Finite Group Databasebeta

← #30617421

Subgroups — #30617421

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462611C211yesmaximal, characteristic, complemented
2#21462545C211yesmaximal, characteristic
3#21462006C211yesmaximal, characteristic, complemented
4#21461956C211yesmaximal, characteristic
5#21461915C211yesmaximal, characteristic, complemented
6#21461914C211yesmaximal, characteristic, complemented
7#21461913C211yesmaximal, characteristic, complemented
8#21461912C211yesmaximal, characteristic, complemented
9#21461743C211yesmaximal, characteristic, complemented
10#21461600C211yesmaximal, characteristic, complemented
11#21461598C211yesmaximal, characteristic, complemented
12#21461596C211yesmaximal, characteristic, complemented
13#21461594C211yesmaximal, characteristic, complemented
14#21461475C211yesmaximal, characteristic, complemented
15#21456140C211yesmaximal, characteristic
16C2×#14519865C2211yescharacteristic
17C2×#14519864C2211yescharacteristic
18#16840814—21no—
19#16840814—21no—
20#16840812—21no—
21#16840812—21no—
22#16840794—21no—
23#16840794—21no—
24#16840791—21no—
25#16840791—21no—
26#16840769C2211yescharacteristic
27#16840769C2211yescharacteristic
28#16840766C2211yescharacteristic
29#16840766C2211yescharacteristic
30C2×#14519842C2211yescharacteristic
31C2×#14519842C2211yescharacteristic
32C2×#14519841C2211yescharacteristic
33C2×#14519841C2211yescharacteristic
34#16840737C2211yescharacteristic
35#16840736C2211yescharacteristic
36#16840727C2211yescharacteristic
37#16840727C2211yescharacteristic
38#16840721C2211yescharacteristic
39C2×#14519831C2211yescharacteristic
40C2×#14519831C2211yescharacteristic
41C2×C8×#6923C2211yescharacteristic
42C2×C8×#6923C2211yescharacteristic
43#16840698C2211yescharacteristic
44#16840638C2211yescharacteristic
45#16840638C2211yescharacteristic
46#16840638C2211yescharacteristic
47#16840638C2211yescharacteristic
48#16840632C2211yescharacteristic
49#16840632C2211yescharacteristic
50#16840632C2211yescharacteristic
51#16840632C2211yescharacteristic
52#16840318C2211yescharacteristic
53#16840311C2211yescharacteristic
54#16840119C411yescharacteristic, complemented
55#16840117C411yescharacteristic, complemented
56#16840116C411yescharacteristic, complemented
57#16840111C411yescharacteristic, complemented
58#16840111C411yescharacteristic, complemented
59#16840107C411yescharacteristic, complemented
60#16840107C411yescharacteristic, complemented
61#16839757C2211yescharacteristic
62#16839757C2211yescharacteristic
63#16839755C2211yescharacteristic
64#16839755C2211yescharacteristic
65C4×C8×D61C2211yescharacteristic
66C61×#488C411yescharacteristic, complemented
67#14519865—22no—
68#14519865—22no—
69#14519864—22no—
70#14519864—22no—
71C2×#11758213—21no—
72C2×#11758213C2311yescharacteristic
73C2×#11758213C2311yescharacteristic
74C2×#11758213C2311yescharacteristic
75C2×#11758213C2311yescharacteristic
76C2×#11758217—21no—
77C2×D61⋊C8—21no—
78C2×D61⋊C8C2311yescharacteristic
79C2×D61⋊C8C2311yescharacteristic
80C22×#92420—21no—
81C2×C4×#6923C2311yescharacteristic
