Finite Group Databasebeta

← #29654294

Subgroups — #29654294

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C212yesmaximal, complemented
2#21450992C211yesmaximal, characteristic, complemented
3#21444799C214yesmaximal, complemented
4#21444776C212yesmaximal, complemented
5#21444771C212yesmaximal, complemented
6#21444344C212yesmaximal, complemented
7#21444160C211yesmaximal, characteristic, complemented
8C61×#3459C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic, complemented
10#16840157—22no—
11#16840132—22no—
12#16839972—22no—
13#16839967—22no—
14#16839917C2214yescomplemented
15#16839900C2214yes—
16#16839819C2212yescomplemented
17#16839674C2212yes—
18#16839674C2212yescomplemented
19#16839665—21no—
20#16839627—21no—
21#16839622—21no—
22#16839622—21no—
23C4×C4⋊Dic61C2211yescharacteristic, complemented
24#16839499C2214yescomplemented
25#16839496C2214yes—
26#16839486C2212yes—
27#16839486C2212yescomplemented
28C61×#543C2211yescharacteristic
29C61×#539—22no—
30C61×#539—21no—
31C61×#538—22no—
32C61×#538C2212yes—
33C61×#534—21no—
34C61×#534C2211yescharacteristic
35C61×#529—22no—
36C61×#529C2212yes—
37C61×#525—21no—
38C61×#525—22no—
39C2×C4×D4×C61—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×Dic122C2312yes—
42C2×#11758202—24no—
43C2×#11758200—24no—
44#14519777—22no—
45#14519777—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49C2×C22⋊Dic61—22no—
50C2×C22⋊Dic61—22no—
51C2×C4×Dic61—21no—
52C4⋊#92418—22no—
53#14519700—24no—
54#14519700—24no—
55C22⋊#92418C2314yes—
56#14519688—22no—
57#14519684—22no—
58#14519684—22no—
59#14519684C2312yes—
60C2×#41×C61—21no—
61C22×D4×C61—21no—
62C22×D4×C61D411yescharacteristic
63C23×C4×C61—21no—
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66#126×C61—22no—
67#126×C61—22no—
68#126×C61—22no—
69#125×C61—22no—
70#125×C61D412yes—
71#124×C61—22no—
72#124×C61D412yes—
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—21no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61C2312yes—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61—21no—
88#121×C61—22no—
89#121×C61—21no—
90#121×C61C2312yes—
91#120×C61—22no—
92#120×C61—22no—
93#120×C61—22no—
94C4×Q8×C61D412yes—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61D412yes—
103#117×C61—21no—
104#117×C61—22no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61C2311yescharacteristic
110C2×C22⋊C4×C61C2311yescharacteristic
111C2×C42×C61—21no—
112C2×C42×C61C2311yescharacteristic
113C24×C61—21no—
114C24×C61C2×D411yescharacteristic
115C24×C61C2×D411yescharacteristic
116#41×C61—82no—
117C2×Q8×C61—21no—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—21no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—41no—
127C2×D4×C61—42no—
128C2×D4×C61—42no—
129C2×D4×C61C2×D412yes—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61—82no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—22no—
148C22×Dic61—22no—
149C2×Dic122—24no—
150C2×Dic122—82no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D412yes—
161C4⋊C4×C61C2×D412yes—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C42×C61—22no—
175C42×C61—22no—
176C42×C61C2×D412yes—
177C42×C61C2×D412yes—
178C22⋊Dic61—82no—
179C22⋊Dic61—82no—
180#11758202—84no—
181#11758200—84no—
182C4⋊Dic61—42no—
183C4⋊Dic61—42no—
184C4⋊Dic61—41no—
185C4⋊Dic61—41no—
186C4×Dic61—42no—
187C2×#92418—24no—
188C23×C61—21no—
189C23×C61—21no—
190C23×C61—21no—
191C23×C61—22no—
192C23×C61—21no—
193C23×C61—21no—
194C23×C61—41no—
195C23×C61—41no—
196C23×C61—41no—
197C23×C61—41no—
198C23×C61—42no—
199C23×C61—81no—
200C23×C61—81no—
201C23×C61#13712yes—
202C23×C61C22×D411yescharacteristic
203Q8×C61—42no—
204Q8×C61—81no—
205D4×C61—44no—
206D4×C61—42no—
207D4×C61—44no—
208D4×C61—42no—
209D4×C61—42no—
210D4×C61—82no—
211D4×C61—81no—
212D4×C61—82no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—44no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—81no—
229C2×C4×C61—81no—
230C2×C4×C61—81no—
231C2×C4×C61—82no—
232C2×C4×C61—81no—
233C2×C4×C61#12012yes—
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61C22×D412yes—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243Dic122—44no—
244Dic122—44no—
245#92418—44no—
246#18336—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—22no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—41no—
255C22×C61—42no—
256C22×C61—41no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—41no—
260C22×C61—81no—
261C22×C61—81no—
262C22×C61—84no—
263C22×C61C2×#12011yescharacteristic
264C22×C61C2×#13711yescharacteristic
265C22×C61#57511yescharacteristic
266C4×C61—22no—
267C4×C61—21no—
268C4×C61—22no—
269C4×C61—42no—
270C4×C61—41no—
271C4×C61—41no—
272C4×C61—42no—
273C4×C61—42no—
274C4×C61#45512yes—
275Dic61—82no—
276Dic61—82no—
