Finite Group Databasebeta

← #29690628

Subgroups — #29690628

Order 15616.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839821C211yesmaximal, characteristic
2#21448440C211yesmaximal, characteristic
3#21448427C211yesmaximal, characteristic
4#21446590C211yesmaximal, characteristic, complemented
5#21446586C211yesmaximal, characteristic, complemented
6#21446581C211yesmaximal, characteristic, complemented
7#21446576C211yesmaximal, characteristic, complemented
8#21446021C211yesmaximal, characteristic
9#21446019C211yesmaximal, characteristic
10#21446013C211yesmaximal, characteristic
11#21446008C211yesmaximal, characteristic
12#21445985C211yesmaximal, characteristic, complemented
13#21445983C211yesmaximal, characteristic, complemented
14#21445954C211yesmaximal, characteristic, complemented
15C61×#1834C211yesmaximal, characteristic, complemented
16C2×#14519735C2211yescharacteristic
17C2×#14519734C2211yescharacteristic
18C2×C4⋊C4×D61C2211yescharacteristic
19#16839821C418yescomplemented
20C2×#14519711C2211yescharacteristic
21C2×#14519711C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33#16839657—21no—
34#16839657—21no—
35#16839652—21no—
36#16839652—21no—
37#16839643—21no—
38#16839643—21no—
39#16839633—21no—
40#16839633—21no—
41#16839631C2211yescharacteristic
42#16839628C2211yescharacteristic
43#16839628C2211yescharacteristic
44#16839627C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×C4×C8×C61C2211yescharacteristic
54C61×#384C2211yescharacteristic
55C61×#384C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519735—41no—
65#14519735—41no—
66#14519734C2×C414yes—
67#14519734Q814yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196D411yescharacteristic
76C2×#11758196D411yescharacteristic
77C4⋊C4×D61—41no—
78C4⋊C4×D61—41no—
79C2×#11758198—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×#11758198D411yescharacteristic
83C2×#11758198D411yescharacteristic
84C2×C4×Dic61—21no—
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61C2311yescharacteristic
88#14519711—41no—
89#14519711—41no—
90#14519710—41no—
91#14519710—41no—
92#14519709—41no—
93#14519709—41no—
94#14519708—41no—
95#14519708—41no—
96#14519706C2×C414yescomplemented
97#14519706C2×C414yescomplemented
98#14519689C2×C414yes—
99#14519689C2×C414yes—
100#14519688C2×C414yescomplemented
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C4×C8×C61C2×C414yescomplemented
123C23×D61C2×D411yescharacteristic
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C22×Dic61C2×D411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—42no—
143C2×C4×D61—81no—
144C2×C4×D61—81no—
145C2×C4×D61C22×C412yes—
146C2×C4×D61C2×Q812yes—
147C2×C8×C61—21no—
148C2×C8×C61—21no—
149C2×C8×C61—21no—
150C2×C8×C61—21no—
151C2×C8×C61C22×C412yes—
152C2×C8×C61C22×C412yes—
153C4⋊C4×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—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—81no—
162C4⋊C4×C61—81no—
163C4⋊C4×C61—81no—
164C4⋊C4×C61—81no—
165C42×C61C22×C412yes—
166C42×C61C2×Q812yes—
167#11758198—42no—
168#11758198—81no—
169#11758198—81no—
170#11758198C4⋊C414yes—
171#11758198C4⋊C414yes—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180#11758196—42no—
181#11758196—81no—
182#11758196—81no—
183#11758196C4⋊C414yes—
184#11758196C4⋊C414yes—
185C4×Dic61—42no—
186C4×Dic61—81no—
187C4×Dic61—81no—
188C4×Dic61C22×C412yes—
189C4×Dic61C2×Q812yes—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418C22×C412yes—
195C2×#92418C22×C412yes—
196C23×C61C22×D411yescharacteristic
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61C2×C4⋊C412yes—
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×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61C2×C4⋊C412yes—