82C2×C4×#6923C2311yescharacteristic
83#14519844—21no—
84#14519844—21no—
85C4⋊#92420—21no—
86C4⋊#92420—21no—
87C4⋊#92420—21no—
88C4⋊#92420—21no—
89C4⋊#92420—21no—
90C4⋊#92420—21no—
91#14519842—22no—
92#14519842—22no—
93#14519841—22no—
94#14519841—22no—
95#14519834—21no—
96#14519834—21no—
97C22⋊#92420—21no—
98C22⋊#92420—21no—
99#14519832—21no—
100#14519832—21no—
101#14519832D411yescharacteristic
102#14519832D411yescharacteristic
103#14519832D411yescharacteristic
104#14519832D411yescharacteristic
105#14519831—22no—
106#14519831—22no—
107C4×#92420—21no—
108C4×#92420—21no—
109C8×#6923—22no—
110C8×#6923—22no—
111(C2×#6923)⋊C4—21no—
112(C2×#6923)⋊C4—21no—
113C2×#11758226C2311yescharacteristic
114C2×#11758224C2311yescharacteristic
115C2×C8×D61C2311yescharacteristic
116C2×C8×D61C2311yescharacteristic
117#14519776C2×C411yescharacteristic
118#14519776C2×C411yescharacteristic
119#14519775C2×C411yescharacteristic
120#14519773C2×C411yescharacteristic
121#14519769C2×C411yescharacteristic
122#14519765C2×C411yescharacteristic
123#14519755—21no—
124#14519755—21no—
125#14519750—21no—
126#14519750—21no—
127#14519742—21no—
128#14519742—21no—
129#14519737—21no—
130#14519737—21no—
131#14519735C2311yescharacteristic
132#14519735C2311yescharacteristic
133C42×D61C2311yescharacteristic
134#14519702C2×C411yescharacteristic
135#14519702C2×C411yescharacteristic
136#14519701C2×C411yescharacteristic
137#14519701C2×C411yescharacteristic
138#14519700C2×C411yescharacteristic
139#14519700C2×C411yescharacteristic
140#14519699C2×C411yescharacteristic
141#14519699C2×C411yescharacteristic
142#14519698C2×C411yescharacteristic
143#14519698C2×C411yescharacteristic
144#14519697C2×C411yescharacteristic
145#14519697C2×C411yescharacteristic
146C4×#92418C2×C411yescharacteristic
147C8×Dic61C2×C411yescharacteristic
148C8×Dic61C2×C411yescharacteristic
149#125×C61C2×C411yescharacteristic
150#124×C61C2×C411yescharacteristic
151#103×C61C2×C411yescharacteristic
152#103×C61C2×C411yescharacteristic
153#102×C61C2×C411yescharacteristic
154#102×C61C2×C411yescharacteristic
155C4×C8×C61C2×C411yescharacteristic
156C22×#6923—21no—
157C22×#6923—21no—
158C2×Q8×C61C22×C411yescharacteristic
159C2×D4×C61C22×C411yescharacteristic
160C2×C22⋊D61—21no—
161C2×C22⋊D61—21no—
162C2×C22⋊D61—21no—
163C22×Dic61—21no—
164#11758226—21no—
165#11758226—21no—
166#11758226—21no—
167#11758226—21no—
168#11758224—21no—
169#11758224—21no—
170#11758224—21no—
171#11758224—21no—
172C2×D244—21no—
173C2×D244C22×C411yescharacteristic
174C2×C4×D61—21no—
175C2×C4×D61#4111yescharacteristic
176C2×C4×D61#4111yescharacteristic
177C2×C4×D61C2411yescharacteristic
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122C22×C411yescharacteristic
181#11758217—42no—
182C2×#92420—21no—
183C2×#92420—21no—
184C2×#92420—21no—
185C2×#92420—21no—
186C2×#92420—42no—
187C2×#92420C2×D411yescharacteristic
188C2×#92420C2×D411yescharacteristic
189C2×#92420C2×D411yescharacteristic
190C2×#92420C2×D411yescharacteristic
191C4×#6923—21no—
192C4×#6923—21no—
193C4×#6923#4112yes—
194C4×#6923#4112yes—
