Finite Group Databasebeta

← #29753118

Subgroups — #29753118

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#21450252C211yesmaximal, characteristic
2#21448844C211yesmaximal, characteristic
3#21447229C211yesmaximal, characteristic, complemented
4#21447213C211yesmaximal, characteristic, complemented
5#21447016C211yesmaximal, characteristic
6#21447016C211yesmaximal, characteristic
7#21446583C211yesmaximal, characteristic, complemented
8#21446562C211yesmaximal, characteristic, complemented
9#21446405C211yesmaximal, characteristic, complemented
10#21446405C211yesmaximal, characteristic, complemented
11#21446090C211yesmaximal, characteristic, complemented
12#21446020C211yesmaximal, characteristic
13#21445915C211yesmaximal, characteristic
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1924C211yesmaximal, characteristic
16C2×#14519756C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27#16839704—21no—
28#16839685—21no—
29#16839668C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839656—21no—
34#16839655—21no—
35#16839650—21no—
36#16839649—21no—
37#16839644—21no—
38#16839632C2211yescharacteristic
39#16839622C2211yescharacteristic
40#16839622C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839613—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C4×C22⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60#14519756—22no—
61#14519756—22no—
62C4×Dic122—22no—
63C4×Dic122—41no—
64#14519736—22no—
65#14519736—41no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C22⋊Dic61—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88#14519711—41no—
89#14519709—22no—
90#14519709—22no—
91#14519709—22no—
92#14519702—22no—
93#14519702—22no—
94#14519702—22no—
95#14519702—41no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C22⋊C4×C61D411yescharacteristic
111C2×C22⋊C4×C61D411yescharacteristic
112C2×C42×C61C2311yescharacteristic
113C2×C42×C61C2311yescharacteristic
114C4⋊C8×C61—22no—
115C22⋊C8×C61—22no—
116C22⋊C8×C61—41no—
117#97×C61—41no—
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×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2411yescharacteristic
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
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—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61#4112yes—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—41no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—21no—
165C22⋊Dic61—24no—
166C22⋊Dic61—42no—
167C22⋊Dic61—81no—
168C22⋊Dic61—81no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—81no—
182C4⋊Dic61—81no—
183C4⋊Dic61#4112yes—
184C4⋊Dic61#4112yes—
185#11758196—24no—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189#11758196—81no—
190C4×Dic61—42no—
191C4×Dic61—42no—
192C4×Dic61—42no—
193C4×Dic61—81no—
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#13712yes—
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—41no—
231C2×Dic61—41no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—41no—
235C2×Dic61—41no—
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—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251Dic122—22no—
252Dic122—22no—
253Dic122—41no—
254Dic122—41no—
255C8×C61—22no—
256C8×C61—41no—
257C8×C61—41no—
258C8×C61—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#52911yescharacteristic
280C4×C61#53911yescharacteristic
281C4×C61#55311yescharacteristic
282C4×C61#55611yescharacteristic
283Dic61—42no—
284Dic61—42no—
285Dic61—42no—
286Dic61—81no—
287Dic61—81no—
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#1924D6111yescharacteristic
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⋊C8C2×D6111yescharacteristic
322C2×#103—611no—
323C2×#103—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×C22⋊C8C2×D6111yescharacteristic
327C2×C22⋊C8C2×D6111yescharacteristic
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—611no—
345C4×C4⋊C4—1221no—
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#336C2×D6111yescharacteristic
356#336C2×D6111yescharacteristic
357C61#2706011yescharacteristic, complemented, Sylow
358C22×Q8—611no—
359C23×C4C22×D6111yescharacteristic
360C22×C8C22×D6111yescharacteristic
361C22×C8C22×D6111yescharacteristic
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—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
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⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C22⋊C4D24411yescharacteristic
389C2×C22⋊C4D24411yescharacteristic
390C2×C42—1221no—
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—22no—
401#103—1222no—
402#103—1222no—
403#103—1222no—
404#103—2441no—
405C22⋊C8—22no—
406C22⋊C8—41no—
407#97—41no—
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#95—1221no—
417#95D24411yescharacteristic
418#95D24411yescharacteristic
419C24C2×D24411yescharacteristic
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—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—1221no—
440C22×C4C2×D24411yescharacteristic
441C22×C4C2×D24411yescharacteristic
442C22×C4C2×D24411yescharacteristic
443C22×C4C2×D24411yescharacteristic
444C22×C4C2×D24411yescharacteristic
445C22×C4C23×D6111yescharacteristic
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8#1175822412yes—
458C4⋊C4—612no—
459C4⋊C4—612no—
460C4⋊C4—1224no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
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—2442no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—1224no—
482C22⋊C4—4881no—
483C22⋊C4—4881no—
484C22⋊C4—2442no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—4881no—
490C42—2442no—
491C42—2442no—
492C42—2442no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23C22×D24411yescharacteristic
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—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—2441no—
527C2×C4—2441no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
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#1451981012yes—
548C2×C4#41×D6111yescharacteristic
549C2×C4#41×D6111yescharacteristic
550C2×C4#41×D6111yescharacteristic
551C2×C4C2×#1175822411yescharacteristic
552C2×C4#1451982211yescharacteristic
553C2×C4#1451982411yescharacteristic
554C8—22no—
555C8—41no—
556C8—41no—
557C8—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22#1684027411yescharacteristic
566C22#1684040711yescharacteristic
567C22#1684041611yescharacteristic
568C22C2×#1451981011yescharacteristic
569C22#1684056211yescharacteristic
570C22#1684057411yescharacteristic
571C22#1684057411yescharacteristic
572C4—22no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—4881no—
578C4—4881no—
579C4—4881no—
580C4—4881no—
581C4—4881no—
582C4—2442no—
583C4—2442no—
584C4—2442no—
585C4#1684029111yescharacteristic
586C4#1684035511yescharacteristic
587C4#1684045511yescharacteristic
588C4#1684047911yescharacteristic
589C2—81no—
590C2#2144985311yescharacteristic
591C2#2145598211yescharacteristic
592C2#2145611111yescharacteristic
593C2#2145800711yescharacteristic
594C2#2145813611yescharacteristic
595C2#2145827611yescharacteristic
596C2#2145836611yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1924C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1924D61
hypocenterC61#27060
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1924

Complements

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

NH ≅ G/Ntype
3. #21447229C2semidirect
4. #21447213C2semidirect
7. #21446583C2semidirect
8. #21446562C2semidirect
9. #21446405C2semidirect
10. #21446405C2semidirect
11. #21446090C2semidirect
14. #21445650C2semidirect
357. C61#27060semidirect

Maximal subgroups

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

subgroupindex
#214502522
#214488442
#214472292
#214472132
#214470162
#214465832
#214465622
#214464052
#214460902
#214460202
#214459152
#214456502
C61×#19242
#2706061