Finite Group Databasebeta

← #29753092

Subgroups — #29753092

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#21450242C211yesmaximal, characteristic
2#21448842C211yesmaximal, characteristic
3#21447230C211yesmaximal, characteristic, complemented
4#21447196C211yesmaximal, characteristic, complemented
5#21447021C211yesmaximal, characteristic
6#21447020C211yesmaximal, characteristic
7#21446576C211yesmaximal, characteristic, complemented
8#21446554C211yesmaximal, characteristic, complemented
9#21446430C211yesmaximal, characteristic
10#21446412C211yesmaximal, characteristic, complemented
11#21446392C211yesmaximal, characteristic, complemented
12#21446015C211yesmaximal, characteristic
13#21445913C211yesmaximal, characteristic
14#21445649C211yesmaximal, characteristic, complemented
15C61×#1863C211yesmaximal, characteristic, complemented
16C2×#14519772—21no—
17C2×#14519753C2211yescharacteristic
18C2×#14519735C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519689C2211yescharacteristic
26C2×#14519689C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839704—21no—
29#16839691—21no—
30#16839669C2211yescharacteristic
31#16839661C2211yescharacteristic
32#16839660C2211yescharacteristic
33#16839653—21no—
34#16839652—21no—
35#16839643—21no—
36#16839643—21no—
37#16839631C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×#11758196—21no—
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×C4×D61C2311yescharacteristic
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519772—44no—
63#14519753—22no—
64#14519753—22no—
65#14519735—22no—
66#14519735—41no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
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×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86#14519711—22no—
87#14519710—22no—
88#14519708—41no—
89#14519706—22no—
90#14519701—22no—
91#14519701—22no—
92#14519689—22no—
93#14519689—41no—
94#14519688—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
105C22×Q8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C4×Q8×C61—22no—
109C4×Q8×C61—41no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61C2311yescharacteristic
117#107×C61—22no—
118#103×C61—22no—
119#103×C61—41no—
120#95×C61—21no—
121C23×D61C2×D411yescharacteristic
122C2×Q8×C61—21no—
123C2×Q8×C61—21no—
124C2×Q8×C61—21no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61#4112yes—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—21no—
146C2×C4×D61—21no—
147C2×C4×D61—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—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—42no—
160C4⋊C4×C61—42no—
161C4⋊C4×C61#4112yes—
162C42×C61—21no—
163C42×C61—21no—
164C42×C61—42no—
165#11758198—24no—
166#11758198—22no—
167#11758198—41no—
168#11758198—42no—
169#11758198—42no—
170#11758198—42no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—42no—
179C4⋊Dic61#4112yes—
180#11758196—24no—
181#11758196—42no—
182#11758196—42no—
183#11758196—42no—
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#4112yes—
195C23×C61C22×D411yescharacteristic
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203Q8×C61—22no—
204Q8×C61—22no—
205Q8×C61—41no—
206Q8×C61—41no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
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—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×C61—41no—
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×C4×C612−1+411yescharacteristic
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—22no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
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—
251C2×Dic61AGL(1,ℤ/8)12yes—
252C4×D61—41no—
253C4×D61—41no—
254C4×D61—41no—
255C4×D61—41no—
256C8×C61—41no—
257C8×C61—41no—
258#92418—22no—
259#92418—41no—
260#27058—611nomaximal, Sylow
261C22×C61#52511yescharacteristic
262C22×C61#54611yescharacteristic
263C22×C61#54711yescharacteristic
264C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C61#57511yescharacteristic
266C22×C61#57811yescharacteristic
267C22×C61#57811yescharacteristic
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C2×D61—41no—
274C2×D61—41no—
275C4×C61—41no—
276C4×C61—42no—
277C4×C61—42no—
278C4×C61—41no—
279C4×C61—81no—
280C4×C61—81no—
281C4×C61#52911yescharacteristic
282C4×C61#53911yescharacteristic
283C4×C61#55311yescharacteristic
284C4×C61#55611yescharacteristic
285Dic61—22no—
286Dic61—41no—
287Dic61—42no—
288Dic61—41no—
289Dic61—81no—
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#1863D6111yescharacteristic, complemented
304#1854—611no—
305#1825—611no—
306#1763—611no—
307C2×C61#246611yescharacteristic
308C2×C61#311511yescharacteristic
309C2×C61#312711yescharacteristic
310C2×C61#331311yescharacteristic
311C2×C61#333011yescharacteristic
312C2×C61#334311yescharacteristic
