Finite Group Databasebeta

← #29753126

Subgroups — #29753126

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450253C211yesmaximal, characteristic
2#21448845C211yesmaximal, characteristic
3#21447231C211yesmaximal, characteristic, complemented
4#21447209C211yesmaximal, characteristic, complemented
5#21447018C211yesmaximal, characteristic
6#21447017C211yesmaximal, characteristic
7#21446579C211yesmaximal, characteristic, complemented
8#21446432C211yesmaximal, characteristic
9#21446407C211yesmaximal, characteristic, complemented
10#21446401C211yesmaximal, characteristic, complemented
11#21446086C211yesmaximal, characteristic, complemented
12#21446015C211yesmaximal, characteristic
13#21445912C211yesmaximal, characteristic
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1944C211yesmaximal, characteristic, complemented
16C2×#14519757C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519708C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28#16839700—21no—
29#16839684—21no—
30#16839667C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839652—21no—
33#16839651—21no—
34#16839651—21no—
35#16839645—21no—
36#16839645—21no—
37#16839631C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40C4⋊C4×Dic61—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#107×C61C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×C4×D61C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63#14519757—22no—
64#14519757—22no—
65C4×Dic122—22no—
66C4×Dic122—41no—
67#14519735—22no—
68#14519735—41no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198D411yescharacteristic
82C2×#11758198D411yescharacteristic
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88#14519708—22no—
89#14519708—22no—
90#14519706—22no—
91#14519702—22no—
92#14519702—22no—
93#14519700—22no—
94#14519700—41no—
95#14519689—22no—
96#14519689—41no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C42×C61C2311yescharacteristic
115#107×C61—41no—
116#106×C61—22no—
117#97×C61—41no—
118#95×C61—21no—
119#95×C61—21no—
120C23×D61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122#4112yes—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61#4112yes—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61—81no—
162C4⋊C4×C61#4112yes—
163C42×C61—21no—
164C42×C61—21no—
165#11758198—24no—
166#11758198—22no—
167#11758198—41no—
168#11758198—42no—
169#11758198—81no—
170#11758198—81no—
171C4⋊Dic61—24no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—42no—
178C4⋊Dic61#4112yes—
179#11758196—42no—
180#11758196—81no—
181#11758196—81no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—42no—
185C4×Dic61—42no—
186C4×Dic61—42no—
187C4×Dic61—81no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C23×C61C22×D411yescharacteristic
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—41no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C612−1+411yescharacteristic
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61#13712yes—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251Dic122—22no—
252Dic122—22no—
253Dic122—41no—
254Dic122—41no—
255C8×C61—22no—
256C8×C61—41no—
257#92418—41no—
258#92418—41no—
259#27060—611nomaximal, Sylow
260C22×C61#52511yescharacteristic
261C22×C61#54611yescharacteristic
262C22×C61#54711yescharacteristic
263C22×C61C2×#13711yescharacteristic
264C22×C61#57511yescharacteristic
265C22×C61#57711yescharacteristic
266C22×C61#57711yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C4×C61—41no—
275C4×C61—42no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61#52911yescharacteristic
281C4×C61#53911yescharacteristic
282C4×C61#55311yescharacteristic
283C4×C61#55611yescharacteristic
284Dic61—22no—
285Dic61—42no—
286Dic61—41no—
287Dic61—42no—
288Dic61—41no—
289Dic61—81no—
290Dic61—81no—
291#2497—611no—
292#2362—611no—
293#2072—611no—
294#2068—611no—
295#2035—611no—
296#2035—611no—
297#1948—611no—
298#1944D6111yescharacteristic, complemented
299#1924—611no—
300#1920—611no—
301#1920—611no—
302#1865—611no—
303#1854—611no—
304#1825—611no—
305#1763—611no—
306C2×C61#246611yescharacteristic
307C2×C61#311611yescharacteristic
308C2×C61#312611yescharacteristic
309C2×C61#332111yescharacteristic
310C2×C61#333111yescharacteristic
311C2×C61#334411yescharacteristic
312C2×C61#335011yescharacteristic
313D61—81no—
314C2×C22⋊Q8—611no—
315C2×C4×Q8—611no—
316C2×#117—611no—
317C2×#107C2×D6111yescharacteristic
318C2×#106—611no—
319C2×#106—611no—