277#3459D6111yescharacteristic
278#3218—612no—
279#2563—611no—
280#1606—614no—
281#1602—612no—
282#1601—612no—
283#1509—612no—
284#1466—611no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—42no—
288C2×C61—81no—
289C2×C61—82no—
290C2×C61#300711yescharacteristic
291C2×C61C2×#45511yescharacteristic
292C2×C61#302711yescharacteristic
293#543C2×D6111yescharacteristic
294#539—22no—
295#539—21no—
296#538—22no—
297#538C2×D6112yes—
298#534—21no—
299#534C2×D6111yescharacteristic
300C2×C4⋊Q8—611no—
301#529—22no—
302#529C2×D6112yes—
303#525—21no—
304#525—22no—
305C2×C4×D4—21no—
306C2×C4×D4C2×D6111yescharacteristic
307#497—1222no—
308#491—1222no—
309#462—1222no—
310#461—1222no—
311#451—614no—
312#448—614no—
313#432—612no—
314#393—612no—
315#393—612no—
316#390—1221no—
317#382—1221no—
318#380—1221no—
319#380—1221no—
320C4×C4⋊C4—611no—
321#328—614no—
322#327—614no—
323#323—612no—
324#323—612no—
325C61#1833611yescharacteristic, complemented, Sylow
326C2×#41—21no—
327C22×Q8—612no—
328C22×D4—21no—
329C22×D4C22⋊D6111yescharacteristic
330C23×C4—21no—
331C23×C4C22⋊D6111yescharacteristic
332C23×C4C22⋊D6111yescharacteristic
333C2×Q16—1224no—
334C2×SD16—1224no—
335C4⋊Q8—1222no—
336C4⋊Q8—1222no—
337#126—22no—
338#126—22no—
339#126—22no—
340#125—22no—
341#125C22⋊D6112yes—
342#124—22no—
343#124C22⋊D6112yes—
344#123—22no—
345#123—22no—
346#123—22no—
347#123—22no—
348#123—22no—
349#123—22no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—21no—
354C22⋊Q8—22no—
355C22⋊Q8C22×D6112yes—
356#121—22no—
357#121—21no—
358#121—22no—
359#121—22no—
360#121—21no—
361#121C22×D6112yes—
362#120—22no—
363#120—22no—
364#120—22no—
365C4×Q8C22⋊D6112yes—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—22no—
373C4×D4C22⋊D6112yes—
374#117—21no—
375#117—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22⋊D6111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—1222no—
384C2×C22⋊C4—1222no—
385C2×C22⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4C22×D6111yescharacteristic
387C2×C42—21no—
388C2×C42—1221no—
389C2×C42C22×D6111yescharacteristic
390C4⋊C8—1222no—
391#103—1224no—
392#103—1224no—
393C22⋊C8—614no—
394#97—1222no—
395#95—612no—
396#95—1222no—
397#95—1222no—
398C24—21no—
399C24C2×C22⋊D6111yescharacteristic
400C24C2×C22⋊D6111yescharacteristic
401#41—82no—
402C2×Q8—21no—
403C2×Q8—1224no—
404C2×Q8—4882no—
405C2×Q8C2×C22⋊D6112yes—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—41no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4C2×C22⋊D6112yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—82no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436Q16—4884no—
437SD16—4884no—
438C2×C8—1224no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—2442no—
449C4⋊C4—2442no—
450C4⋊C4—2441no—
451C4⋊C4—2441no—
452C4⋊C4C2×C22⋊D6112yes—
453C4⋊C4C2×C22⋊D6112yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—4882no—
467C22⋊C4—4882no—
468C42—22no—
469C42—22no—
470C42—2442no—
471C42C2×C22⋊D6112yes—
472C42C2×C22⋊D6112yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—21no—
478C23—21no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—81no—
485C23—81no—
486C23#1451980712yes—
487C23C22×C22⋊D6111yescharacteristic
488Q8—42no—
489Q8—81no—
490Q8—2444no—
491Q8—2444no—
492D4—44no—
493D4—42no—
494D4—44no—
495D4—42no—
496D4—42no—
497D4—82no—
498D4—81no—
499D4—82no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4#1451974612yes—
527C2×C4#1451974611yescharacteristic
528C2×C4#1451974611yescharacteristic
529C2×C4C22×C22⋊D6112yes—
530C8—2444no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—84no—
547C22C2×#1451974611yescharacteristic
548C22C2×#1451980711yescharacteristic
549C22#1684056011yescharacteristic
550C4—21no—
551C4—22no—
552C4—22no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—4882no—
559C4—4882no—
560C4#1683994012yes—
561C2—21no—
562C2—21no—
563C2—42no—
564C2—81no—
565C2—82no—
566C2#2145507911yescharacteristic
567C2C2×#1683994011yescharacteristic
568C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3459C2
socleC22×C61C2×#120
hypercenter#3459D61
hypocenterC61#18336
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3459

Complements

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

NH ≅ G/Ntype
1. #21456843C2semidirect
2. #21450992C2semidirect
3. #21444799C2semidirect
4. #21444776C2semidirect
5. #21444771C2semidirect
6. #21444344C2semidirect
7. #21444160C2semidirect
9. C2×#14519777C22semidirect
14. #16839917C22semidirect
16. #16839819C22semidirect
18. #16839674C22semidirect
23. C4×C4⋊Dic61C22semidirect
24. #16839499C22semidirect
27. #16839486C22semidirect
325. C61#18336semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214509922
#214447992
#214447762
#214447712
#214443442
#214441602
C61×#34592
#1833661