Finite Group Databasebeta

← #29738212

Subgroups — #29738212

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#21450840C211yesmaximal, characteristic, complemented
2#21448790C211yesmaximal, characteristic
3#21447206C211yesmaximal, characteristic
4#21447192C211yesmaximal, characteristic
5#21446966C211yesmaximal, characteristic
6#21446956C211yesmaximal, characteristic
7#21446774C211yesmaximal, characteristic, complemented
8#21446766C211yesmaximal, characteristic, complemented
9#21446589C211yesmaximal, characteristic, complemented
10#21446398C211yesmaximal, characteristic, complemented
11#21446391C211yesmaximal, characteristic, complemented
12#21445950C211yesmaximal, characteristic, complemented
13#21445830C211yesmaximal, characteristic
14C4×#14519699C211yesmaximal, characteristic
15C61×#1939C211yesmaximal, characteristic, complemented
16C2×#14519777C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519709C2211yescharacteristic
21C2×#14519699C2211yescharacteristic
22C2×#14519699C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25#16839695—21no—
26#16839682—21no—
27#16839666—21no—
28#16839664C2211yescharacteristic
29#16839664C2211yescharacteristic
30#16839660C2211yescharacteristic
31#16839660C2211yescharacteristic
32#16839659—21no—
33#16839652—21no—
34#16839652—21no—
35#16839628C2211yescharacteristic
36#16839628C2211yescharacteristic
37#16839624—21no—
38C4×C4⋊Dic61C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41#16839609—21no—
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C2×C4⋊C8×C61C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122—21no—
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62#14519777—22no—
63#14519777—41no—
64#14519769—22no—
65#14519769—41no—
66C4×D244—22no—
67C4×D244Q814yescomplemented
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88#14519711—22no—
89#14519709—22no—
90#14519699—22no—
91#14519699—22no—
92#14519699—41no—
93#14519699—41no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×C8×C61C2311yescharacteristic
104C22×C8×C61C2311yescharacteristic
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112C4⋊C8×C61—22no—
113C4⋊C8×C61—41no—
114C4⋊C8×C61—41no—
115C4×C8×C61—22no—
116#95×C61D411yescharacteristic
117#95×C61D411yescharacteristic
118C23×D61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C22×Dic61—21no—
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—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—41no—
159C42×C61—21no—
160C42×C61—21no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61C2×Q812yes—
164C42×C61#4112yes—
165#11758198—24no—
166#11758198—42no—
167#11758198—42no—
168#11758198—81no—
169#11758198—81no—
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—42no—
182C4⋊Dic61C2×Q812yes—
183C4⋊Dic61#4112yes—
184C4⋊Dic61#4112yes—
185#11758196—42no—
186#11758196—42no—
187#11758196—42no—
188#11758196—81no—
189C4×Dic61—81no—
190C4×Dic61—81no—
191C4×Dic61—81no—
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—21no—
201C2×C4×C61—21no—
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×C61C22⋊Q812yes—
216C2×C4×C61C22⋊Q812yes—
217C2×C4×C61#13512yes—
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—22no—
225C2×Dic61—22no—
226C2×Dic61—22no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
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—
245D244—22no—
246D244—22no—
247D244—41no—
248D244—41no—
249C4×D61—84no—
250Dic122—84no—
251C8×C61—41no—
252C8×C61—41no—
253C8×C61—41no—
254C8×C61—41no—
255#26030—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—22no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61#53311yescharacteristic
277C4×C61#54411yescharacteristic
278C4×C61#55511yescharacteristic
279C4×C61#55611yescharacteristic
280Dic61—42no—
281Dic61—42no—
282Dic61—42no—
283Dic61—81no—
284Dic61—81no—
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#1987—611no—
295#1949—611no—
296#1939D6111yescharacteristic, complemented
297#1919—611no—
298#1918—611no—
299#1835—611no—
300#1804—611no—
301C4×#102—611no—
302C2×C61#243911yescharacteristic
303C2×C61#307611yescharacteristic
304C2×C61#308111yescharacteristic
305C2×C61#330111yescharacteristic
306C2×C61#332211yescharacteristic
307C2×C61#336111yescharacteristic
308C2×C61#337111yescharacteristic
309D61—82no—
310C2×C4⋊Q8—611no—
311C2×#125—611no—
312C2×C4×D4—611no—
313C2×#107—611no—
314C2×#106—611no—
315C2×C4⋊C8C2×D6111yescharacteristic
316C2×C4⋊C8C2×D6111yescharacteristic
317C2×C4⋊C8C2×D6111yescharacteristic
318C2×#102—611no—
319C2×#102—611no—
320C2×#102—611no—
321C2×#102—611no—
322C2×C4×C8C2×D6111yescharacteristic
323#398—1221no—
324#395—1221no—
325#391—1221no—
326#390—611no—
327#390—611no—
328#389—611no—
329#389—611no—
330#389—1221no—
