋊C4×C61—21no—
235C22⋊C4×C61—21no—
236C22⋊C4×C61—21no—
237C22⋊C4×C61—21no—
238C42×C61—22no—
239C42×C61—22no—
240C42×C61—22no—
241C42×C61—22no—
242C42×C61—21no—
243C42×C61—21no—
244C42×C61—21no—
245C42×C61C2×D411yescharacteristic
246C42×C61C2×D411yescharacteristic
247C42×C61C2×D411yescharacteristic
248C42×C61C2411yescharacteristic
249#11758202—81no—
250#11758201—44no—
251#11758201—81no—
252#11758201—81no—
253#11758201—81no—
254#11758200—81no—
255#11758199—81no—
256#11758198—41no—
257#11758198—41no—
258C4⋊Dic61—21no—
259C4⋊Dic61—21no—
260#11758194—42no—
261C2×#92418—21no—
262C2×#92418—21no—
263C2×#92418—21no—
264C2×#92418—21no—
265C2×#92418—21no—
266C2×#92418—21no—
267C2×#92418—81no—
268C23×C61—21no—
269C23×C61#12011yescharacteristic
270C22×D61—41no—
271Q8×C61—21no—
272Q8×C61—21no—
273Q8×C61—21no—
274Q8×C61—21no—
275Q8×C61—21no—
276Q8×C61—21no—
277Q8×C61—42no—
278Q8×C61—41no—
279Q8×C61—41no—
280Q8×C61—81no—
281Q8×C61—82no—
282D4×C61—21no—
283D4×C61—21no—
284D4×C61—81no—
285C2×C4×C61—21no—
286C2×C4×C61—21no—
287C2×C4×C61—21no—
288C2×C4×C61—21no—
289C2×C4×C61—21no—
290C2×C4×C61—21no—
291C2×C4×C61—21no—
292C2×C4×C61—22no—
293C2×C4×C61—21no—
294C2×C4×C61—21no—
295C2×C4×C61—21no—
296C2×C4×C61—22no—
297C2×C4×C61—21no—
298C2×C4×C61—41no—
299C2×C4×C61—41no—
300C2×C4×C61—41no—
301C2×C4×C61—82no—
302C2×C4×C61#12011yescharacteristic
303C2×C4×C61#12011yescharacteristic
304C2×C4×C61#12011yescharacteristic
305C2×C4×C61C22×D411yescharacteristic
306C2×C4×C61C22×D411yescharacteristic
307C2×C4×C61C22×D411yescharacteristic
308C22⋊D61—82no—
309C2×Dic61—41no—
310D244—42no—
311D244—41no—
312D244—41no—
313D244—41no—
314D244—41no—
315C4×D61—82no—
316Dic122—41no—
317Dic122—41no—
318#92418—22no—
319#92418—41no—
320#92418—41no—
321#92418—41no—
322#92418—41no—
323#92418—41no—
324#18744—611nomaximal, Sylow
325C22×C61—21no—
326C22×C61—21no—
327C22×C61—21no—
328C22×C61—41no—
329C22×C61—41no—
330C22×C61—41no—
331C22×C61C2×#12011yescharacteristic
332C2×D61—81no—
333C2×D61—81no—
334C2×D61—81no—
335C4×C61—21no—
336C4×C61—21no—
337C4×C61—21no—
338C4×C61—21no—
339C4×C61—41no—
340C4×C61—42no—
341C4×C61—41no—
342C4×C61—41no—
343C4×C61—41no—
344C4×C61—41no—
345C4×C61—42no—
346C4×C61—41no—
347C4×C61—41no—
348C4×C61#57711yescharacteristic
349C4×C61#57711yescharacteristic
350C4×C61#57711yescharacteristic
351C4×C61#57811yescharacteristic
352Dic61—81no—
353#3555D6111yescharacteristic, complemented
354#3246—611no—
355#3212—611no—
356#3182—611no—
357#1650—611no—
358#1648—611no—
359#1645—611no—
360#1643—611no—
361#1628—611no—
362#1626—611no—
363#1623—611no—
364#1622—611no—
365#1542—611no—
366#1485—611no—
367#1483—611no—
368C2×C61—41no—
369C2×C61—81no—
370C2×C61#301011yescharacteristic
371C2×C61#301311yescharacteristic
372C2×C61#301811yescharacteristic
373D61—161no—
374#563—22no—
375Q82C2×D6111yescharacteristic
376#557—21no—
377#557C2×D6111yescharacteristic
378#556—21no—
379#556—21no—
380#556C2×D6111yescharacteristic
381#554C2×D6111yescharacteristic
382#552—21no—
383#544—21no—
384#542—21no—
385#541—21no—
386#541—21no—
387#541—22no—
388#533C2×D6111yescharacteristic
389#533C2×D6111yescharacteristic
390#532—611no—
391#529C2×D6111yescharacteristic
392#500—611no—
393#495—1221no—
394#492—611no—
395#488—1221no—
396#487—611no—
397#486—611no—
398#476—1221no—
399#475—611no—
