Finite Group Databasebeta

← #29646881

Subgroups — #29646881

Order 15616.

980 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×#92418C211yesmaximal, characteristic, complemented
2#21454485C211yesmaximal, characteristic, complemented
3#21444279C212yesmaximal, complemented
4#21444274C211yesmaximal, characteristic, complemented
5#21444253C212yesmaximal, complemented
6#21444244C211yesmaximal, characteristic, complemented
7#21444227C212yesmaximal, complemented
8#21444222C211yesmaximal, characteristic, complemented
9#21444201C212yesmaximal, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#3475C211yesmaximal, characteristic
12Q8×#92418—21no—
13D4×#92418—22no—
14D4×#92418—21no—
15#16839823—22no—
16#16839819C2212yescomplemented
17#16839815C2211yescharacteristic
18#16839815C2212yescomplemented
19#16839815C2212yescomplemented
20#16839786C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22#16839729—21no—
23#16839718—21no—
24C2×C4×#92418C2211yescharacteristic
25C2×C4×#92418C2212yescomplemented
26Q8⋊#92418C2212yescomplemented
27Q8⋊#92418C2212yescomplemented
28Q8⋊#92418C2212yescomplemented
29Q8⋊#92418C2212yescomplemented
30D4⋊#92418C2211yescharacteristic
31D4⋊#92418C2211yescharacteristic
32D4⋊#92418C2211yescharacteristic
33D4⋊#92418C2211yescharacteristic
34D4⋊#92418C2212yes—
35D4⋊#92418C2212yes—
36C61×#556C411yescharacteristic
37C61×#555C411yescharacteristic
38C61×#554C411yescharacteristic
39C61×#553—21no—
40C61×#552—21no—
41C61×#552C412yes—
42C61×#551—22no—
43C61×#547C411yescharacteristic
44C61×#546C412yes—
45C61×#533C2211yescharacteristic
46C61×#529—22no—
47C61×#529C2212yes—
48C61×#525—21no—
49C4×#41×C61C2212yes—
50C4×#41×C61C2211yescharacteristic
51C2×C4×D4×C61C2211yescharacteristic
52C2×#117×C61—21no—
53C2×#11758194—21no—
54C22×#92418—21no—
55C22×#92418—22no—
56C22×#92418—21no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60C4⋊#92418—22no—
61C4⋊#92418C2311yescharacteristic
62C4⋊#92418C2311yescharacteristic
63C4⋊#92418C2312yes—
64C22⋊#92418—22no—
65C22⋊#92418—22no—
66C22⋊#92418—21no—
67C22⋊#92418—22no—
68#14519688—22no—
69C4×#92418—21no—
70C4×#92418—22no—
71C4×#92418—22no—
72C4×#92418C2311yescharacteristic
73C4×#92418C2311yescharacteristic
74C4×#92418C2312yes—
75C2×#41×C61C2×C412yes—
76C2×#41×C61D411yescharacteristic
77C22×D4×C61C2×C411yescharacteristic
78C23×C4×C61—21no—
79C4⋊Q8×C61C2×C411yescharacteristic
80C4⋊Q8×C61C2×C411yescharacteristic
81#126×C61—22no—
82#126×C61—22no—
83#126×C61—22no—
84#125×C61—22no—
85#125×C61—21no—
86#125×C61C2×C412yes—
87#124×C61—22no—
88#124×C61C2×C412yes—
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61C2×C412yes—
98C22⋊Q8×C61C2×C412yes—
99#121×C61—21no—
100#121×C61—21no—
101#121×C61—22no—
102#121×C61C2×C412yes—
103#120×C61—22no—
104C4×Q8×C61—21no—
105C4×Q8×C61C2×C412yes—
106C4×Q8×C61C2×C411yescharacteristic
107C4×Q8×C61D411yescharacteristic
108C4×Q8×C61C2312yes—
109C4×D4×C61—22no—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C4×D4×C61—21no—
113C4×D4×C61—21no—
114C4×D4×C61C2×C412yes—
115C4×D4×C61C2×C411yescharacteristic
116C4×D4×C61C2×C411yescharacteristic
117C4×D4×C61C2×C412yes—
118C4×D4×C61C2×C411yescharacteristic
119C4×D4×C61D412yes—
120C4×D4×C61D411yescharacteristic
121C4×D4×C61C2311yescharacteristic
122C4×D4×C61C2311yescharacteristic
123#117×C61—21no—
124#117×C61—21no—
125#117×C61—21no—
126#117×C61—22no—
127#117×C61—22no—
128#117×C61—22no—
129#117×C61C2×C412yes—
