Finite Group Databasebeta

← #29753116

Subgroups — #29753116

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#21448844C211yesmaximal, characteristic
2#21447228C211yesmaximal, characteristic, complemented
3#21447211C211yesmaximal, characteristic, complemented
4#21447015C211yesmaximal, characteristic
5#21447015C211yesmaximal, characteristic
6#21446580C211yesmaximal, characteristic, complemented
7#21446559C211yesmaximal, characteristic, complemented
8#21446434C211yesmaximal, characteristic
9#21446404C211yesmaximal, characteristic, complemented
10#21446401C211yesmaximal, characteristic, complemented
11#21446086C211yesmaximal, characteristic, complemented
12#21446016C211yesmaximal, characteristic
13#21445911C211yesmaximal, characteristic
14#21445652C211yesmaximal, characteristic, complemented
15C61×#2497C211yesmaximal, characteristic
16C2×#14519756C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519708C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519688C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839656—21no—
34#16839650—21no—
35#16839644—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37#16839613—21no—
38#16839524C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48C61×#400—21no—
49C61×#396—21no—
50C61×#389C2211yescharacteristic
51C61×#389C2211yescharacteristic
52C61×#388—21no—
53C61×#387—21no—
54C61×#384C2211yescharacteristic
55C61×#380C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519756—22no—
65#14519756—22no—
66C4×Dic122—22no—
67C4×Dic122—41no—
68#14519736—22no—
69#14519736—41no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519710—41no—
82#14519708—22no—
83#14519708—22no—
84#14519708—22no—
85C4⋊#92418—22no—
86#14519700—22no—
87#14519700—22no—
88#14519700—22no—
89#14519700—41no—
90C22⋊#92418—22no—
91C22⋊#92418—41no—
92#14519688—41no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97C23×C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C22⋊C4×C61—21no—
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61D411yescharacteristic
109C2×C22⋊C4×C61D411yescharacteristic
110C2×C42×C61—21no—
111C2×C42×C61C2311yescharacteristic
112C2×C42×C61C2311yescharacteristic
113#95×C61—21no—
114#95×C61—21no—
115#95×C61—21no—
116#95×C61—21no—
117#95×C61—21no—
118#95×C61D411yescharacteristic
119#95×C61D411yescharacteristic
120C24×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
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×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C2×Dic122—21no—
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122#4112yes—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—42no—
156C4⋊C4×C61—81no—
157C4⋊C4×C61—81no—
158C4⋊C4×C61—81no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—41no—
161C22⋊C4×C61—81no—
162C22⋊C4×C61—81no—
163C42×C61—21no—
164C42×C61—21no—
165C42×C61—21no—
166C42×C61—21no—
167C42×C61—81no—
168C22⋊Dic61—24no—
169C22⋊Dic61—42no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61#4112yes—
175C4⋊Dic61#4112yes—
176#11758196—24no—
177#11758196—42no—
178#11758196—42no—
179C4×Dic61—42no—
180C4×Dic61—42no—
181C4×Dic61—42no—
182C2×#92418—21no—
183C2×#92418—21no—
184C2×#92418—21no—
185C2×#92418—21no—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418#4112yes—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61C22×D411yescharacteristic
202C2×C4×C61—21no—
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—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
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×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61#13712yes—
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
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—41no—
251Dic122—22no—
252Dic122—22no—
253Dic122—41no—
254Dic122—41no—
255#92418—22no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#27060—611nomaximal, Sylow
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61#52511yescharacteristic
268C22×C61#54611yescharacteristic
269C22×C61#54711yescharacteristic
270C22×C61C2×#13711yescharacteristic
271C22×C61#57511yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57711yescharacteristic
274C4×C61—22no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61—81no—
283C4×C61#52911yescharacteristic
284C4×C61#53911yescharacteristic
285C4×C61#55311yescharacteristic
286C4×C61#55611yescharacteristic
287Dic61—42no—
288Dic61—42no—
289Dic61—42no—
290Dic61—81no—