400#475—611no—
401#468—1222no—
402#465—1221no—
403#465—1221no—
404#462—611no—
405#461—611no—
406#461—1221no—
407#460—611no—
408#460—611no—
409#460—611no—
410#459—611no—
411#452—1221no—
412#449—1221no—
413#440—1222no—
414#431—611no—
415C2×C4⋊C8—611no—
416C4×M16—611no—
417#333—611no—
418#333—611no—
419#332—611no—
420#332—611no—
421Q8⋊C8—611no—
422Q8⋊C8—611no—
423Q8⋊C8—611no—
424Q8⋊C8—611no—
425Q8⋊C8—611no—
426Q8⋊C8—611no—
427D4⋊C8—611no—
428D4⋊C8—611no—
429C61#1874411yescharacteristic, complemented, Sylow
430C2×#41—1221no—
431C2×Q16—1221no—
432C2×SD16—1221no—
433C2×SD16—1221no—
434C2×SD16—1221no—
435C2×SD16—1221no—
436C2×SD16—1222no—
437C2×D8—1221no—
438C2×M16—1221no—
439C22×C8—1221no—
440C4⋊Q8—22no—
441C4⋊Q8—21no—
442C4⋊Q8—22no—
443C4⋊Q8—21no—
444C4⋊Q8—21no—
445C4⋊Q8—611no—
446C4⋊Q8C22⋊D6111yescharacteristic
447C4⋊Q8C22⋊D6111yescharacteristic
448C4⋊Q8C22×D6111yescharacteristic
449C4⋊Q8C22×D6111yescharacteristic
450#127—611no—
451#126—22no—
452#126—22no—
453#126—22no—
454#126—22no—
455#126—22no—
456#126—22no—
457#126—21no—
458#125—22no—
459#125—22no—
460#125—22no—
461#125—22no—
462#125—21no—
463#125—22no—
464#125—21no—
465#125—21no—
466#125—21no—
467#125C22⋊D6111yescharacteristic
468#124—22no—
469#124—21no—
470#124—1221no—
471#124C22⋊D6111yescharacteristic
472#123—22no—
473#123—21no—
474C22⋊Q8—22no—
475C22⋊Q8—22no—
476C22⋊Q8—21no—
477C22⋊Q8—21no—
478C22⋊Q8—22no—
479C22⋊Q8—21no—
480C22⋊Q8—21no—
481C22⋊Q8C22⋊D6111yescharacteristic
482C22⋊Q8C22⋊D6111yescharacteristic
483C22⋊Q8C22⋊D6111yescharacteristic
484C22⋊Q8C22⋊D6112yes—
485#121—1221no—
486#121—1221no—
487#121C22⋊D6111yescharacteristic
488C4×Q8—22no—
489C4×Q8—21no—
490C4×Q8—21no—
491C4×Q8—21no—
492C4×Q8—21no—
493C4×Q8—21no—
494C4×Q8—21no—
495C4×Q8—21no—
496C4×Q8C22⋊D6112yes—
497C4×Q8C22×D6111yescharacteristic
498C4×Q8C22×D6111yescharacteristic
499C4×Q8C22×D6111yescharacteristic
500C4×D4—21no—
501C4×D4—21no—
502C4×D4—1221no—
503C4×D4—1221no—
504C4×D4C22×D6111yescharacteristic
505#117—22no—
506#117—21no—
507#117—21no—
508#117—21no—
509#117—21no—
510C2×C4⋊C4—21no—
511C2×C42C22×D6111yescharacteristic
512#107—1222no—
513#106—1221no—
514#106—1221no—
515C4⋊C8—611no—
516C4⋊C8—611no—
517C4⋊C8—611no—
518C4⋊C8—611no—
519C4⋊C8—1221no—
520#103—1222no—
521#103—1221no—
522#103—1221no—
523#103—1221no—
524#103—1221no—
525#102—1222no—
526#102—1221no—
527#102—1221no—
528#102—1221no—
529#102—1221no—
530C22⋊C8—1221no—
531#97—1221no—
532C4×C8—611no—
533C4×C8—611no—
534C4×C8—1221no—
535#41—4881no—
536#41—4881no—
537C2×Q8—22no—
538C2×Q8—21no—
539C2×Q8—21no—
540C2×Q8—1221no—
541C2×Q8C2×C22⋊D6111yescharacteristic
542C2×Q8C2×C22⋊D6111yescharacteristic
543C2×Q8C2×C22⋊D6111yescharacteristic
544C2×Q8C2×C22⋊D6112yes—
545C2×D4—21no—
546C2×D4—1221no—
547C2×D4—1221no—
548C2×D4—1221no—
549C2×D4—2441no—
550C2×D4C2×C22⋊D6111yescharacteristic
551C22×C4—22no—
552C22×C4—21no—
553C22×C4—2441no—
554C22×C4C2×C22⋊D6111yescharacteristic
555C22×C4C2×C22⋊D6111yescharacteristic
556C22×C4C2×C22⋊D6111yescharacteristic
557Q16—4881no—
558SD16—4881no—
559SD16—4881no—
560SD16—4881no—
561SD16—4881no—
562SD16—2444no—
563D8—4881no—
564M16—2442no—
565C2×C8—1221no—
566C2×C8—1221no—
567C2×C8—1221no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8—1221no—
571C2×C8—4881no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—22no—