130#117×C61D411yescharacteristic
131#117×C61D412yes—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61C2×C411yescharacteristic
135C2×C22⋊C4×C61—21no—
136C2×C22⋊C4×C61—22no—
137C2×C42×C61—22no—
138C2×C42×C61C2311yescharacteristic
139C2×C42×C61C2312yes—
140C24×C61—21no—
141#41×C61—22no—
142#41×C61—22no—
143#41×C61—24no—
144#41×C61—42no—
145C2×Q8×C61—21no—
146C2×Q8×C61C22⋊C411yescharacteristic
147C2×Q8×C61C22×C412yes—
148C2×D4×C61—22no—
149C2×D4×C61—22no—
150C2×D4×C61—22no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C2×D4×C61C22⋊C412yes—
154C2×D4×C61C22⋊C411yescharacteristic
155C2×D4×C61C22×C411yescharacteristic
156C2×D4×C61C22×C411yescharacteristic
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—21no—
163C22×C4×C61—22no—
164C22×C4×C61—42no—
165C22×C4×C61—42no—
166C22×C4×C61C22×C411yescharacteristic
167C22×C4×C61C22×C412yes—
168C22×C4×C61C2×D412yes—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D412yes—
171C22×C4×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C22⋊C411yescharacteristic
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22⋊C412yes—
185C4⋊C4×C61C22⋊C412yes—
186C4⋊C4×C61C22×C411yescharacteristic
187C4⋊C4×C61C22×C411yescharacteristic
188C4⋊C4×C61C22×C412yes—
189C22⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—21no—
195C22⋊C4×C61—21no—
196C22⋊C4×C61—22no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—21no—
199C22⋊C4×C61—21no—
200C22⋊C4×C61C22⋊C412yes—
201C22⋊C4×C61C22⋊C412yes—
202C22⋊C4×C61C22⋊C412yes—
203C42×C61—22no—
204C42×C61—22no—
205C42×C61—22no—
206C42×C61C22×C411yescharacteristic
207C42×C61C22×C412yes—
208C42×C61C2×D412yes—
209C42×C61C2×D412yes—
210C42×C61C2×D411yescharacteristic
211C42×C61C2×D411yescharacteristic
212C42×C61C2411yescharacteristic
213#11758194—42no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418—21no—
220C2×#92418—42no—
221C2×#92418—44no—
222C2×#92418—42no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—42no—
226C23×C61—42no—
227C23×C61—81no—
228C23×C61C2×C22⋊C412yes—
229C23×C61C2×C22⋊C411yescharacteristic
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—81no—
234D4×C61—24no—
235D4×C61—21no—
236D4×C61—21no—
237D4×C61—21no—
238D4×C61—22no—
239D4×C61—42no—
240D4×C61—44no—
241D4×C61—82no—
242D4×C61#13112yes—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—42no—
254C2×C4×C61—41no—
255C2×C4×C61—44no—
256C2×C4×C61—42no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—42no—
262C2×C4×C61—42no—
263C2×C4×C61—82no—
264C2×C4×C61C2×C22⋊C412yes—
265C2×C4×C61C2×C22⋊C412yes—
266C2×C4×C61C2×C22⋊C411yescharacteristic
267C2×C4×C61C2×C22⋊C412yes—
268C2×C4×C61C2×C22⋊C411yescharacteristic
269C2×C4×C61C2×C22⋊C411yescharacteristic
270C2×C4×C61#13512yes—
271C2×C4×C61#13512yes—
272C2×C4×C61C23×C411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×D411yescharacteristic
275#92418—22no—
276#92418—41no—
277#92418—42no—
278#92418—41no—
279#92418—41no—
280#92418—42no—
281#17508—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—22no—
285C22×C61—22no—
286C22×C61—22no—
287C22×C61—22no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—84no—
294C22×C61#41612yes—
295C22×C61C22×C22⋊C411yescharacteristic
296C4×C61—22no—
297C4×C61—21no—
298C4×C61—22no—
299C4×C61—21no—
300C4×C61—22no—
301C4×C61—21no—
302C4×C61—42no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61—42no—
306C4×C61#41612yes—
307C4×C61C2×#13111yescharacteristic
308C4×C61#56811yescharacteristic
