Finite Group Databasebeta

← #29654110

Subgroups — #29654110

Order 15616.

868 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456983C212yesmaximal, complemented
2#21451695C211yesmaximal, characteristic, complemented
3#21444861C211yesmaximal, characteristic, complemented
4#21444856C211yesmaximal, characteristic
5#21444850C212yesmaximal, complemented
6#21444827C211yesmaximal, characteristic, complemented
7#21444822C211yesmaximal, characteristic
8#21444816C212yesmaximal, complemented
9#21444396C212yesmaximal, complemented
10#21444143C211yesmaximal, characteristic, complemented
11C61×#3461C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840161—22no—
14#16840116—22no—
15#16840095C2212yes—
16#16840090C2212yes—
17#16840078C2212yes—
18#16840073C2212yes—
19#16840069—22no—
20#16840062—22no—
21#16839815C2212yes—
22#16839701C2211yescharacteristic
23#16839701C2211yescharacteristic
24#16839700—21no—
25#16839692C2211yescharacteristic
26#16839692C2211yescharacteristic
27#16839681—21no—
28#16839646—21no—
29#16839646—21no—
30#16839630C2211yescharacteristic
31#16839499C2212yes—
32#16839499C2212yes—
33#16839496C2212yes—
34#16839496C2212yes—
35#16839488C2212yes—
36#16839488C2211yescharacteristic
37#16839488C2211yescharacteristic
38C61×#544C2211yescharacteristic
39C61×#542—22no—
40C61×#541—22no—
41C61×#539—21no—
42C61×#539C2212yes—
43C61×#537C2211yescharacteristic
44C61×#536—21no—
45C61×#533—22no—
46C61×#529—22no—
47C61×#529—21no—
48C61×#529C2212yes—
49C4×#41×C61—21no—
50C2×C4×Q8×C61C2211yescharacteristic
51#14519769—22no—
52#14519769—22no—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59#14519710—22no—
60#14519710—22no—
61#14519708—22no—
62#14519708—22no—
63C4⋊#92418—22no—
64#14519701—22no—
65#14519701—22no—
66#14519698—22no—
67#14519698—22no—
68C22⋊#92418C2312yes—
69C22⋊#92418C2312yes—
70C4×#92418—22no—
71#14519684—21no—
72#14519684—21no—
73#14519684—21no—
74#14519684—21no—
75#14519684—21no—
76#14519684—21no—
77#14519684—21no—
78#14519684—21no—
79#14519684C2311yescharacteristic
80#14519684C2311yescharacteristic
81C2×#41×C61—21no—
82C22×Q8×C61D411yescharacteristic
83C4⋊Q8×C61—22no—
84C4⋊Q8×C61—22no—
85#126×C61—22no—
86#126×C61—22no—
87#126×C61—22no—
88#126×C61—22no—
89#125×C61—22no—
90#125×C61—21no—
91#125×C61—22no—
92#125×C61—22no—
93#125×C61D412yes—
94#124×C61—22no—
95#124×C61—21no—
96#124×C61D412yes—
97#123×C61—22no—
98#123×C61—22no—
99#123×C61—22no—
100#123×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61C2312yes—
108#121×C61—22no—
109#121×C61—21no—
110#121×C61C2312yes—
111C4×Q8×C61—21no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—22no—
115C4×Q8×C61—22no—
116C4×Q8×C61D412yes—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61—21no—
120C4×D4×C61—22no—
121C4×D4×C61—22no—
122C4×D4×C61D412yes—
123#117×C61—22no—
124#117×C61—22no—
125#117×C61—21no—
126#117×C61—22no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61—22no—
131C2×C42×C61D411yescharacteristic
132C2×C42×C61D411yescharacteristic
133C2×C42×C61C2311yescharacteristic
134#41×C61—42no—
135#41×C61—42no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—22no—
139C2×Q8×C61—22no—
140C2×Q8×C61—41no—
141C2×Q8×C61C2×D412yes—
142C2×D4×C61—21no—
143C2×D4×C61—22no—
144C2×D4×C61—21no—
145C2×D4×C61C2×D412yes—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—22no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D412yes—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—21no—
189C42×C61—22no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61C2×D412yes—
194C42×C61C2×D412yes—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197#11758196—42no—
198#11758196—41no—
199#11758196—41no—
200#11758196—42no—
201#11758196—42no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—22no—
205C23×C61C22×D411yescharacteristic
206Q8×C61—44no—
207Q8×C61—42no—
208Q8×C61—42no—
209Q8×C61—81no—
210D4×C61—42no—
211D4×C61—42no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—81no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—44no—
230C2×C4×C61—44no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—82no—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12012yes—
237C2×C4×C61#13512yes—
238C2×C4×C61C22×D412yes—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—42no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250#92418—42no—
251#92418—42no—
252#18315—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61C2×#12011yescharacteristic
259C22×C61C2×#13511yescharacteristic
260C22×C61#57511yescharacteristic
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—21no—
264C4×C61—21no—
265C4×C61—22no—
266C4×C61—41no—
267C4×C61—42no—
268C4×C61—42no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—42no—
272Dic61—81no—
273Dic61—81no—
274Dic61—81no—
275Dic61—81no—