221C2×C4×C61C2×C4⋊C412yes—
222C2×C4×C61C23×C411yescharacteristic
223C2×C4×C61C22×Q811yescharacteristic
224C2×C4×C61H(5,𝔽2)11yescharacteristic
225C2×C4×C61H(5,𝔽2)11yescharacteristic
226C2×C4×C612−1+411yescharacteristic
227C2×C4×C612−1+411yescharacteristic
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
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×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61C2×C4⋊C412yes—
247C2×Dic61C2×C4⋊C412yes—
248C2×Dic61C2×C4⋊C412yes—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—84no—
254C8×C61—22no—
255C8×C61—22no—
256#92418—22no—
257#92418—22no—
258#21992—611nomaximal, Sylow
259C22×C61C22×C4⋊C411yescharacteristic
260C22×C61#53511yescharacteristic
261C22×C61#53611yescharacteristic
262C22×C61#57611yescharacteristic
263C22×C61#57711yescharacteristic
264C22×C61#57711yescharacteristic
265C22×C61#57811yescharacteristic
266C2×D61—22no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61—41no—
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C4×C61—22no—
274C4×C61—22no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61#52011yescharacteristic
280C4×C61#52011yescharacteristic
281C4×C61#54411yescharacteristic
282C4×C61#54411yescharacteristic
283Dic61—22no—
284Dic61—22no—
285Dic61—22no—
286Dic61—81no—
287Dic61—81no—
288Dic61—81no—
289Dic61—81no—
290C2×#433—611no—
291#2317—611no—
292#2314—611no—
293#1949—611no—
294#1949—611no—
295#1948—611no—
296#1947—611no—
297#1854—611no—
298#1854—611no—
299#1853—611no—
300#1852—611no—
301#1847—611no—
302#1846—611no—
303#1836—611no—
304#1834D6111yescharacteristic, complemented
305C2×C61#231711yescharacteristic
306C2×C61#291211yescharacteristic
307C2×C61#291311yescharacteristic
308C2×C61#322411yescharacteristic
309C2×C61#322511yescharacteristic
310C2×C61#324111yescharacteristic
311C2×C61#324211yescharacteristic
312D61—42no—
313C2×#117—611no—
314C2×#117—611no—
315C22×C4⋊C4—611no—
316#433—618no—
317C2×#107—611no—
318C2×#107—611no—
319C2×#107—611no—
320C2×#107—611no—
321C2×#106—611no—
322C2×#106—611no—
323C2×#106—611no—
324C2×#106—611no—
325C2×C4⋊C8—611no—
326C2×C4⋊C8—611no—
327C2×C22⋊C8—611no—
328C2×C22⋊C8—611no—
329C2×#97—611no—
330C2×C4×C8C2×D6111yescharacteristic
331#389—1221no—
332#389—1221no—
333#388—1221no—
334#388—1221no—
335#387—1221no—
336#387—1221no—
337#385—1221no—
338#385—1221no—
339#384—611no—
340#384C2×D6111yescharacteristic
341#384C2×D6111yescharacteristic
342#382—611no—
343#382—611no—
344#382—611no—
345#380—611no—
346#380—611no—
347C4×C4⋊C4—611no—
348C4×C4⋊C4—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340C2×D6111yescharacteristic
354#340C2×D6111yescharacteristic
355#340C2×D6111yescharacteristic
356#340C2×D6111yescharacteristic
357C61#2199211yescharacteristic, complemented, Sylow
358C23×C4—611no—
359C23×C4—1221no—
360C22×C8—611no—
361C22×C8—611no—
362C22×C8C22×D6111yescharacteristic
363C22×C8C22×D6111yescharacteristic
364#117—614no—
365#117—614no—
366#117—2441no—
367#117—2441no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—611no—
373C2×C4⋊C4—611no—
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—2441no—
382C2×C4⋊C4—2441no—
383C2×C4⋊C4C22×D6111yescharacteristic
384C2×C4⋊C4C22×D6111yescharacteristic
385C2×C4⋊C4C22×D6111yescharacteristic
386C2×C4⋊C4C22×D6111yescharacteristic
387C2×C22⋊C4—611no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—1221no—
390C2×C22⋊C4—1221no—
391C2×C22⋊C4—1221no—
392C2×C42—611no—
393C2×C42—1221no—
394C2×C42—1221no—
395C2×C42—1221no—
396C2×C42C22×D6111yescharacteristic
397#107—2441no—
398#107—2441no—
399#107—2441no—
400#107—2441no—
