Finite Group Databasebeta

← #29680587

Subgroups — #29680587

Order 15616.

810 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
1C2×#16839720C211yesmaximal, characteristic
2#21448426C211yesmaximal, characteristic
3#21448426C211yesmaximal, characteristic
4#21445986C211yesmaximal, characteristic, complemented
5#21445984C211yesmaximal, characteristic, complemented
6#21445948C211yesmaximal, characteristic, complemented
7#21445945C211yesmaximal, characteristic, complemented
8#21445852C211yesmaximal, characteristic
9#21445846C211yesmaximal, characteristic
10#21445842C211yesmaximal, characteristic
11#21445715C211yesmaximal, characteristic, complemented
12#21445715C211yesmaximal, characteristic, complemented
13#107×Dic61C211yesmaximal, characteristic, complemented
14#106×Dic61C211yesmaximal, characteristic, complemented
15C61×#1809C211yesmaximal, characteristic
16C23×#92418C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C2×#14519711C2211yescharacteristic
19C2×#14519710C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24#16839720C418yescomplemented
25C2×#14519687C2211yescharacteristic
26C2×#14519687C2211yescharacteristic
27C2×C8×Dic61C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839656—21no—
30#16839656—21no—
31#16839650—21no—
32#16839650—21no—
33#16839647—21no—
34#16839647—21no—
35#16839645—21no—
36#16839645—21no—
37#16839631C2211yescharacteristic
38#16839631C2211yescharacteristic
39#16839627C2211yescharacteristic
40#16839627C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839622C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4⋊C4×Dic61C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C22×M16×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C61×#380C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×#92418—22no—
61C22×#92418—22no—
62C22×#92418C2×C414yes—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519736C2×C414yes—
66#14519736Q814yes—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
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×Dic61—21no—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519711—22no—
88#14519710—22no—
89#14519710—22no—
90#14519709—22no—
91#14519708—22no—
92#14519708—22no—
93#14519687C2×C414yes—
94#14519687C2×C414yes—
95C8×Dic61C2×C414yes—
96C8×Dic61C2×C414yes—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107C23×C4×C61C2311yescharacteristic
108C2×M16×C61—22no—
109C2×M16×C61—22no—
110C2×M16×C61C2×C414yes—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119#107×C61—22no—
120#106×C61—22no—
121#95×C61—21no—
122#95×C61—21no—
123C24×C61C2×D411yescharacteristic
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×C61C22×C412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×Q812yes—
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61C2×D411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144M16×C61—28no—
145C2×C8×C61—21no—
146C2×C8×C61—21no—
147C2×C8×C61C22×C412yes—
148C2×C8×C61C22×C412yes—
149C2×C8×C61#4112yes—
150C2×C8×C61#4112yes—
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—21no—
158C4⋊C4×C61—21no—
159C22⋊C4×C61—81no—
160C22⋊C4×C61—81no—
161C22⋊Dic61—81no—
162C22⋊Dic61—81no—
163C22⋊Dic61C4⋊C414yes—
164C22⋊Dic61C4⋊C414yes—
165C4⋊Dic61—21no—
166C4⋊Dic61—21no—
167C4⋊Dic61—21no—
168C4⋊Dic61—21no—
169C4⋊Dic61—21no—
170C4⋊Dic61—21no—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61C4⋊C414yes—
174C4⋊Dic61C4⋊C414yes—
175#11758196—81no—
176#11758196—81no—
177#11758196—81no—
178#11758196—81no—
179C4×Dic61—81no—
180C4×Dic61—81no—
181C4×Dic61—81no—
182C4×Dic61—81no—
183C4×Dic61C22×C412yes—
184C4×Dic61C22×C412yes—
185C4×Dic61C2×Q812yes—
