Finite Group Databasebeta

← #29661802

Subgroups — #29661802

Order 15616.

886 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456822C211yesmaximal, characteristic, complemented
2#21456529C211yesmaximal, characteristic, complemented
3#21456341C211yesmaximal, characteristic, complemented
4#21445153C212yesmaximal, complemented
5#21445136C212yesmaximal, complemented
6#21445136C212yesmaximal, complemented
7#21445130C212yesmaximal, complemented
8#21444439C211yesmaximal, characteristic, complemented
9#21444434C211yesmaximal, characteristic, complemented
10C2×#16839502C211yesmaximal, characteristic, complemented
11C61×#3561C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840172C2211yescharacteristic, complemented
14#16840172C2212yescomplemented
15#16840172C2212yescomplemented
16#16840168—21no—
17#16840168—21no—
18#16840157C2212yes—
19#16840157C2211yescharacteristic, complemented
20#16840132C2211yescharacteristic, complemented
21#16840121C2212yes—
22#16840121C2211yescharacteristic, complemented
23#16840001—21no—
24#16839989—21no—
25#16839981—22no—
26#16839918—21no—
27#16839913—21no—
28#16839867—21no—
29#16839862—21no—
30#16839819C2211yescharacteristic
31#16839798—22no—
32C2×#14519688C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34#16839504C2212yescomplemented
35#16839504C2212yescomplemented
36#16839504C2212yescomplemented
37#16839502C2212yes—
38#16839502C2212yes—
39#16839502C2212yescomplemented
40C61×#563C2212yes—
41C61×#562C2212yes—
42C61×#556—21no—
43C61×#551—22no—
44D4×Q8×C61—21no—
45C61×#548—21no—
46C61×#546—21no—
47C61×#543—21no—
48C61×#542—21no—
49C61×#542—22no—
50C61×#538—21no—
51C61×#536—21no—
52C61×#533C2211yescharacteristic
53C61×#529C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C2×#11758202—21no—
56C2×#11758202—22no—
57C2×#11758200—21no—
58C2×#11758200—22no—
59C22×#92418—21no—
60C22×#92418—21no—
61#14519777C2311yescharacteristic
62#14519777C2311yescharacteristic
63#14519769—21no—
64#14519756—21no—
65#14519756—21no—
66C4×Dic122—21no—
67C4×Dic122—21no—
68#14519736—21no—
69#14519710—21no—
70#14519710—22no—
71#14519710—22no—
72#14519708—21no—
73#14519708—22no—
74#14519708—22no—
75C4⋊#92418—21no—
76#14519701—22no—
77#14519701—21no—
78#14519700—21no—
79#14519700—21no—
80#14519698—22no—
81#14519698—21no—
82C22⋊#92418—21no—
83#14519688—21no—
84#14519688—21no—
85#14519688C2312yes—
86#14519688C2311yescharacteristic
87#14519688C2311yescharacteristic
88#14519688C2312yes—
89C2×#41×C61—21no—
90C22×Q8×C61—21no—
91C22×Q8×C61D411yescharacteristic
92C22×D4×C61D411yescharacteristic
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61—22no—
95C4⋊Q8×C61D411yescharacteristic
96C4⋊Q8×C61D411yescharacteristic
97#126×C61—22no—
98#126×C61—22no—
99#126×C61—22no—
100#126×C61—21no—
101#125×C61C2312yes—
102#125×C61C2311yescharacteristic
103#124×C61—21no—
104#124×C61—22no—
105#124×C61—22no—
106#124×C61—21no—
107#124×C61—21no—
108#124×C61D412yes—
109#124×C61C2311yescharacteristic
110#124×C61C2312yes—
111#123×C61—22no—
112#123×C61—22no—
113#123×C61—21no—
114#123×C61—22no—
115#123×C61—22no—
116C22⋊Q8×C61—22no—