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#340—611no—
349#340—611no—
350#340—611no—
351#336C2×D6111yescharacteristic
352#336C2×D6111yescharacteristic
353C61#2603011yescharacteristic, complemented, Sylow
354C22×Q8—1221no—
355C22×D4—611no—
356C23×C4—1221no—
357C22×C8C22×D6111yescharacteristic
358C22×C8C22×D6111yescharacteristic
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—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—611no—
371C2×C4⋊C4—611no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
374C2×C4⋊C4—1221no—
375C2×C4⋊C4—1221no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1221no—
378C2×C4⋊C4—1221no—
379C2×C4⋊C4D24411yescharacteristic
380C2×C4⋊C4D24411yescharacteristic
381C2×C22⋊C4—1221no—
382C2×C22⋊C4—1221no—
383C2×C22⋊C4—1221no—
384C2×C22⋊C4—1221no—
385C2×C22⋊C4—1221no—
386C2×C42—1221no—
387C2×C42—1221no—
388C2×C42—1221no—
389C2×C42C22×D6111yescharacteristic
390C2×C42C22×D6111yescharacteristic
391C2×C42C22×D6111yescharacteristic
392#107—1222no—
393#106—1222no—
394C4⋊C8—22no—
395C4⋊C8—41no—
396C4⋊C8—41no—
397#102—1222no—
398#102—1222no—
399#102—2441no—
400#102—2441no—
401C4×C8—22no—
402#95—1221no—
403#95—1221no—
404#95—1221no—
405#95—1221no—
406#95—1221no—
407#95—1221no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95D24411yescharacteristic
412#95D24411yescharacteristic
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—1221no—
424C22×C4—1221no—
425C22×C4—1221no—
426C22×C4—1224no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1221no—
430C22×C4—1221no—
431C22×C4—4881no—
432C22×C4—4881no—
433C22×C4C2×D24411yescharacteristic
434C22×C4C2×D24411yescharacteristic
435C22×C4C2×D24411yescharacteristic
436C22×C4C2×D24411yescharacteristic
437C22×C4C2×D24411yescharacteristic
438C22×C4C2×D24411yescharacteristic
439C22×C4C23×D6111yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C4⋊C4—22no—
453C4⋊C4—41no—
454C4⋊C4—612no—
455C4⋊C4—612no—
456C4⋊C4—612no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—4881no—
469C4⋊C4—2442no—
470C4⋊C4—2442no—
471C4⋊C4—2442no—
472C4⋊C4—2442no—
473C22⋊C4—1224no—
474C22⋊C4—4881no—
475C22⋊C4—4881no—
476C22⋊C4—2442no—
477C22⋊C4—2442no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—4881no—
483C42—4881no—
484C42—4881no—
485C42#1175822412yes—
486C42Q8×D6112yes—
487C23—1222no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23C22×D24411yescharacteristic
495Q8—4884no—
496D4—1222no—
497D4—1222no—
498D4—2441no—
499D4—2441no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
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—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—4884no—
519C2×C4—2441no—
520C2×C4—2441no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
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#1451975712yes—
538C2×C4#1451975712yes—
539C2×C4#1451979912yes—
540C2×C4C2×Q8×D6111yescharacteristic
541C2×C4#41×D6111yescharacteristic
542C2×C4#41×D6111yescharacteristic
543C2×C4C2×#1175822411yescharacteristic
544C2×C4#1451982211yescharacteristic
545C2×C4#1451982511yescharacteristic
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C22—4881no—
551C22—4881no—
552C22—4881no—
553C22—4881no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22C2×#1451975711yescharacteristic
558C22#1684040711yescharacteristic
559C22#1684042411yescharacteristic
560C22C2×#1451979911yescharacteristic
561C22#1684056211yescharacteristic
562C22#1684056711yescharacteristic
563C22#1684057411yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—4881no—
571C4—4881no—
572C4—4881no—
573C4—4881no—
574C4—2442no—
575C4—2442no—
576C4—2442no—
577C4#1684031811yescharacteristic
578C4#1684038911yescharacteristic
579C4#1684047611yescharacteristic
580C4#1684048311yescharacteristic
581C2—4882no—
582C2#2144950811yescharacteristic
583C2#2145560911yescharacteristic
584C2#2145565111yescharacteristic
585C2#2145772311yescharacteristic
586C2#2145802211yescharacteristic
587C2#2145848111yescharacteristic
588C2#2145858911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1939C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1939D61
hypocenterC61#26030
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
1. #21450840C2semidirect
7. #21446774C2semidirect
8. #21446766C2semidirect
9. #21446589C2semidirect
10. #21446398C2semidirect
11. #21446391C2semidirect
12. #21445950C2semidirect
15. C61×#1939C2semidirect
67. C4×D244Q8semidirect
296. #1939D61semidirect
353. C61#26030semidirect

Maximal subgroups

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

subgroupindex
#214508402
#214487902
#214472062
#214471922
#214469662
#214469562
#214467742
#214467662
#214465892
#214463982
#214463912
#214459502
#214458302
C4×#145196992
C61×#19392
#2603061