Finite Group Databasebeta

← #29753087

Subgroups — #29753087

Order 15616.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450243C211yesmaximal, characteristic
2#21448846C211yesmaximal, characteristic
3#21447231C211yesmaximal, characteristic, complemented
4#21447195C211yesmaximal, characteristic, complemented
5#21447023C211yesmaximal, characteristic
6#21447022C211yesmaximal, characteristic
7#21446576C211yesmaximal, characteristic, complemented
8#21446555C211yesmaximal, characteristic, complemented
9#21446435C211yesmaximal, characteristic
10#21446409C211yesmaximal, characteristic, complemented
11#21446392C211yesmaximal, characteristic, complemented
12#21446070C211yesmaximal, characteristic, complemented
13#21445914C211yesmaximal, characteristic
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1854C211yesmaximal, characteristic
16C2×#14519770—21no—
17C2×#14519756C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C2×#14519687C2211yescharacteristic
28#16839704—21no—
29#16839690—21no—
30#16839667C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839658C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839656—21no—
35#16839650—21no—
36#16839646—21no—
37#16839644—21no—
38#16839631C2211yescharacteristic
39#16839622C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196—21no—
43C4×C22⋊Dic61C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519770—44no—
63#14519756—22no—
64#14519756—22no—
65#14519742—22no—
66#14519742—41no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61D411yescharacteristic
80C2×C22⋊Dic61D411yescharacteristic
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519710—22no—
88#14519706—22no—
89#14519702—22no—
90#14519702—22no—
91#14519700—22no—
92#14519700—41no—
93C22⋊#92418—41no—
94#14519687—41no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108#117×C61—22no—
109#117×C61—41no—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115C2×C22⋊C4×C61—21no—
116C2×C42×C61—21no—
117#107×C61—22no—
118#106×C61—41no—
119C22⋊C8×C61—22no—
120#95×C61—21no—
121C24×C61C2×D411yescharacteristic
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×C61—21no—
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
141C22×Dic61C2×D411yescharacteristic
142C2×Dic122—21no—
143C2×Dic122—21no—
144C2×Dic122—21no—
145C2×Dic122—21no—
146C2×Dic122—21no—
147C2×Dic122#4112yes—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61#4112yes—
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—21no—
161C4⋊C4×C61#4112yes—
162C22⋊C4×C61—42no—
163C22⋊C4×C61—42no—
164C42×C61—42no—
165C22⋊Dic61—24no—
166C22⋊Dic61—22no—
167C22⋊Dic61—41no—
168C22⋊Dic61—42no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—42no—
175C4⋊Dic61#4112yes—
176#11758196—24no—
177#11758196—42no—
178#11758196—42no—
179#11758196—42no—
180#11758196—42no—
181#11758196—42no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C4×Dic61—21no—
186C4×Dic61—42no—
187C4×Dic61—42no—
188C4×Dic61—42no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61C22×D411yescharacteristic
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
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×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61C2×#4111yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
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—
251C2×Dic61AGL(1,ℤ/8)12yes—
252Dic122—22no—
253Dic122—22no—
254Dic122—41no—
255Dic122—41no—
256C8×C61—22no—
257C8×C61—41no—
258#92418—41no—
259#92418—41no—
260#27058—611nomaximal, Sylow
261C22×C61—41no—
262C22×C61—41no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61#52511yescharacteristic
269C22×C61#54611yescharacteristic
270C22×C61#54711yescharacteristic
271C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
272C22×C61#57511yescharacteristic
273C22×C61#57811yescharacteristic
274C22×C61#57811yescharacteristic
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—41no—
286Dic61—41no—
287Dic61—42no—
288Dic61—42no—
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292#2496—611no—
293#2362—611no—
294#2072—611no—
295#2066—611no—
296#2036—611no—
297#2036—611no—
298#1947—611no—
299#1943—611no—
300#1924—611no—
301#1921—611no—
302#1918—611no—
303#1863—611no—
304#1854D6111yescharacteristic
305#1825—611no—
306#1763—611no—
307C2×C61—81no—
308C2×C61#246611yescharacteristic
309C2×C61#311511yescharacteristic
310C2×C61#312711yescharacteristic
311C2×C61#331311yescharacteristic
312C2×C61#333011yescharacteristic
313C2×C61#334311yescharacteristic
314C2×C61#334911yescharacteristic
315C2×#126—1221no—
316C2×C22⋊Q8—611no—