401#106—2441no—
402#106—2441no—
403#106—2441no—
404#106—2441no—
405C4⋊C8—614no—
406C4⋊C8—614no—
407C22⋊C8—614no—
408C22⋊C8—614no—
409#97—614no—
410C4×C8C4×D6114yescomplemented
411#95—1221no—
412#95—1221no—
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—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—611no—
429C22×C4—612no—
430C22×C4—612no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4—1221no—
436C22×C4—1221no—
437C22×C4—1221no—
438C22×C4—1221no—
439C22×C4—4881no—
440C22×C4—4881no—
441C22×C4—2442no—
442C22×C4D4×D6111yescharacteristic
443C22×C4D4×D6111yescharacteristic
444C22×C4C23×D6111yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—612no—
450C2×C8—612no—
451C2×C8—1221no—
452C2×C8—1221no—
453C2×C8—1221no—
454C2×C8—1221no—
455C2×C8C2×C4×D6112yes—
456C2×C8C2×C4×D6112yes—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—614no—
470C4⋊C4—614no—
471C4⋊C4—1221no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—4881no—
480C4⋊C4—4881no—
481C4⋊C4—2442no—
482C22⋊C4—614no—
483C22⋊C4—614no—
484C22⋊C4—4881no—
485C22⋊C4—4881no—
486C22⋊C4—2442no—
487C42—612no—
488C42—612no—
489C42—4881no—
490C42—4881no—
491C42—2442no—
492C42C2×C4×D6112yes—
493C42Q8×D6112yes—
494C23—612no—
495C23—1221no—
496C23—1221no—
497C23—1221no—
498C23—1221no—
499C23—1221no—
500C23—1221no—
501C23C2×D4×D6111yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
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—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
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×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4C4⋊C4×D6112yes—
545C2×C4C4⋊C4×D6112yes—
546C2×C4C22×C4×D6111yescharacteristic
547C2×C4C2×Q8×D6111yescharacteristic
548C2×C4#1451982311yescharacteristic
549C2×C4#1451982311yescharacteristic
550C2×C4#1451982411yescharacteristic
551C2×C4#1451982411yescharacteristic
552C8—22no—
553C8—22no—
554C8—1222no—
555C8—1222no—
556C22—1222no—
557C22—1222no—
558C22—1222no—
559C22—2441no—
560C22—2441no—
561C22—2441no—
562C22—2441no—
563C22C2×C4⋊C4×D6111yescharacteristic
564C22#1684033011yescharacteristic
565C22#1684033211yescharacteristic
566C22#1684056311yescharacteristic
567C22#1684056811yescharacteristic
568C22#1684056811yescharacteristic
569C22#1684057511yescharacteristic
570C4—22no—
571C4—22no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—1222no—
577C4—1222no—
578C4—1222no—
579C4—4881no—
580C4—4881no—
581C4—4881no—
582C4—4881no—
583C4#1684024211yescharacteristic
584C4#1684024211yescharacteristic
585C4#1684039211yescharacteristic
586C4#1684039211yescharacteristic
587C2—2442no—
588C2#2144844411yescharacteristic
589C2#2145441011yescharacteristic
590C2#2145441511yescharacteristic
591C2#2145691611yescharacteristic
592C2#2145693111yescharacteristic
593C2#2145706511yescharacteristic
594C2#2145708011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1834C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1834D61
hypocenterC61#21992
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
4. #21446590C2semidirect
5. #21446586C2semidirect
6. #21446581C2semidirect
7. #21446576C2semidirect
12. #21445985C2semidirect
13. #21445983C2semidirect
14. #21445954C2semidirect
15. C61×#1834C2semidirect
19. #16839821C4semidirect
96. #14519706C2×C4semidirect
97. #14519706C2×C4semidirect
100. #14519688C2×C4semidirect
122. C4×C8×C61C2×C4semidirect
304. #1834D61semidirect
357. C61#21992semidirect
410. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398212
#214484402
#214484272
#214465902
#214465862
#214465812
#214465762
#214460212
#214460192
#214460132
#214460082
#214459852
#214459832
#214459542
C61×#18342
#2199261