195#11758213—21no—
196#11758213—21no—
197#11758213—21no—
198#11758213—21no—
199#11758213—21no—
200#11758213—21no—
201#11758213—21no—
202#11758213—21no—
203#11758213—42no—
204D61⋊C8—21no—
205D61⋊C8—21no—
206D61⋊C8—42no—
207D61⋊C8#4112yes—
208D61⋊C8#4112yes—
209C2×C8×C61C22×C411yescharacteristic
210C2×C8×C61C22×C411yescharacteristic
211C4⋊C4×C61—21no—
212C4⋊C4×C61—21no—
213C4⋊C4×C61C22×C411yescharacteristic
214C4⋊C4×C61C22×C411yescharacteristic
215C22⋊C4×C61—21no—
216C22⋊C4×C61—21no—
217C42×C61C22×C411yescharacteristic
218#11758198—21no—
219#11758198—21no—
220#11758198—21no—
221#11758198—21no—
222C4⋊Dic61C22×C411yescharacteristic
223C4⋊Dic61C22×C411yescharacteristic
224#11758196—21no—
225#11758196—21no—
226#11758196—21no—
227#11758196—21no—
228#11758196—21no—
229#11758196—21no—
230C4×Dic61—21no—
231C4×Dic61—21no—
232C4×Dic61C22×C411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C2×#92418C22×C411yescharacteristic
236C2×#92418C22×C411yescharacteristic
237C8×D61—21no—
238C8×D61—21no—
239C8×D61—21no—
240C8×D61—21no—
241C23×C61—21no—
242C22×D61—21no—
243C22×D61C2×#4111yescharacteristic
244C2×#6923—22no—
245C2×#6923—22no—
246C2×#6923—41no—
247C2×#6923—41no—
248C2×#6923—41no—
249C2×#6923—41no—
250Q8×C61—21no—
251Q8×C61—21no—
252D4×C61—21no—
253D4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61C4×D411yescharacteristic
258C2×C4×C61C4×D411yescharacteristic
259C2×C4×C61C23×C411yescharacteristic
260C22⋊D61—41no—
261C22⋊D61—41no—
262C22⋊D61—81no—
263C22⋊D61—81no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61C4×D411yescharacteristic
270C2×Dic61C4×D411yescharacteristic
271C2×Dic61C22×D411yescharacteristic
272D244—21no—
273D244—21no—
274D244—41no—
275D244—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61C2×#4111yescharacteristic
281C4×D61C2×#4111yescharacteristic
282C4×D61C2×#4111yescharacteristic
283C4×D61C2×#4111yescharacteristic
284Dic122—21no—
285Dic122—21no—
286Dic122—81no—
287Dic122—81no—
288#92420—21no—
289#92420—21no—
290#92420—41no—
291#92420—41no—
292#92420—41no—
293#92420—41no—
294#92420#12412yes—
295#92420#12412yes—
296C8×C61—21no—
297C8×C61—21no—
298#92418—21no—
299#92418—21no—
300#21659—611nomaximal, Sylow
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61C2×C4×D411yescharacteristic
305C2×D61—21no—
306C2×D61—21no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61#52911yescharacteristic
311C2×D61#52911yescharacteristic
312#6923—42no—
313#6923—42no—
314C4×C61—21no—
315C4×C61—21no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61C4×#4111yescharacteristic
320C4×C61C4×#4111yescharacteristic
321Dic61—21no—
322Dic61—21no—
323Dic61—41no—
324Dic61—41no—
325Dic61—41no—
326Dic61C2×#12411yescharacteristic
327Dic61#53211yescharacteristic
328#3131—611no—
329#2961—611no—
330#2932—611no—
331#1995—611no—
332#1963—611no—
333#1938—611no—
334#1938—611no—
335#1937—611no—
336#1937—611no—
337#1837—611no—
338#1765—611no—
339#1765—611no—
340#1764—611no—
341#1764—611no—
342#1726—611no—
343C2×C61—81no—
344C2×C61C4×#12411yescharacteristic
345C2×C61#295711yescharacteristic