117C22⋊Q8×C61—22no—
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—22no—
120C22⋊Q8×C61—22no—
121C22⋊Q8×C61—22no—
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—22no—
125C22⋊Q8×C61—22no—
126C22⋊Q8×C61D411yescharacteristic
127#121×C61—21no—
128#121×C61—22no—
129#121×C61—22no—
130#121×C61D411yescharacteristic
131#120×C61—22no—
132C4×Q8×C61—21no—
133C4×Q8×C61—21no—
134C4×Q8×C61—21no—
135C4×Q8×C61D411yescharacteristic
136C4×D4×C61—22no—
137C4×D4×C61—21no—
138C4×D4×C61—22no—
139C4×D4×C61—21no—
140C4×D4×C61D411yescharacteristic
141#117×C61—22no—
142#117×C61—21no—
143#117×C61D412yes—
144C2×C22⋊C4×C61—22no—
145C2×C22⋊C4×C61—21no—
146C2×C22⋊C4×C61—21no—
147C2×C42×C61C2311yescharacteristic
148C24×C61—21no—
149#41×C61—82no—
150C2×Q8×C61—22no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—82no—
155C2×Q8×C61—81no—
156C2×Q8×C61C2×D411yescharacteristic
157C2×Q8×C61C2×D412yes—
158C2×D4×C61—21no—
159C2×D4×C61—22no—
160C2×D4×C61—22no—
161C2×D4×C61—21no—
162C2×D4×C61—81no—
163C2×D4×C61C2×D411yescharacteristic
164C2×D4×C61C2×D412yes—
165C22×C4×C61—22no—
166C22×C4×C61—22no—
167C22×C4×C61—22no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C2×Dic122—21no—
174C2×Dic122—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—22no—
187C4⋊C4×C61C2×D412yes—
188C4⋊C4×C61C2×D412yes—
189C4⋊C4×C61C2×D411yescharacteristic
190C4⋊C4×C61C2×D411yescharacteristic
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C22⋊C4×C61—21no—
197C22⋊C4×C61—22no—
198C22⋊C4×C61—22no—
199C22⋊C4×C61—22no—
200C22⋊C4×C61—21no—
201C22⋊C4×C61—21no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—21no—
205C22⋊C4×C61—21no—
206C42×C61—22no—
207C42×C61—22no—
208C42×C61—21no—
209C42×C61C2×D411yescharacteristic
210C42×C61C2×D411yescharacteristic
211C42×C61C2×D411yescharacteristic
212C42×C61C2411yescharacteristic
213C22⋊Dic61—41no—
214#11758202—42no—
215#11758202—82no—
216#11758200—42no—
217#11758200—82no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222#11758196—41no—
223#11758196—41no—
224C4×Dic61—41no—
225C2×#92418—24no—
226C2×#92418—22no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—22no—
230C2×#92418—42no—
231C23×C61—22no—
232C23×C61—21no—
233C23×C61—42no—
234C23×C61—81no—
235C23×C61—81no—
236C23×C61#12011yescharacteristic
237Q8×C61—42no—
238Q8×C61—42no—
239Q8×C61—41no—
240Q8×C61—42no—
241Q8×C61—81no—
242Q8×C61—82no—
243D4×C61—41no—
244D4×C61—42no—
245D4×C61—42no—
246D4×C61—42no—
247D4×C61—82no—
248D4×C61—82no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—42no—
259C2×C4×C61—81no—
260C2×C4×C61—82no—
261C2×C4×C61—82no—
262C2×C4×C61—82no—
263C2×C4×C61#12011yescharacteristic
264C2×C4×C61#12011yescharacteristic
265C2×C4×C61#12011yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×Dic61—41no—
270C2×Dic61—41no—
271Dic122—41no—
272Dic122—42no—
273Dic122—41no—
274#92418—24no—
275#92418—22no—
276#92418—41no—
277#92418—42no—
278#19049—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—41no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—41no—
285C22×C61—41no—
286C22×C61—42no—
287C22×C61—162no—
288C22×C61#45612yes—
