Finite Group Databasebeta

← #29753120

Subgroups — #29753120

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#21450246C211yesmaximal, characteristic
2#21448842C211yesmaximal, characteristic
3#21447230C211yesmaximal, characteristic, complemented
4#21447212C211yesmaximal, characteristic, complemented
5#21447014C211yesmaximal, characteristic
6#21447013C211yesmaximal, characteristic
7#21446581C211yesmaximal, characteristic, complemented
8#21446562C211yesmaximal, characteristic, complemented
9#21446432C211yesmaximal, characteristic
10#21446404C211yesmaximal, characteristic, complemented
11#21446402C211yesmaximal, characteristic, complemented
12#21446087C211yesmaximal, characteristic, complemented
13#21446021C211yesmaximal, characteristic
14#21445913C211yesmaximal, characteristic
15C61×#1763C211yesmaximal, characteristic, complemented
16C2×#14519753C2211yescharacteristic
17C2×#14519735C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27#16839703—21no—
28#16839686—21no—
29#16839669C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839653—21no—
34#16839653—21no—
35#16839652—21no—
36#16839646—21no—
37#16839643—21no—
38#16839631C2211yescharacteristic
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196—21no—
43C4×#11758198C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839517C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519753—22no—
63#14519753—22no—
64#14519735—22no—
65#14519735—41no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×#11758198D411yescharacteristic
81C2×#11758198D411yescharacteristic
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85#14519711—41no—
86#14519709—22no—
87#14519708—22no—
88#14519708—22no—
89#14519706—22no—
90#14519701—22no—
91#14519701—41no—
92#14519689—22no—
93#14519689—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104C22×Q8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C4×Q8×C61—22no—
108C4×Q8×C61—41no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#103×C61—22no—
117#103×C61—22no—
118#97×C61—41no—
119#95×C61—21no—
120C23×D61C2×D411yescharacteristic
121C2×Q8×C61—21no—
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61#4112yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
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—42no—
159C4⋊C4×C61#4112yes—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—42no—
163C42×C61—81no—
164#11758198—24no—
165#11758198—22no—
166#11758198—41no—
167#11758198—42no—
168#11758198—81no—
169#11758198—81no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61#4112yes—
178#11758196—24no—
179#11758196—42no—
180#11758196—42no—
181#11758196—81no—
182#11758196—81no—
183#11758196—81no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—42no—
187C4×Dic61—42no—
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—
202Q8×C61—22no—
203Q8×C61—22no—
204Q8×C61—41no—
205Q8×C61—41no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
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×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61H(5,𝔽2)11yescharacteristic
229C2×C4×C612−1+411yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—22no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—41no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61#13712yes—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—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—42no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61#52911yescharacteristic
281C4×C61#53911yescharacteristic
282C4×C61#55311yescharacteristic
283C4×C61#55611yescharacteristic
284Dic61—22no—
285Dic61—41no—
286Dic61—42no—
287Dic61—41no—
288Dic61—81no—
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#1944—611no—
299#1924—611no—
300#1920—611no—
301#1920—611no—
302#1865—611no—
303#1854—611no—
304#1825—611no—
305#1763D6111yescharacteristic, complemented
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×Q8C2×D6111yescharacteristic
316C2×#117—611no—
317C2×#107—611no—
318C2×#106—611no—
319C2×#106—611no—
320C2×#106—611no—
321C2×C4⋊C8—611no—
322C2×#103—611no—
323C2×#103—611no—
324C2×#103C2×D6111yescharacteristic
325C2×#103C2×D6111yescharacteristic
326C2×C22⋊C8—611no—
327C2×C22⋊C8—611no—
328C2×#97C2×D6111yescharacteristic
329#400—1221no—
330#396—1221no—
331#391—611no—
332#389—611no—
333#389—611no—
334#389—611no—
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—1221no—
345C4×C4⋊C4C2×D6111yescharacteristic
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×Q8C22×D6111yescharacteristic
359C23×C4—611no—
360C22×C8—611no—
361C22×C8—611no—
362C22×C8C22×D6111yescharacteristic
363C22×C8C22×D6111yescharacteristic
364C22⋊Q8—1222no—
365C22⋊Q8—1222no—
366C4×Q8—22no—
367C4×Q8—41no—
368#117—1222no—
369#117—2441no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—611no—
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⋊C4—1221no—
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—21no—
391C2×C42—21no—
392C2×C42—611no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42C22×D6111yescharacteristic
396#107—2441no—
397#106—1222no—
398#106—1222no—
399#106—1222no—
400C4⋊C8—1222no—
401#103—22no—
402#103—22no—
403#103—1222no—
404#103—2441no—
405C22⋊C8—1222no—
406C22⋊C8—2441no—
407#97—41no—
408#95—21no—
409#95—611no—
410#95—611no—
411#95—1221no—
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—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8#1175822212yes—
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#1175822212yes—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—612no—
465C4⋊C4—1224no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—4881no—
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—42no—
488C42—81no—
489C42—1221no—
490C42—1221no—
491C42—2442no—
492C42—2442no—
493C23—1221no—
494C23—1221no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23C2×D4×D6111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
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—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—2441no—
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×C4C2×#1175822211yescharacteristic
550C2×C4C2×#1175822211yescharacteristic
551C2×C4C2×#1175822211yescharacteristic
552C2×C4#1451982211yescharacteristic
553C2×C4#1451982411yescharacteristic
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#1684040211yescharacteristic
567C22#1684041711yescharacteristic
568C22#137×D6111yescharacteristic
569C22#1684056111yescharacteristic
570C22#1684057111yescharacteristic
571C22#1684057211yescharacteristic
572C4—41no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—81no—
577C4—81no—
578C4—1222no—
579C4—4881no—
580C4—4881no—
581C4—4881no—
582C4—2442no—
583C4—2441no—
584C4—2441no—
585C4#1684029611yescharacteristic
586C4#1684035411yescharacteristic
587C4#1684046111yescharacteristic
588C4#1684048011yescharacteristic
589C2—4881no—
590C2#2144985511yescharacteristic
591C2#2145597911yescharacteristic
592C2#2145610911yescharacteristic
593C2#2145800611yescharacteristic
594C2#2145813111yescharacteristic
595C2#2145827911yescharacteristic
596C2#2145836911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1763D61
hypocenterC61#27060
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
3. #21447230C2semidirect
4. #21447212C2semidirect
7. #21446581C2semidirect
8. #21446562C2semidirect
10. #21446404C2semidirect
11. #21446402C2semidirect
12. #21446087C2semidirect
15. C61×#1763C2semidirect
305. #1763D61semidirect
357. C61#27060semidirect

Maximal subgroups

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

subgroupindex
#214502462
#214488422
#214472302
#214472122
#214470142
#214470132
#214465812
#214465622
#214464322
#214464042
#214464022
#214460872
#214460212
#214459132
C61×#17632
#2706061