291#2497D6111yescharacteristic
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#1763—611no—
306C2×C61—81no—
307C2×C61#246611yescharacteristic
308C2×C61#311611yescharacteristic
309C2×C61#312611yescharacteristic
310C2×C61#332111yescharacteristic
311C2×C61#333111yescharacteristic
312C2×C61#334411yescharacteristic
313C2×C61#335011yescharacteristic
314C2×C22⋊Q8—611no—
315C2×C4×Q8—611no—
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×#103—611no—
325C2×#103—611no—
326C2×C22⋊C8—611no—
327C2×C22⋊C8—611no—
328C2×#97—611no—
329#400—21no—
330#396—21no—
331#391—611no—
332#389—611no—
333#389C2×D6111yescharacteristic
334#389C2×D6111yescharacteristic
335#388—21no—
336#388—1221no—
337#388—1221no—
338#387—21no—
339#387—1221no—
340#384C2×D6111yescharacteristic
341#380C2×D6111yescharacteristic
342#380C2×D6111yescharacteristic
343C4×C4⋊C4—611no—
344C4×C4⋊C4—1221no—
345C4×C4⋊C4C2×D6111yescharacteristic
346C4×C22⋊C4C2×D6111yescharacteristic
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340—611no—
355#336—611no—
356#336—611no—
357C61#2706011yescharacteristic, complemented, Sylow
358C22×Q8—611no—
359C23×C4C22×D6111yescharacteristic
360C22×C8—611no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8—611no—
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—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
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—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4C22⋊D6111yescharacteristic
389C2×C22⋊C4C22⋊D6111yescharacteristic
390C2×C42—21no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394C2×C42C22×D6111yescharacteristic
395C2×C42C22×D6111yescharacteristic
396#107—2441no—
397#106—1222no—
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—2441no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95C22⋊D6111yescharacteristic
418#95C22⋊D6111yescharacteristic
419C24C2×C22⋊D6111yescharacteristic
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—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6111yescharacteristic
445C22×C4C23×D6111yescharacteristic
446C2×C8—612no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—612no—
471C4⋊C4—612no—
472C4⋊C4—1224no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—1224no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—81no—
490C42—2442no—
491C42—2442no—
492C42—2442no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×C22⋊D6111yescharacteristic
501Q8—1222no—
502Q8—1222no—
503Q8—2441no—
504Q8—2441no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
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#1451980812yes—
548C2×C4#41×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C2×C4#41×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C2×C4#1451982311yescharacteristic
553C2×C4#1451982411yescharacteristic
554C8—1222no—
555C8—2441no—
556C8—2441no—
557C8—2441no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1684026911yescharacteristic
566C22#1684040011yescharacteristic
567C22#1684041111yescharacteristic
568C22C2×#1451980811yescharacteristic
569C22#1684056011yescharacteristic
570C22#1684057311yescharacteristic
571C22#1684057311yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—81no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—4881no—
582C4—2442no—
583C4—2442no—
584C4—2442no—
585C4#1684029511yescharacteristic
586C4#1684035311yescharacteristic
587C4#1684046311yescharacteristic
588C4#1684047911yescharacteristic
589C2—81no—
590C2#2144985111yescharacteristic
591C2#2145597511yescharacteristic
592C2#2145610511yescharacteristic
593C2#2145799911yescharacteristic
594C2#2145812911yescharacteristic
595C2#2145826811yescharacteristic
596C2#2145835811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2497C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2497D61
hypocenterC61#27060
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2497

Complements

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

NH ≅ G/Ntype
2. #21447228C2semidirect
3. #21447211C2semidirect
6. #21446580C2semidirect
7. #21446559C2semidirect
9. #21446404C2semidirect
10. #21446401C2semidirect
11. #21446086C2semidirect
14. #21445652C2semidirect
357. C61#27060semidirect

Maximal subgroups

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

subgroupindex
#214488442
#214472282
#214472112
#214470152
#214465802
#214465592
#214464342
#214464042
#214464012
#214460862
#214460162
#214459112
#214456522
C61×#24972
#2706061