186C4×Dic61C2×Q812yes—
187C2×#92418—21no—
188C2×#92418—21no—
189C2×#92418—28no—
190C2×#92418C22×C412yes—
191C2×#92418C22×C412yes—
192C2×#92418#4112yes—
193C2×#92418#4112yes—
194C23×C61—21no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61C2×C4⋊C412yes—
201C23×C61C22×D411yescharacteristic
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
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×C61C23×C411yescharacteristic
222C2×C4×C61C22×Q811yescharacteristic
223C2×C4×C61C2×#4111yescharacteristic
224C2×C4×C61C2×#4111yescharacteristic
225C2×C4×C61C2×#4111yescharacteristic
226C2×C4×C61C2×#4111yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
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×Dic61—41no—
247C2×Dic61C2×C4⋊C412yes—
248C2×Dic61C2×C4⋊C412yes—
249C2×Dic61C2×C4⋊C412yes—
250C2×Dic61C2×C4⋊C412yes—
251C8×C61—22no—
252C8×C61#11714yes—
253#92418—22no—
254#92418#11714yes—
255#20822—611nomaximal, Sylow
256C22×C61—22no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—41no—
260C22×C61—41no—
261C22×C61—41no—
262C22×C61—41no—
263C22×C61C22×C4⋊C411yescharacteristic
264C22×C61#53211yescharacteristic
265C22×C61#53211yescharacteristic
266C22×C61#57611yescharacteristic
267C22×C61#57711yescharacteristic
268C22×C61#57711yescharacteristic
269C22×C61#57811yescharacteristic
270C4×C61—22no—
271C4×C61—81no—
272C4×C61—81no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61C2×#11711yescharacteristic
276C4×C61C2×#11711yescharacteristic
277C4×C61#53311yescharacteristic
278C4×C61#53311yescharacteristic
279Dic61—22no—
280Dic61—22no—
281Dic61—22no—
282Dic61—22no—
283Dic61—81no—
284Dic61—81no—
285Dic61—81no—
286Dic61—81no—
287C2×#405—611no—
288#2314—611no—
289#2314—611no—
290#1847—611no—
291#1847—611no—
292#1835—611no—
293#1834—611no—
294#1809—611no—
295#1809D6111yescharacteristic
296#1808—611no—
297#1807—611no—
298#1776—611no—
299#1776—611no—
300C4×#107—611no—
301C4×#106—611no—
302C2×C61—42no—
303C2×C61#228511yescharacteristic
304C2×C61#291211yescharacteristic
305C2×C61#291311yescharacteristic
306C2×C61#317811yescharacteristic
307C2×C61#317811yescharacteristic
308C2×C61#317911yescharacteristic
309C2×C61#317911yescharacteristic
310C22×M16C2×D6111yescharacteristic
311C23×C8—611no—
312C2×#117—611no—
313C2×#107—611no—
314C2×#107—611no—
315C2×#107—611no—
316C2×#107C2×D6111yescharacteristic
317C2×#106—611no—
318C2×#106—611no—
319C2×#106—611no—
320C2×#106C2×D6111yescharacteristic
321#405—618no—
322C2×#97—611no—
323C2×#97—611no—
324C2×C4×C8—611no—
325C2×C4×C8—611no—
326#388—1221no—
327#388—1221no—
328#388—1221no—
329#388—1221no—
330#387—1221no—
331#387—1221no—
332#387—1221no—
333#387—1221no—
334#384—611no—
335#384—611no—
336#382—611no—
337#382—611no—
338#380—611no—
339#380—611no—
340#380C2×D6111yescharacteristic
341#380C2×D6111yescharacteristic
342C4×C4⋊C4—611no—
343C4×C4⋊C4—611no—
344C4×C4⋊C4—611no—
345C4×C4⋊C4—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340C2×D6111yescharacteristic
353#340C2×D6111yescharacteristic
354C61#2082211yescharacteristic, complemented, Sylow
355C23×C4C22×D6111yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C4×D6114yes—
359C22×C8—611no—
360C22×C8—614no—
361C22×C8—611no—
362C22×C8—1222no—
363C22×C8—1222no—
364C22×C8C22×D6111yescharacteristic
365C22×C8C22×D6111yescharacteristic
366#117—614no—
367#117—614no—
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⋊C4C22×D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C4⋊C4C22×D6111yescharacteristic
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—1221no—
391C2×C42—1221no—
392C2×C42—1221no—
393C2×C42—1221no—
394#107—22no—
395#107—1222no—