576C4⋊C4—22no—
577C4⋊C4—22no—
578C4⋊C4—22no—
579C4⋊C4—22no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—21no—
584C4⋊C4—21no—
585C4⋊C4—21no—
586C4⋊C4—21no—
587C4⋊C4—21no—
588C4⋊C4—21no—
589C4⋊C4—21no—
590C4⋊C4—21no—
591C4⋊C4—21no—
592C4⋊C4—21no—
593C4⋊C4—22no—
594C4⋊C4—21no—
595C4⋊C4—22no—
596C4⋊C4—1221no—
597C4⋊C4—1221no—
598C4⋊C4C2×C22⋊D6111yescharacteristic
599C4⋊C4C2×C22⋊D6111yescharacteristic
600C4⋊C4C2×C22⋊D6111yescharacteristic
601C4⋊C4C2×C22⋊D6111yescharacteristic
602C4⋊C4C2×C22⋊D6112yes—
603C22⋊C4—22no—
604C22⋊C4—22no—
605C22⋊C4—22no—
606C22⋊C4—21no—
607C22⋊C4—22no—
608C22⋊C4—21no—
609C22⋊C4—21no—
610C22⋊C4—21no—
611C22⋊C4—21no—
612C22⋊C4—21no—
613C22⋊C4—21no—
614C22⋊C4—2441no—
615C22⋊C4—2441no—
616C42—22no—
617C42—22no—
618C42—22no—
619C42—21no—
620C42—21no—
621C42—21no—
622C42—22no—
623C42C2×C22⋊D6111yescharacteristic
624C42C2×C22⋊D6111yescharacteristic
625C42C2×C22⋊D6111yescharacteristic
626C42C23×D6111yescharacteristic
627C23—21no—
628C23—2441no—
629C23#1451974611yescharacteristic
630Q8—21no—
631Q8—21no—
632Q8—21no—
633Q8—21no—
634Q8—21no—
635Q8—21no—
636Q8—42no—
637Q8—41no—
638Q8—41no—
639Q8—81no—
640Q8—82no—
641Q8—2441no—
642Q8—2441no—
643D4—21no—
644D4—21no—
645D4—81no—
646D4—4882no—
647D4—2442no—
648D4—2441no—
649D4—2441no—
650D4—2441no—
651D4—2441no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—22no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—22no—
664C2×C4—21no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—82no—
669C2×C4—4882no—
670C2×C4—2441no—
671C2×C4#1451974611yescharacteristic
672C2×C4#1451974611yescharacteristic
673C2×C4#1451974611yescharacteristic
674C2×C4C22×C22⋊D6111yescharacteristic
675C2×C4C22×C22⋊D6111yescharacteristic
676C2×C4C22×C22⋊D6111yescharacteristic
677C8—1222no—
678C8—2441no—
679C8—2441no—
680C8—2441no—
681C8—2441no—
682C8—2441no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—4881no—
690C22—4881no—
691C22—4881no—
692C22C2×#1451974611yescharacteristic
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—41no—
698C4—42no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—42no—
704C4—41no—
705C4—41no—
706C4—4881no—
707C4#1684057011yescharacteristic
708C4#1684057011yescharacteristic
709C4#1684057011yescharacteristic
710C4#1684057711yescharacteristic
711C2—41no—
712C2—81no—
713C2—9761no—
714C2#2145510511yescharacteristic
715C2#2145513411yescharacteristic
716C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3555C2
socleC22×C61C2×#120
prime radical#21456791C2
hypercenter#3555D61
hypocenterC61#18744
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3555

Complements

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

NH ≅ G/Ntype
1. #21457131C2semidirect
3. #21456538C2semidirect
4. #21445075C2semidirect
5. #21445061C2semidirect
6. #21445040C2semidirect
7. #21445026C2semidirect
8. #21444929C2semidirect
9. #21444915C2semidirect
10. #21444897C2semidirect
11. #21444891C2semidirect
12. #21444476C2semidirect
13. #21444240C2semidirect
14. #21444232C2semidirect
15. C61×#3555C2semidirect
353. #3555D61semidirect
429. C61#18744semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214567912
#214565382
#214450752
#214450612
#214450402
#214450262
#214449292
#214449152
#214448972
#214448912
#214444762
#214442402
#214442322
C61×#35552
#1874461