317C2×C4×Q8—611no—
318C2×#117C2×D6111yescharacteristic
319C2×#107—611no—
320C2×#107—611no—
321C2×#107C2×D6111yescharacteristic
322C2×#106C2×D6111yescharacteristic
323C2×C4⋊C8—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103—611no—
327C2×#103—611no—
328C2×C22⋊C8—611no—
329C2×C22⋊C8C2×D6111yescharacteristic
330C2×#97—611no—
331#400—1221no—
332#397—1221no—
333#391—611no—
334#389—611no—
335#389—611no—
336#389—611no—
337#388—1221no—
338#388—1221no—
339#387—1221no—
340#387—1221no—
341#384—611no—
342#380—611no—
343#380C2×D6111yescharacteristic
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346C4×C4⋊C4—1221no—
347C4×C22⋊C4—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340C2×D6111yescharacteristic
355#340C2×D6111yescharacteristic
356#336—611no—
357#336—611no—
358C61#2705811yescharacteristic, complemented, Sylow
359C22×Q8—611no—
360C23×C4C22×D6111yescharacteristic
361C22×C8—611no—
362C22×C8—611no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365#126—2444no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C4×Q8—1222no—
369C4×Q8—2441no—
370#117—22no—
371#117—41no—
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⋊C4—1221no—
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C42—21no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42—1221no—
398#107—22no—
399#107—1222no—
400#107—1222no—
401#106—41no—
402C4⋊C8—1222no—
403#103—1222no—
404#103—1222no—
405#103—1222no—
406#103—2441no—
407C22⋊C8—22no—
408C22⋊C8—2441no—
409#97—2441no—
410#95—21no—
411#95—611no—
412#95—611no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421C24D4×D6111yescharacteristic
422C2×Q8—612no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—1221no—
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—21no—
437C22×C4—21no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8#1175822212yes—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—612no—
468C4⋊C4—1224no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—2442no—
475C4⋊C4—2442no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4—2442no—
480C4⋊C4#1175822412yes—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—1222no—
484C22⋊C4—1224no—
485C22⋊C4—2441no—
486C22⋊C4—2442no—
487C42—42no—
488C42—1221no—
489C42—1221no—
490C42—1221no—
491C42—1221no—
492C42—2442no—
493C42—2442no—
494C42—2442no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C2×D4×D6111yescharacteristic
503Q8—1222no—
504Q8—1222no—
505Q8—2441no—
506Q8—2441no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—41no—
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—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—612no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
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—2441no—
549C2×C4—2441no—
550C2×C4#41×D6111yescharacteristic
551C2×C4#41×D6111yescharacteristic
552C2×C4C2×#1175822411yescharacteristic
553C2×C4C2×#1175822211yescharacteristic
554C2×C4#1451982211yescharacteristic
555C2×C4#1451982511yescharacteristic
556C8—22no—
557C8—41no—
558C8—2441no—
559C8—2441no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#1684027011yescharacteristic
568C22#1684039911yescharacteristic
569C22#1684041311yescharacteristic
570C22AGL(1,ℤ/8)×D6111yescharacteristic
571C22#1684055911yescharacteristic
572C22#1684057611yescharacteristic
573C22#1684057811yescharacteristic
574C4—41no—
575C4—42no—
576C4—81no—
577C4—81no—
578C4—81no—
579C4—1222no—
580C4—4881no—
581C4—4881no—
582C4—2442no—
583C4—2442no—
584C4—2441no—
585C4—2441no—
586C4—2441no—
587C4#1684029411yescharacteristic
588C4#1684035811yescharacteristic
589C4#1684046111yescharacteristic
590C4#1684048311yescharacteristic
591C2—81no—
592C2#2144985211yescharacteristic
593C2#2145596511yescharacteristic
594C2#2145611511yescharacteristic
595C2#2145788911yescharacteristic
596C2#2145811711yescharacteristic
597C2#2145826011yescharacteristic
598C2#2145834511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1854C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1854D61
hypocenterC61#27058
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. #21447231C2semidirect
4. #21447195C2semidirect
7. #21446576C2semidirect
8. #21446555C2semidirect
10. #21446409C2semidirect
11. #21446392C2semidirect
12. #21446070C2semidirect
14. #21445648C2semidirect
358. C61#27058semidirect

Maximal subgroups

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

subgroupindex
#214502432
#214488462
#214472312
#214471952
#214470232
#214470222
#214465762
#214465552
#214464352
#214464092
#214463922
#214460702
#214459142
#214456482
C61×#18542
#2705861