276#3461D6111yescharacteristic
277#3232—612no—
278#2634—611no—
279#1617—611no—
280#1616—611no—
281#1615—612no—
282#1611—611no—
283#1610—611no—
284#1609—612no—
285#1521—612no—
286#1461—611no—
287C2×C61—21no—
288C2×C61—21no—
289C2×C61—82no—
290C2×C61#300211yescharacteristic
291C2×C61#302011yescharacteristic
292C2×C61#302811yescharacteristic
293#544C2×D6111yescharacteristic
294#542—22no—
295#541—22no—
296#539—21no—
297#539C2×D6112yes—
298#537C2×D6111yescharacteristic
299#536—21no—
300#533—22no—
301#529—22no—
302#529—21no—
303#529C2×D6112yes—
304C2×#125—611no—
305C4×#41—21no—
306C2×C4×Q8C2×D6111yescharacteristic
307#499—1222no—
308#488—1222no—
309#484—612no—
310#483—612no—
311#481—612no—
312#480—612no—
313#479—1222no—
314#478—1222no—
315#431—612no—
316#400—611no—
317#400—611no—
318#400—1221no—
319#398—611no—
320#398—611no—
321#395—1221no—
322#387—1221no—
323#387—1221no—
324#383—611no—
325#328—612no—
326#328—612no—
327#327—612no—
328#327—612no—
329#324—612no—
330#324—611no—
331#324—611no—
332C61#1831511yescharacteristic, complemented, Sylow
333C2×#41—21no—
334C22×Q8C22⋊D6111yescharacteristic
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337#126—22no—
338#126—22no—
339#126—22no—
340#126—22no—
341#125—21no—
342#125—22no—
343#125—22no—
344#125—22no—
345#125—1222no—
346#125—1222no—
347#125C22⋊D6112yes—
348#124—22no—
349#124—21no—
350#124C22⋊D6112yes—
351#123—22no—
352#123—22no—
353#123—22no—
354#123—22no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—21no—
360C22⋊Q8—22no—
361C22⋊Q8C22×D6112yes—
362#121—22no—
363#121—21no—
364#121C22×D6112yes—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8C22⋊D6112yes—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4C22⋊D6112yes—
377#117—22no—
378#117—22no—
379#117—21no—
380#117—22no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—1221no—
384C2×C4⋊C4—1221no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22⋊D6111yescharacteristic
388C2×C4⋊C4C22×D6111yescharacteristic
389C2×C4⋊C4C22×D6111yescharacteristic
390C2×C42—22no—
391C2×C42C22⋊D6111yescharacteristic
392C2×C42C22⋊D6111yescharacteristic
393C2×C42C22×D6111yescharacteristic
394#107—1222no—
395#107—1222no—
396#106—1222no—
397#106—1222no—
398C4⋊C8—1222no—
399#103—1222no—
400#103—1222no—
401#102—1222no—
402#102—1222no—
403C22⋊C8—612no—
404C22⋊C8—612no—
405C4×C8—1222no—
406#95—611no—
407#95—611no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#41—42no—
417#41—42no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—41no—
423C2×Q8C2×C22⋊D6112yes—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4C2×C22⋊D6112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438C22×C4C2×C22⋊D6111yescharacteristic
439C22×C4C2×C22⋊D6111yescharacteristic
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C23×D6111yescharacteristic
442C2×C8—1222no—
443C2×C8—1222no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—1222no—
460C4⋊C4—1222no—
461C4⋊C4—2442no—
462C4⋊C4—2441no—
463C4⋊C4—2441no—
464C4⋊C4—2442no—
465C4⋊C4—2442no—
466C4⋊C4C2×C22⋊D6112yes—
467C4⋊C4C2×C22⋊D6112yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42C2×C22⋊D6112yes—
485C42C2×C22⋊D6112yes—
486C23—22no—
487C23C22×C22⋊D6111yescharacteristic
488Q8—44no—
489Q8—42no—
490Q8—42no—
491Q8—81no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—81no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—2441no—
517C2×C4—2441no—
518C2×C4—2441no—
519C2×C4—2441no—
520C2×C4—2442no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4#1451974611yescharacteristic
528C2×C4#1451974611yescharacteristic
529C2×C4#1451974612yes—
530C2×C4#1451979712yes—
531C2×C4C22×C22⋊D6112yes—
532C8—2442no—
533C8—2442no—
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×#1451974611yescharacteristic
540C22C2×#1451979711yescharacteristic
541C22#1684056011yescharacteristic
542C4—21no—
543C4—21no—
544C4—22no—
545C4—22no—
546C4—22no—
547C4—42no—
548C4—41no—
549C4—41no—
550C4—42no—
551C4—41no—
552C4—42no—
553C4—4881no—
554C4—4881no—
555C4—4881no—
556C4—4881no—
557C2—21no—
558C2—21no—
559C2—82no—
560C2#2145503711yescharacteristic
561C2#2145518311yescharacteristic
562C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3461C2
socleC22×C61C2×#120
prime radicalC61×#537C22
hypercenter#3461D61
hypocenterC61#18315
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. #21456983C2semidirect
2. #21451695C2semidirect
3. #21444861C2semidirect
5. #21444850C2semidirect
6. #21444827C2semidirect
8. #21444816C2semidirect
9. #21444396C2semidirect
10. #21444143C2semidirect
332. C61#18315semidirect

Maximal subgroups

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

subgroupindex
#214569832
#214516952
#214448612
#214448562
#214448502
#214448272
#214448222
#214448162
#214443962
#214441432
C61×#34612
#1831561