Finite Group Databasebeta

← #29738191

Subgroups — #29738191

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21450843C211yesmaximal, characteristic, complemented
2#21448788C211yesmaximal, characteristic
3#21447204C211yesmaximal, characteristic
4#21447188C211yesmaximal, characteristic
5#21446966C211yesmaximal, characteristic
6#21446954C211yesmaximal, characteristic
7#21446776C211yesmaximal, characteristic, complemented
8#21446769C211yesmaximal, characteristic, complemented
9#21446588C211yesmaximal, characteristic, complemented
10#21446536C211yesmaximal, characteristic, complemented
11#21446414C211yesmaximal, characteristic, complemented
12#21445948C211yesmaximal, characteristic, complemented
13#21445828C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#1920C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×#14519739C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519706C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28#16839696—21no—
29#16839682—21no—
30#16839666—21no—
31#16839664C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839658—21no—
35#16839653—21no—
36#16839651—21no—
37#16839627C2211yescharacteristic
38#16839627C2211yescharacteristic
39#16839624—21no—
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196C2211yescharacteristic
43C4×#11758198—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#106×C61C2211yescharacteristic
53C61×#389C2211yescharacteristic
54C4×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122—21no—
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519776—22no—
65#14519776—41no—
66#14519769—22no—
67#14519769—41no—
68#14519739—22no—
69#14519739Q814yescomplemented
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196D411yescharacteristic
78C2×#11758196D411yescharacteristic
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61C2311yescharacteristic
87C2×C4×Dic61C2311yescharacteristic
88#14519710—22no—
89C4⋊#92418—41no—
90#14519706—22no—
91#14519706—41no—
92#14519699—22no—
93#14519699—41no—
94#14519697—22no—
95#14519697—41no—
96C8×Dic61—22no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684D411yescharacteristic
105#14519684D411yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C42×C61—21no—
114C2×C42×C61C2311yescharacteristic
115#106×C61—22no—
116#95×C61—21no—
117#95×C61—21no—
118C23×D61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61C2×D411yescharacteristic
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C2×D244—21no—
134C2×D244—21no—
135C2×D244—21no—
136C2×D244—21no—
137C2×D244—21no—
138C2×D244C2×Q812yes—
139C2×C4×D61—24no—
140C2×C4×D61—81no—
141C2×C4×D61—81no—
142C2×Dic122—42no—
143C2×Dic122—42no—
144C2×Dic122—81no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—21no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—81no—
158C4⋊C4×C61C2×Q812yes—
159C4⋊C4×C61#4112yes—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—81no—
163#11758198—24no—
164#11758198—42no—
165#11758198—42no—
166#11758198—81no—
167#11758198—81no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—42no—
174C4⋊Dic61#4112yes—
175#11758196—22no—
176#11758196—41no—
177#11758196—42no—
178#11758196—42no—
179#11758196—42no—
180C4×Dic61—21no—
181C4×Dic61—21no—
182C4×Dic61—81no—
183C4×Dic61—81no—
184C4×Dic61C2×Q812yes—
185C4×Dic61#4112yes—
186C2×#92418—21no—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—21no—
190C2×#92418—21no—
191C2×#92418—21no—
192C23×C61C22×D411yescharacteristic
193C22×D61—22no—
194C22×D61—41no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
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×C61C22×Q811yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61C2×#4111yescharacteristic
221C2×C4×C61C2×#4111yescharacteristic
222C2×C4×C61H(5,𝔽2)11yescharacteristic
223C2×C4×C612−1+411yescharacteristic
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
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×Dic61C22⋊Q812yes—
243C2×Dic61C22⋊Q812yes—
244C2×Dic61#13512yes—
245D244—22no—
246D244—22no—
247D244—41no—
248D244—41no—
249C4×D61—84no—
250Dic122—84no—
251C8×C61—41no—
252C8×C61—41no—
253#92418—41no—
254#92418—41no—
255#26028—611nomaximal, Sylow
256C22×C61C2×C22⋊Q811yescharacteristic
257C22×C61#54611yescharacteristic
258C22×C61#54811yescharacteristic
259C22×C61C2×#13511yescharacteristic
260C22×C61#57511yescharacteristic
261C22×C61#57611yescharacteristic
262C22×C61#57711yescharacteristic
263C2×D61—42no—
264C2×D61—42no—
265C2×D61—42no—
266C2×D61—81no—
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—42no—
271C4×C61—42no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61#53311yescharacteristic
277C4×C61#54411yescharacteristic
278C4×C61#55511yescharacteristic
279C4×C61#55611yescharacteristic
280Dic61—22no—
281Dic61—22no—
282Dic61—22no—
283Dic61—42no—
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287#2550—611no—
288#2357—611no—
289#2067—611no—
290#2065—611no—
291#2025—611no—
292#2023—611no—
293#1989—611no—
294#1988—611no—
295#1949—611no—
296#1939—611no—
297#1921—611no—
298#1920D6111yescharacteristic, complemented
299#1835—611no—
300#1804—611no—
301C4×#102—611no—
302C2×C61#243911yescharacteristic