313C2×C61#334911yescharacteristic
314D61—81no—
315C2×#126—1221no—
316C2×C22⋊Q8—611no—
317C2×C4×Q8C2×D6111yescharacteristic
318C2×#117—611no—
319C2×#107—611no—
320C2×#107—611no—
321C2×#107C2×D6111yescharacteristic
322C2×#106—611no—
323C2×C4⋊C8—611no—
324C2×#103—611no—
325C2×#103—611no—
326C2×#103C2×D6111yescharacteristic
327C2×#103C2×D6111yescharacteristic
328C2×C22⋊C8—611no—
329C2×C22⋊C8—611no—
330C2×#97—611no—
331#400—1221no—
332#397—1221no—
333#391—611no—
334#389—611no—
335#389—611no—
336#389C2×D6111yescharacteristic
337#388—1221no—
338#388—1221no—
339#387—1221no—
340#387—1221no—
341#384—611no—
342#380—611no—
343#380—611no—
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#340—611no—
355#340C2×D6111yescharacteristic
356#336—611no—
357#336C2×D6111yescharacteristic
358C61#2705811yescharacteristic, complemented, Sylow
359C22×Q8C22×D6111yescharacteristic
360C23×C4—611no—
361C22×C8—611no—
362C22×C8—611no—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365#126—2444no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C4×Q8—22no—
369C4×Q8—41no—
370#117—1222no—
371#117—2441no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—611no—
375C2×C4⋊C4—611no—
376C2×C4⋊C4—611no—
377C2×C4⋊C4—611no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C22⋊C4—611no—
387C2×C22⋊C4—611no—
388C2×C22⋊C4—1221no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C42—21no—
393C2×C42—611no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42—1221no—
397C2×C42C22×D6111yescharacteristic
398#107—22no—
399#107—1222no—
400#107—1222no—
401#106—2441no—
402C4⋊C8—1222no—
403#103—22no—
404#103—41no—
405#103—1222no—
406#103—1222no—
407C22⋊C8—1222no—
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—
421C24—611no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8#1175822212yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—611no—
433C22×C4—611no—
434C22×C4—611no—
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×C4—1221no—
444C22×C4—1221no—
445C22×C4D4×D6111yescharacteristic
446C22×C4D4×D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—612no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—1221no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—612no—
468C4⋊C4—1224no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—2442no—
477C4⋊C4—2442no—
478C4⋊C4—2442no—
479C4⋊C4—2442no—
480C4⋊C4#1175822412yes—
481C22⋊C4—1222no—
482C22⋊C4—1224no—
483C22⋊C4—2441no—
484C22⋊C4—2442no—
485C22⋊C4—2442no—
486C22⋊C4—2442no—
487C42—21no—
488C42—21no—
489C42—42no—
490C42—1221no—
491C42—1221no—
492C42—2442no—
493C42—2442no—
494C42—2442no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23—1221no—
501C23—1221no—
502C23C2×D4×D6111yescharacteristic
503Q8—22no—
504Q8—22no—
505Q8—41no—
506Q8—41no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
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—612no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1222no—
529C2×C4—1221no—
530C2×C4—1221no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
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—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—41no—
557C8—41no—
558C8—1222no—
559C8—2441no—
560C22—2441no—
561C22—2441no—
562C22—2441no—
563C22—2441no—
564C22—2441no—
565C22—2441no—
566C22—2441no—
567C22#1684027111yescharacteristic
568C22#1684039911yescharacteristic
569C22#1684041411yescharacteristic
570C22AGL(1,ℤ/8)×D6111yescharacteristic
571C22#1684056111yescharacteristic
572C22#1684057511yescharacteristic
573C22#1684057611yescharacteristic
574C4—42no—
575C4—41no—
576C4—42no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—1222no—
581C4—4881no—
582C4—4881no—
583C4—4881no—
584C4—2442no—
585C4—2441no—
586C4—2441no—
587C4#1684029211yescharacteristic
588C4#1684035911yescharacteristic
589C4#1684045711yescharacteristic
590C4#1684048311yescharacteristic
591C2—4881no—
592C2#2144985711yescharacteristic
593C2#2145596811yescharacteristic
594C2#2145611611yescharacteristic
595C2#2145787811yescharacteristic
596C2#2145811711yescharacteristic
597C2#2145825611yescharacteristic
598C2#2145835611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1863C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1863D61
hypocenterC61#27058
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
3. #21447230C2semidirect
4. #21447196C2semidirect
7. #21446576C2semidirect
8. #21446554C2semidirect
10. #21446412C2semidirect
11. #21446392C2semidirect
14. #21445649C2semidirect
15. C61×#1863C2semidirect
303. #1863D61semidirect
358. C61#27058semidirect

Maximal subgroups

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

subgroupindex
#214502422
#214488422
#214472302
#214471962
#214470212
#214470202
#214465762
#214465542
#214464302
#214464122
#214463922
#214460152
#214459132
#214456492
C61×#18632
#2705861