289C22×C61C2×#12011yescharacteristic
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—42no—
299C4×C61—41no—
300C4×C61—42no—
301C4×C61—42no—
302Dic61—81no—
303Dic61—81no—
304#3561D6111yescharacteristic
305#3215—611no—
306#3181—611no—
307#3162—611no—
308#1664—612no—
309#1661—612no—
310#1661—612no—
311#1660—612no—
312#1532—611no—
313#1531—611no—
314C2×#329—611no—
315C2×C61—22no—
316C2×C61—81no—
317C2×C61—82no—
318C2×C61C2×#45611yescharacteristic
319C2×C61#301811yescharacteristic
320C2×C61#301911yescharacteristic
321#563C2×D6112yes—
322#562C2×D6112yes—
323#556—21no—
324#551—22no—
325D4×Q8—21no—
326#548—21no—
327#546—21no—
328#543—21no—
329#542—21no—
330#542—22no—
331#538—21no—
332#536—21no—
333#533—611no—
334#533C2×D6111yescharacteristic
335#529C2×D6111yescharacteristic
336C2×#124C2×D6111yescharacteristic
337#501—612no—
338#501—612no—
339#501—611no—
340#501—1221no—
341#501—1221no—
342#497—611no—
343#497—612no—
344#491—611no—
345#489—611no—
346#489—612no—
347#467—1221no—
348#465—1221no—
349#464—1222no—
350#451—1221no—
351#450—1221no—
352#441—1221no—
353#440—1221no—
354#432—611no—
355#427—1222no—
356C2×#97—611no—
357C2×#97—611no—
358#330—612no—
359#330—612no—
360#330—612no—
361#329—612no—
362#329—612no—
363#329—612no—
364C61#1904911yescharacteristic, complemented, Sylow
365C2×#41—21no—
366C22×Q8—21no—
367C22×Q8C22⋊D6111yescharacteristic
368C22×D4C22⋊D6111yescharacteristic
369C2×Q16—1222no—
370C2×Q16—1221no—
371C2×SD16—1222no—
372C2×SD16—1221no—
373C22×C8—1221no—
374C22×C8—1221no—
375C4⋊Q8—21no—
376C4⋊Q8—22no—
377C4⋊Q8—611no—
378C4⋊Q8—611no—
379C4⋊Q8C22⋊D6111yescharacteristic
380C4⋊Q8C22⋊D6111yescharacteristic
381#126—22no—
382#126—22no—
383#126—22no—
384#126—21no—
385#125—1221no—
386#125C22×D6111yescharacteristic
387#125C22×D6112yes—
388#124—21no—
389#124—22no—
390#124—22no—
391#124—21no—
392#124—21no—
393#124C22⋊D6112yes—
394#124C22×D6112yes—
395#124C22×D6111yescharacteristic
396#123—22no—
397#123—22no—
398#123—21no—
399#123—22no—
400#123—22no—
401C22⋊Q8—22no—
402C22⋊Q8—22no—
403C22⋊Q8—21no—
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C22⋊Q8—22no—
410C22⋊Q8—22no—
411C22⋊Q8—1221no—
412C22⋊Q8—1221no—
413C22⋊Q8C22⋊D6111yescharacteristic
414#121—21no—
415#121—22no—
416#121—22no—
417#121C22⋊D6111yescharacteristic
418#120—22no—
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—1221no—
423C4×Q8—1221no—
424C4×Q8C22⋊D6111yescharacteristic
425C4×D4—22no—
426C4×D4—21no—
427C4×D4—22no—
428C4×D4—21no—
429C4×D4C22⋊D6111yescharacteristic
430#117—22no—
431#117—21no—
432#117—1221no—
433#117C22⋊D6112yes—
434C2×C22⋊C4—22no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C42C22×D6111yescharacteristic
438#107—1222no—
439#107—1222no—
440#107—1221no—
441#106—1222no—
442#106—1222no—
443#106—1221no—
444C4⋊C8—1221no—
445#103—1221no—
446#103—1222no—
447#103—1221no—
448#103—1221no—
449#102—1221no—
450#102—1222no—
451C22⋊C8—1221no—
452#97—612no—
453#97—611no—
454#97—611no—
455#97—612no—
456#97—1221no—
457#97—1221no—
458C24—21no—
459#41—82no—
460C2×Q8—22no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—82no—