303C2×C61#307611yescharacteristic
304C2×C61#308111yescharacteristic
305C2×C61#329711yescharacteristic
306C2×C61#332611yescharacteristic
307C2×C61#336511yescharacteristic
308C2×C61#337011yescharacteristic
309D61—82no—
310C2×C4⋊Q8—611no—
311C2×#125—611no—
312C2×C4×D4—611no—
313C2×#107—611no—
314C2×#106C2×D6111yescharacteristic
315C2×C4⋊C8—611no—
316C2×C4⋊C8—611no—
317C2×C4⋊C8—611no—
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—611no—
321C2×#102—611no—
322C2×C4×C8—611no—
323#398—1221no—
324#395—1221no—
325#391—1221no—
326#390—611no—
327#390—611no—
328#389—611no—
329#389—1221no—
330#389C2×D6111yescharacteristic
331#388—1221no—
332#388—1221no—
333#382—611no—
334#382—611no—
335#381—1221no—
336C4×C4⋊C4—611no—
337C4×C4⋊C4—611no—
338C4×C4⋊C4—611no—
339C4×C4⋊C4C2×D6111yescharacteristic
340C4×C22⋊C4—1221no—
341#340—611no—
342#340—611no—
343#340—611no—
344#340—611no—
345#340—611no—
346#340—611no—
347#340—611no—
348#340C2×D6111yescharacteristic
349#340C2×D6111yescharacteristic
350#340C2×D6111yescharacteristic
351#336—611no—
352#336C2×D6111yescharacteristic
353C61#2602811yescharacteristic, complemented, Sylow
354C22×Q8—1221no—
355C22×D4—611no—
356C23×C4—1221no—
357C22×C8—611no—
358C22×C8—611no—
359C22×C8C22×D6111yescharacteristic
360C22×C8C22×D6111yescharacteristic
361C4⋊Q8—1222no—
362C4⋊Q8—2441no—
363#125—1222no—
364#125—2441no—
365C4×D4—614no—
366C4×D4—1222no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—1221no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4C22×D6111yescharacteristic
378C2×C4⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C42—21no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—1221no—
390C2×C42—1221no—
391C2×C42C22×D6111yescharacteristic
392#107—1222no—
393#106—22no—
394C4⋊C8—1222no—
395C4⋊C8—2441no—
396C4⋊C8—2441no—
397#102—1222no—
398#102—1222no—
399#102—2441no—
400#102—2441no—
401C4×C8—1222no—
402#95—21no—
403#95—21no—
404#95—611no—
405#95—611no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413C24—1221no—
414C2×Q8—4881no—
415C2×Q8—2442no—
416C2×Q8—2442no—
417C2×D4—612no—
418C2×D4—1221no—
419C2×D4—1221no—
420C2×D4—1221no—
421C2×D4—1221no—
422C2×D4—1221no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—1224no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—4881no—
436C22×C4—4881no—
437C22×C4D4×D6111yescharacteristic
438C22×C4D4×D6111yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—1221no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8—1221no—
450C2×C8—1221no—
451C2×C8—1221no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—81no—
459C4⋊C4—612no—
460C4⋊C4—1222no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—2441no—
467C4⋊C4—2442no—
468C4⋊C4—2442no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4#1175822212yes—
472C4⋊C4Q8×D6112yes—
473C22⋊C4—1224no—
474C22⋊C4—4881no—
475C22⋊C4—4881no—
476C22⋊C4—2442no—
477C22⋊C4—2442no—
478C42—21no—
479C42—21no—
480C42—81no—
481C42—612no—
482C42—612no—
483C42—1221no—
484C42—1221no—
485C42—4881no—
486C42—4881no—
487C23—1222no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23C2×D4×D6111yescharacteristic
495Q8—4884no—
496D4—1222no—
497D4—1222no—
498D4—2441no—
499D4—2441no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1222no—
531C2×C4—4884no—
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×C4C2×Q8×D6111yescharacteristic
541C2×C4#41×D6111yescharacteristic
542C2×C4#41×D6111yescharacteristic
543C2×C4C2×#1175822211yescharacteristic
544C2×C4#1451982311yescharacteristic
545C2×C4#1451982411yescharacteristic
546C8—41no—
547C8—41no—
548C8—2441no—
549C8—2441no—
550C22—4881no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22C22⋊Q8×D6111yescharacteristic
558C22#1684040411yescharacteristic
559C22#1684042111yescharacteristic
560C22#135×D6111yescharacteristic
561C22#1684056111yescharacteristic
562C22#1684056611yescharacteristic
563C22#1684056811yescharacteristic
564C4—41no—
565C4—42no—
566C4—42no—
567C4—41no—
568C4—81no—
569C4—81no—
570C4—1222no—
571C4—1222no—
572C4—1222no—
573C4—4881no—
574C4—4881no—
575C4—2442no—
576C4—2441no—
577C4#1684031611yescharacteristic
578C4#1684039211yescharacteristic
579C4#1684047511yescharacteristic
580C4#1684047911yescharacteristic
581C2—4882no—
582C2#2144950911yescharacteristic
583C2#2145560611yescharacteristic
584C2#2145564511yescharacteristic
585C2#2145767211yescharacteristic
586C2#2145807511yescharacteristic
587C2#2145851611yescharacteristic
588C2#2145857311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1920C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1920D61
hypocenterC61#26028
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
1. #21450843C2semidirect
7. #21446776C2semidirect
8. #21446769C2semidirect
9. #21446588C2semidirect
10. #21446536C2semidirect
11. #21446414C2semidirect
12. #21445948C2semidirect
15. C61×#1920C2semidirect
69. #14519739Q8semidirect
298. #1920D61semidirect
353. C61#26028semidirect

Maximal subgroups

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

subgroupindex
#214508432
#214487882
#214472042
#214471882
#214469662
#214469542
#214467762
#214467692
#214465882
#214465362
#214464142
#214459482
#214458282
#214456332
C61×#19202
#2602861