346C2×C61#295711yescharacteristic
347D61—21no—
348D61—21no—
349D61—81no—
350C2×#131—611no—
351C2×#131—611no—
352#533—611no—
353#532—611no—
354#488—611no—
355#488—611no—
356#488—611no—
357#488#692311yescharacteristic, complemented
358#487—611no—
359#487—611no—
360#486—611no—
361#486—611no—
362#446—1221no—
363#446—1221no—
364#446—1221no—
365#446—1221no—
366#435—1221no—
367#435—1221no—
368#435—1221no—
369#435—1221no—
370#424—611no—
371#424—611no—
372#423—611no—
373#423—611no—
374C2×#104—611no—
375C2×#104—611no—
376C2×#104—611no—
377C2×#104—611no—
378#416—611no—
379#416—611no—
380#416—611no—
381#416—611no—
382#408—611no—
383#408—611no—
384#405—611no—
385#405—611no—
386C2×#97—611no—
387C2×#97—611no—
388C2×#97—611no—
389C2×C4×C8—611no—
390C2×C4×C8—611no—
391C2×C4×C8—611no—
392#376—611no—
393#341—611no—
394#341—611no—
395#341—611no—
396#341—611no—
397#339—611no—
398#339—611no—
399#339—611no—
400#339—611no—
401C61#2165911yescharacteristic, complemented, Sylow
402C2×#41—611no—
403C2×#41—611no—
404#131—1222no—
405#131—1222no—
406#131—1222no—
407#131—1222no—
408C2×M16—611no—
409C2×M16—611no—
410C2×M16—611no—
411C2×M16—611no—
412C2×M16—1221no—
413C2×M16—1221no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—1221no—
419C22×C8—1221no—
420C4⋊Q8—611no—
421C4⋊Q8—611no—
422C4⋊Q8—611no—
423#127—611no—
424#125—611no—
425#125C2×#692311yescharacteristic
426#124—611no—
427#124C2×#692311yescharacteristic
428C22⋊Q8—1221no—
429C22⋊Q8—1221no—
430#121—1221no—
431#121—1221no—
432C4×Q8—1221no—
433C4×Q8—1221no—
434C4×D4—1221no—
435C4×D4—1221no—
436#117—611no—
437#117—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C4⋊C8—1221no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1221no—
446C4⋊C8—1221no—
447C4⋊C8—1221no—
448C4⋊C8—1221no—
449#104—1222no—
450#104—1222no—
451#104—1222no—
452#104—1222no—
453#103—611no—
454#103—611no—
455#103—611no—
456#103—611no—
457#103—611no—
458#103—611no—
459#103C2×#692311yescharacteristic
460#103C2×#692311yescharacteristic
461#102—611no—
462#102—611no—
463#102—611no—
464#102—611no—
465#102—611no—
466#102—611no—
467#102C2×#692311yescharacteristic
468#102C2×#692311yescharacteristic
469C22⋊C8—1221no—
470C22⋊C8—1221no—
471C22⋊C8—1221no—
472C22⋊C8—1221no—
473#97—611no—
474#97—611no—
475#97—611no—
476#97—611no—
477#97—1222no—
478#97—1222no—
479#97—1221no—
480#97—1221no—
481C4×C8—611no—
482C4×C8—611no—
483C4×C8—611no—
484C4×C8—1222no—
485C4×C8—1222no—
486C4×C8—1221no—
487C4×C8—1221no—
488C4×C8C2×#692311yescharacteristic
489#95—1221no—
490#95—1221no—
491#41—1221no—
492#41—1221no—
493#41—1221no—
494#41—1221no—
495#41—1221no—
496#41—1221no—
497#41—1221no—
498#41—1221no—
499C2×Q8—611no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8C22×#692311yescharacteristic
503C2×D4—611no—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4C22×#692311yescharacteristic
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516M16—1221no—
517M16—1221no—
518M16—1221no—
519M16—1221no—
520M16—1221no—
521M16—1221no—
522M16—1221no—
523M16—1221no—
524M16—2442no—
525M16—2442no—
526C2×C8—611no—