309C4×C61C2×#13511yescharacteristic
310C4×C61C2×#13511yescharacteristic
311#3475D6111yescharacteristic
312C8×#41—611no—
313#2925—611no—
314#1494—612no—
315#1493—611no—
316#1488—612no—
317#1486—611no—
318#1482—612no—
319#1481—611no—
320#1476—612no—
321C2×D4⋊C8—611no—
322C2×C61—22no—
323C2×C61—42no—
324C2×C61—42no—
325C2×C61—81no—
326C2×C61#288011yescharacteristic
327C2×C61C2×#41611yescharacteristic
328C2×C61#290111yescharacteristic
329#556Dic6111yescharacteristic
330#555Dic6111yescharacteristic
331#554Dic6111yescharacteristic
332#553—21no—
333#552—21no—
334#552Dic6112yes—
335#551—22no—
336#547Dic6111yescharacteristic
337#546Dic6112yes—
338#533C2×D6111yescharacteristic
339#529—22no—
340#529C2×D6112yes—
341#525—21no—
342C4×#41C2×D6112yes—
343C4×#41C2×D6111yescharacteristic
344C2×C4×D4C2×D6111yescharacteristic
345C2×#117—21no—
346C8×Q8—1221no—
347C8×D4—1222no—
348C8×D4—1221no—
349#433—1222no—
350#432—612no—
351#431—611no—
352#431—612no—
353#431—612no—
354#423—611no—
355C2×C4⋊C8—611no—
356#408—1221no—
357#405—1221no—
358C2×C4×C8—611no—
359C2×C4×C8—612no—
360Q8⋊C8—612no—
361Q8⋊C8—612no—
362Q8⋊C8—612no—
363Q8⋊C8—612no—
364D4⋊C8—611no—
365D4⋊C8—611no—
366D4⋊C8—611no—
367D4⋊C8—611no—
368D4⋊C8—612no—
369D4⋊C8—612no—
370C61#1750811yescharacteristic, complemented, Sylow
371C2×#41C2×Dic6112yes—
372C2×#41C22⋊D6111yescharacteristic
373C22×D4C2×Dic6111yescharacteristic
374C23×C4—21no—
375C2×M16—1221no—
376C22×C8—1221no—
377C22×C8—1222no—
378C22×C8—1221no—
379C4⋊Q8C2×Dic6111yescharacteristic
380C4⋊Q8C2×Dic6111yescharacteristic
381#126—22no—
382#126—22no—
383#126—22no—
384#125—22no—
385#125—21no—
386#125C2×Dic6112yes—
387#124—22no—
388#124C2×Dic6112yes—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393C22⋊Q8—21no—
394C22⋊Q8—21no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8C2×Dic6112yes—
398C22⋊Q8C2×Dic6112yes—
399#121—21no—
400#121—21no—
401#121—22no—
402#121C2×Dic6112yes—
403#120—22no—
404C4×Q8—21no—
405C4×Q8C2×Dic6112yes—
406C4×Q8C2×Dic6111yescharacteristic
407C4×Q8C22⋊D6111yescharacteristic
408C4×Q8C22×D6112yes—
409C4×D4—22no—
410C4×D4—22no—
411C4×D4—22no—
412C4×D4—21no—
413C4×D4—21no—
414C4×D4C2×Dic6111yescharacteristic
415C4×D4C2×Dic6111yescharacteristic
416C4×D4C2×Dic6112yes—
417C4×D4C2×Dic6112yes—
418C4×D4C2×Dic6111yescharacteristic
419C4×D4C22⋊D6112yes—
420C4×D4C22⋊D6111yescharacteristic
421C4×D4C22×D6111yescharacteristic
422C4×D4C22×D6111yescharacteristic
423#117—21no—
424#117—21no—
425#117—21no—
426#117—22no—
427#117—22no—
428#117—22no—
429#117C2×Dic6112yes—
430#117C22⋊D6111yescharacteristic
431#117C22⋊D6112yes—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4C2×Dic6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—22no—
437C2×C42—22no—
438C2×C42C22×D6111yescharacteristic
439C2×C42C22×D6112yes—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—612no—
443C4⋊C8—1221no—
444C4⋊C8—1221no—
445C4⋊C8—1222no—
446C4⋊C8—1221no—
447C22⋊C8—1222no—
448C22⋊C8—1222no—
449C22⋊C8—1221no—
450C22⋊C8—1222no—
451#97—1222no—
452C4×C8—611no—
453C4×C8—611no—
454C4×C8—612no—
455C4×C8—1222no—
456C4×C8—1221no—
457C4×C8—1222no—
458C24—21no—
459#41—22no—
460#41—22no—
461#41—24no—
462#41—42no—
463C2×Q8—21no—
464C2×Q8C22⋊Dic6111yescharacteristic
465C2×Q8C22×Dic6112yes—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—41no—
471C2×D4C22⋊Dic6112yes—
472C2×D4C22⋊Dic6111yescharacteristic
473C2×D4C22×Dic6111yescharacteristic
474C2×D4C22×Dic6111yescharacteristic