320C2×#106C2×D6111yescharacteristic
321C2×C4⋊C8—611no—
322C2×#103—611no—
323C2×#103—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×C22⋊C8—611no—
327C2×C22⋊C8—611no—
328C2×#97C2×D6111yescharacteristic
329#400—1221no—
330#396—1221no—
331#391—611no—
332#389—611no—
333#389C2×D6111yescharacteristic
334#389C2×D6111yescharacteristic
335#388—1221no—
336#388—1221no—
337#388—1221no—
338#387—1221no—
339#387—1221no—
340#384—611no—
341#380—611no—
342#380—611no—
343C4×C4⋊C4—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—1221no—
346C4×C22⋊C4—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340C2×D6111yescharacteristic
354#340C2×D6111yescharacteristic
355#336—611no—
356#336—611no—
357C61#2706011yescharacteristic, complemented, Sylow
358C22×Q8—611no—
359C23×C4—611no—
360C22×C8—611no—
361C22×C8—611no—
362C22×C8C22×D6111yescharacteristic
363C22×C8C22×D6111yescharacteristic
364C22⋊Q8—1222no—
365C22⋊Q8—1222no—
366C4×Q8—1222no—
367C4×Q8—2441no—
368#117—1222no—
369#117—2441no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C42—611no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396#107—41no—
397#106—22no—
398#106—1222no—
399#106—1222no—
400C4⋊C8—1222no—
401#103—1222no—
402#103—1222no—
403#103—1222no—
404#103—2441no—
405C22⋊C8—1222no—
406C22⋊C8—2441no—
407#97—41no—
408#95—21no—
409#95—21no—
410#95—611no—
411#95—611no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419C24—611no—
420C2×Q8—612no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4—1221no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4D4×D6111yescharacteristic
444C22×C4D4×D6111yescharacteristic
445C22×C4C23×D6111yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8#1175822612yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—612no—
468C4⋊C4—1224no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—4881no—
475C4⋊C4—4881no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4#1175822212yes—
479C22⋊C4—1222no—
480C22⋊C4—1224no—
481C22⋊C4—4881no—
482C22⋊C4—4881no—
483C22⋊C4—2441no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—1221no—
488C42—1221no—
489C42—4881no—
490C42—2442no—
491C42—2442no—
492C42—2442no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23C2×D4×D6111yescharacteristic
501Q8—1222no—
502Q8—1222no—
503Q8—2441no—
504Q8—2441no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—612no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1222no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1222no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4#41×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C2×C4C2×#1175822211yescharacteristic
551C2×C4C2×#1175822611yescharacteristic
552C2×C4#1451982211yescharacteristic
553C2×C4#1451982511yescharacteristic
554C8—22no—
555C8—41no—
556C8—2441no—
557C8—2441no—
558C22—2441no—
559C22—2441no—
560C22—2441no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22—2441no—
565C22#1684027111yescharacteristic
566C22#1684040111yescharacteristic
567C22#1684041511yescharacteristic
568C22#137×D6111yescharacteristic
569C22#1684055811yescharacteristic
570C22#1684057111yescharacteristic
571C22#1684057211yescharacteristic
572C4—41no—
573C4—42no—
574C4—41no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—1222no—
579C4—4881no—
580C4—4881no—
581C4—2441no—
582C4—2442no—
583C4—2441no—
584C4—2442no—
585C4#1684029711yescharacteristic
586C4#1684035811yescharacteristic
587C4#1684046011yescharacteristic
588C4#1684048411yescharacteristic
589C2—4881no—
590C2#2144986111yescharacteristic
591C2#2145597711yescharacteristic
592C2#2145610911yescharacteristic
593C2#2145800011yescharacteristic
594C2#2145813011yescharacteristic
595C2#2145827311yescharacteristic
596C2#2145836311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1944C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1944D61
hypocenterC61#27060
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
3. #21447231C2semidirect
4. #21447209C2semidirect
7. #21446579C2semidirect
9. #21446407C2semidirect
10. #21446401C2semidirect
11. #21446086C2semidirect
14. #21445650C2semidirect
15. C61×#1944C2semidirect
298. #1944D61semidirect
357. C61#27060semidirect

Maximal subgroups

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

subgroupindex
#214502532
#214488452
#214472312
#214472092
#214470182
#214470172
#214465792
#214464322
#214464072
#214464012
#214460862
#214460152
#214459122
#214456502
C61×#19442
#2706061