527C2×C8—611no—
528C2×C8—612no—
529C2×C8—612no—
530C2×C8—611no—
531C2×C8—611no—
532C2×C8—611no—
533C2×C8—611no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—2442no—
545C2×C8—2442no—
546C2×C8C22×#692311yescharacteristic
547C2×C8C22×#692311yescharacteristic
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—611no—
551C4⋊C4—611no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4C22×#692311yescharacteristic
559C4⋊C4C22×#692311yescharacteristic
560C22⋊C4—21no—
561C22⋊C4—21no—
562C22⋊C4—1221no—
563C22⋊C4—1221no—
564C22⋊C4—1221no—
565C22⋊C4—1221no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—612no—
570C42—612no—
571C42—1221no—
572C42—1221no—
573C42—1221no—
574C42—1221no—
575C42C22×#692311yescharacteristic
576C23—21no—
577C23—611no—
578C23—1221no—
579Q8—21no—
580Q8—21no—
581Q8—1221no—
582Q8—1221no—
583Q8—4881no—
584Q8—4881no—
585D4—21no—
586D4—21no—
587D4—1221no—
588D4—1221no—
589D4—4881no—
590D4—4881no—
591D4—2441no—
592D4—2441no—
593D4—2441no—
594D4—2441no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—611no—
599C2×C4—611no—
600C2×C4—611no—
601C2×C4—611no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4D4×#692311yescharacteristic
621C2×C4D4×#692311yescharacteristic
622C2×C4C23×#692311yescharacteristic
623C8—21no—
624C8—21no—
625C8—612no—
626C8—612no—
627C8—1221no—
628C8—1221no—
629C8—1221no—
630C8—1221no—
631C8—2441no—
632C8—2441no—
633C8—2441no—
634C8—2441no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—611no—
639C22—611no—
640C22—1221no—
641C22—1221no—
642C22—2441no—
643C22—2441no—
644C22—2441no—
645C22C2×D4×#692311yescharacteristic
646C4—21no—
647C4—21no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—611no—
652C4—611no—
653C4—1221no—
654C4—1221no—
655C4—2442no—
656C4—2442no—
657C4—2441no—
658C4—2441no—
659C4—2441no—
660C4#41×#692311yescharacteristic
661C4#41×#692311yescharacteristic
662C2—81no—
663C2—1221no—
664C2—1221no—
665C2—4881no—
666C2#124×#692311yescharacteristic
667C2#2146259611yescharacteristic
668C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#488C4
socleC22×C61C2×C4×D4
prime radical#16840311C22
cosocleC2×C4×D61C24
hypercenter#488#6923
hypocenterC61#21659
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #21462611C2semidirect
3. #21462006C2semidirect
5. #21461915C2semidirect
6. #21461914C2semidirect
7. #21461913C2semidirect
8. #21461912C2semidirect
9. #21461743C2semidirect
10. #21461600C2semidirect
11. #21461598C2semidirect
12. #21461596C2semidirect
13. #21461594C2semidirect
14. #21461475C2semidirect
54. #16840119C4semidirect
55. #16840117C4semidirect
56. #16840116C4semidirect
57. #16840111C4semidirect
58. #16840111C4semidirect
59. #16840107C4semidirect
60. #16840107C4semidirect
66. C61×#488C4semidirect
357. #488#6923semidirect
401. C61#21659semidirect

Maximal subgroups

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

subgroupindex
#214626112
#214625452
#214620062
#214619562
#214619152
#214619142
#214619132
#214619122
#214617432
#214616002
#214615982
#214615962
#214615942
#214614752
#214561402
#2165961