465C2×Q8—81no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8C2×C22⋊D6111yescharacteristic
469C2×Q8C2×C22⋊D6112yes—
470C2×D4—21no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—81no—
475C2×D4C2×C22⋊D6111yescharacteristic
476C2×D4C2×C22⋊D6112yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4C2×C22⋊D6111yescharacteristic
483C22×C4C2×C22⋊D6111yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485Q16—4882no—
486Q16—2442no—
487SD16—4882no—
488SD16—2442no—
489C2×C8—1224no—
490C2×C8—1222no—
491C2×C8—1221no—
492C2×C8—1221no—
493C2×C8—1222no—
494C2×C8—2442no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—2441no—
512C4⋊C4—2441no—
513C4⋊C4C2×C22⋊D6111yescharacteristic
514C4⋊C4C2×C22⋊D6111yescharacteristic
515C4⋊C4C2×C22⋊D6112yes—
516C4⋊C4C2×C22⋊D6112yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—21no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—2441no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—2441no—
537C42C2×C22⋊D6111yescharacteristic
538C42C2×C22⋊D6111yescharacteristic
539C42C2×C22⋊D6111yescharacteristic
540C42C23×D6111yescharacteristic
541C23—22no—
542C23—21no—
543C23—42no—
544C23—81no—
545C23—81no—
546C23#1451974611yescharacteristic
547Q8—42no—
548Q8—41no—
549Q8—42no—
550Q8—42no—
551Q8—81no—
552Q8—82no—
553Q8—2442no—
554Q8—2441no—
555Q8—2441no—
556D4—42no—
557D4—41no—
558D4—42no—
559D4—42no—
560D4—82no—
561D4—82no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4#1451974611yescharacteristic
579C2×C4#1451974611yescharacteristic
580C2×C4#1451974611yescharacteristic
581C2×C4C22×C22⋊D6111yescharacteristic
582C2×C4C22×C22⋊D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C8—1224no—
585C8—1222no—
586C8—2441no—
587C8—2442no—
588C22—21no—
589C22—21no—
590C22—41no—
591C22—42no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—162no—
597C22#1683994512yes—
598C22C2×#1451974611yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—4881no—
612C4—4881no—
613C2—22no—
614C2—82no—
615C2—81no—
616C2C2×#1683994511yescharacteristic
617C2#2145516811yescharacteristic
618C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3561C2
socleC22×C61C2×#120
hypercenter#3561D61
hypocenterC61#19049
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3561

Complements

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

NH ≅ G/Ntype
1. #21456822C2semidirect
2. #21456529C2semidirect
3. #21456341C2semidirect
4. #21445153C2semidirect
5. #21445136C2semidirect
6. #21445136C2semidirect
7. #21445130C2semidirect
8. #21444439C2semidirect
9. #21444434C2semidirect
10. C2×#16839502C2semidirect
13. #16840172C22semidirect
14. #16840172C22semidirect
15. #16840172C22semidirect
19. #16840157C22semidirect
20. #16840132C22semidirect
22. #16840121C22semidirect
34. #16839504C22semidirect
35. #16839504C22semidirect
36. #16839504C22semidirect
39. #16839502C22semidirect
364. C61#19049semidirect

Maximal subgroups

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

subgroupindex
#214568222
#214565292
#214563412
#214451532
#214451362
#214451302
#214444392
#214444342
C2×#168395022
C61×#35612
#1904961