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4C22×Dic6111yescharacteristic
485C22×C4C22×Dic6112yes—
486C22×C4C2×C22⋊D6112yes—
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6112yes—
489C22×C4C2×C22⋊D6111yescharacteristic
490M16—2442no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1222no—
494C2×C8—1221no—
495C2×C8—1222no—
496C2×C8—1221no—
497C2×C8—2444no—
498C2×C8—2442no—
499C2×C8—2442no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4C22⋊Dic6111yescharacteristic
511C4⋊C4C22⋊Dic6111yescharacteristic
512C4⋊C4C22⋊Dic6112yes—
513C4⋊C4C22⋊Dic6112yes—
514C4⋊C4C22×Dic6111yescharacteristic
515C4⋊C4C22×Dic6111yescharacteristic
516C4⋊C4C22×Dic6112yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4C22⋊Dic6112yes—
529C22⋊C4C22⋊Dic6112yes—
530C22⋊C4C22⋊Dic6112yes—
531C42—22no—
532C42—22no—
533C42—22no—
534C42C22×Dic6111yescharacteristic
535C42C22×Dic6112yes—
536C42C2×C22⋊D6112yes—
537C42C2×C22⋊D6111yescharacteristic
538C42C2×C22⋊D6112yes—
539C42C2×C22⋊D6111yescharacteristic
540C42C23×D6111yescharacteristic
541C23—21no—
542C23—21no—
543C23—42no—
544C23—42no—
545C23—81no—
546C23C2×C22⋊Dic6112yes—
547C23C2×C22⋊Dic6111yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—22no—
551Q8—81no—
552D4—24no—
553D4—21no—
554D4—22no—
555D4—21no—
556D4—21no—
557D4—42no—
558D4—44no—
559D4—82no—
560D4#1451978412yes—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—82no—
582C2×C4C2×C22⋊Dic6112yes—
583C2×C4C2×C22⋊Dic6112yes—
584C2×C4C2×C22⋊Dic6111yescharacteristic
585C2×C4C2×C22⋊Dic6112yes—
586C2×C4C2×C22⋊Dic6111yescharacteristic
587C2×C4C2×C22⋊Dic6111yescharacteristic
588C2×C4#1451979712yes—
589C2×C4#1451979712yes—
590C2×C4C23×Dic6111yescharacteristic
591C2×C4C22×C22⋊D6111yescharacteristic
592C2×C4C22×C22⋊D6111yescharacteristic
593C8—1222no—
594C8—2441no—
595C8—2442no—
596C8—2441no—
597C8—2441no—
598C8—2442no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—84no—
611C22#1683975812yes—
612C22C22×C22⋊Dic6111yescharacteristic
613C4—21no—
614C4—22no—
615C4—21no—
616C4—22no—
617C4—21no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4#1683975812yes—
624C4C2×#1451978411yescharacteristic
625C4#1684052911yescharacteristic
626C4C2×#1451979711yescharacteristic
627C4C2×#1451979711yescharacteristic
628C2—22no—
629C2—42no—
630C2—42no—
631C2—81no—
632C2#2145426811yescharacteristic
633C2C2×#1683975811yescharacteristic
634C2#2145436611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
Fitting, prime radicalC61×#3475C2
socleC22×C61C22×C22⋊C4
cosocleC42×C61C24
hypercenter#3475D61
hypocenterC61#17508
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #3475

Complements

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

NH ≅ G/Ntype
1. #41×#92418C2semidirect
2. #21454485C2semidirect
3. #21444279C2semidirect
4. #21444274C2semidirect
5. #21444253C2semidirect
6. #21444244C2semidirect
7. #21444227C2semidirect
8. #21444222C2semidirect
9. #21444201C2semidirect
10. C2×D4⋊#92418C2semidirect
16. #16839819C22semidirect
18. #16839815C22semidirect
19. #16839815C22semidirect
25. C2×C4×#92418C22semidirect
26. Q8⋊#92418C22semidirect
27. Q8⋊#92418C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
370. C61#17508semidirect

Maximal subgroups

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

subgroupindex
#41×#924182
#214544852
#214442792
#214442742
#214442532
#214442442
#214442272
#214442222
#214442012
C2×D4⋊#924182
C61×#34752
#1750861