396#107—1222no—
397#107—1222no—
398#106—22no—
399#106—1222no—
400#106—1222no—
401#106—1222no—
402#97—614no—
403#97—614no—
404C4×C8—614no—
405C4×C8—614no—
406#95—21no—
407#95—21no—
408#95—1221no—
409#95—1221no—
410#95—1221no—
411#95—1221no—
412#95—1221no—
413#95—1221no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418C24D4×D6111yescharacteristic
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
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—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1221no—
435C22×C4C2×C4×D6112yes—
436C22×C4D4×D6111yescharacteristic
437C22×C4Q8×D6112yes—
438C22×C4C23×D6111yescharacteristic
439M16—28no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—612no—
443C2×C8—612no—
444C2×C8—612no—
445C2×C8—612no—
446C2×C8—1228no—
447C2×C8—1221no—
448C2×C8—1221no—
449C2×C8C2×C4×D6112yes—
450C2×C8C2×C4×D6112yes—
451C2×C8#1175822412yes—
452C2×C8#1175822412yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—614no—
462C4⋊C4—614no—
463C4⋊C4—1221no—
464C4⋊C4—1221no—
465C4⋊C4—1221no—
466C4⋊C4—1221no—
467C4⋊C4—1221no—
468C4⋊C4—1221no—
469C4⋊C4—1221no—
470C4⋊C4—1221no—
471C4⋊C4—4881no—
472C4⋊C4—4881no—
473C4⋊C4—4881no—
474C4⋊C4—4881no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—614no—
478C22⋊C4—614no—
479C22⋊C4—4881no—
480C22⋊C4—4881no—
481C42—612no—
482C42—612no—
483C42—612no—
484C42—612no—
485C42—4881no—
486C42—4881no—
487C42—4881no—
488C42—4881no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C4⋊C4×D6112yes—
496C23C2×D4×D6111yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
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—612no—
516C2×C4—612no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1221no—
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—2441no—
538C2×C4—2441no—
539C2×C4C4⋊C4×D6112yes—
540C2×C4C22×C4×D6111yescharacteristic
541C2×C4C2×Q8×D6111yescharacteristic
542C2×C4#41×D6111yescharacteristic
543C2×C4#41×D6111yescharacteristic
544C2×C4C2×#1175822411yescharacteristic
545C2×C4C2×#1175822411yescharacteristic
546C8—22no—
547C8—614no—
548C8—1222no—
549C8#1451973314yes—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×C4⋊C4×D6111yescharacteristic
558C22#1684030711yescharacteristic
559C22#1684030711yescharacteristic
560C22#1684056611yescharacteristic
561C22#1684057111yescharacteristic
562C22#1684057211yescharacteristic
563C22#1684057811yescharacteristic
564C4—22no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—1222no—
570C4—1222no—
571C4—1222no—
572C4—1222no—
573C4—4881no—
574C4—4881no—
575C4—4881no—
576C4—4881no—
577C4#117×D6111yescharacteristic
578C4C2×#1451973311yescharacteristic
579C4#1684031411yescharacteristic
580C4#1684031411yescharacteristic
581C2—42no—
582C2#2144822111yescharacteristic
583C2#2145441311yescharacteristic
584C2#2145441811yescharacteristic
585C2#2145648011yescharacteristic
586C2#2145648811yescharacteristic
587C2#2145649511yescharacteristic
588C2#2145650311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1809C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1809D61
hypocenterC61#20822
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
4. #21445986C2semidirect
5. #21445984C2semidirect
6. #21445948C2semidirect
7. #21445945C2semidirect
11. #21445715C2semidirect
12. #21445715C2semidirect
13. #107×Dic61C2semidirect
14. #106×Dic61C2semidirect
24. #16839720C4semidirect
354. C61#20822semidirect

Maximal subgroups

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

subgroupindex
C2×#168397202
#214484262
#214459862
#214459842
#214459482
#214459452
#214458522
#214458462
#214458422
#214457152
#107×Dic612
#106×Dic612
C61×#18092
#2082261