Finite Group Databasebeta

← #29648414

Subgroups — #29648414

Order 15616.

972 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic
3#431×D61C211yesmaximal, characteristic
4#21444516C214yesmaximal, complemented
5#21444505C211yesmaximal, characteristic, complemented
6#21444503C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444482C214yesmaximal, complemented
9C61×#1546C211yesmaximal, characteristic, complemented
10#16840317—21no—
11#16840317—21no—
12#16840317—21no—
13#16840317—21no—
14#16840317C412yes—
15#16840317C412yes—
16#16840317C2212yes—
17#16840317C2212yes—
18C2×#14519777C2211yescharacteristic
19C2×#14519768C412yes—
20C2×#14519757—21no—
21C2×#14519753—21no—
22C2×Q8×Dic61—21no—
23C2×#14519742—21no—
24C2×#14519735C412yes—
25C2×C42×D61C2211yescharacteristic
26#16839818—21no—
27#16839818C2212yes—
28#16839815C2211yescharacteristic
29#16839814—21no—
30#16839814C2212yes—
31#16839814C2212yes—
32#16839813—21no—
33#16839812—21no—
34#16839786C2211yescharacteristic
35C4⋊C8×D61C2212yes—
36#16839725—21no—
37C22⋊C8×D61—21no—
38M16×Dic61—21no—
39C4×C8×D61—21no—
40#16839509C2214yescomplemented
41#16839508C2214yescomplemented
42#16839507C2214yescomplemented
43C2×C4⋊Q8×C61C2211yescharacteristic
44C61×#431C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C61×#332C2214yescomplemented
47C22×Dic122—21no—
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61D411yescharacteristic
51C2×#11758194—21no—
52C2×C8×D61—22no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2×C412yes—
56#14519777C2312yes—
57#14519768—22no—
58#14519768—22no—
59#14519768C2×C414yes—
60#14519768C2×C414yes—
61#14519757—22no—
62#14519757—22no—
63#14519757—22no—
64#14519757—22no—
65#14519753—22no—
66#14519753—22no—
67#14519753—22no—
68#14519753—22no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73#14519742—22no—
74#14519742—22no—
75#14519742—22no—
76#14519742—22no—
77#14519735—22no—
78#14519735—22no—
79#14519735C2×C414yes—
80#14519735C2×C414yes—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—22no—
83C2×C4⋊Dic61C2×C412yes—
84C2×#11758196—22no—
85C2×#11758198—22no—
86C2×C4×Dic61—22no—
87C2×C4×Dic61C2×C411yescharacteristic
88C2×C4×Dic61D411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C42×D61C2×C411yescharacteristic
91C42×D61C2×C411yescharacteristic
92C42×D61D411yescharacteristic
93C42×D61D411yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61C2311yescharacteristic
97C42×D61C2311yescharacteristic
98C4⋊#92418—21no—
99C4⋊#92418C2312yes—
100C4⋊#92418C2312yes—
101#14519706—22no—
102C22⋊#92418—21no—
103#14519689—22no—
104#14519689—22no—
105#14519689—22no—
106#14519687—22no—
107C4×#92418—21no—
108C8×Dic61—22no—
109C8×Dic61—22no—
110C22×Q8×C61—21no—
111C2×M16×C61—21no—
112C4⋊Q8×C61—21no—
113C4⋊Q8×C61—21no—
114C4⋊Q8×C61C2×C412yes—
115C4⋊Q8×C61C2312yes—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2×C412yes—
118C2×C42×C61C2311yescharacteristic
119C4⋊C8×C61—21no—
120C4⋊C8×C61C2312yes—
121C4⋊C8×C61C2312yes—
122C22⋊C8×C61—21no—
123C4×C8×C61—21no—
124C23×D61C22⋊C411yescharacteristic
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—42no—
128C2×Q8×C61—42no—
129C22×C4×C61—22no—
130C22×C4×C61C22×C411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×Dic61—22no—
134C22×Dic61C22⋊C411yescharacteristic
135C22×Dic61C22⋊C411yescharacteristic
136C22×Dic61C22⋊C411yescharacteristic
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—24no—
140C2×C4×D61—42no—
141C2×C4×D61C22⋊C411yescharacteristic
142C2×C4×D61C22⋊C411yescharacteristic
143C2×C4×D61C22⋊C411yescharacteristic
144C2×C4×D61C22⋊C411yescharacteristic
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22×C411yescharacteristic
148C2×C4×D61C22×C411yescharacteristic
149C2×C4×D61C2×D411yescharacteristic
150C2×C4×D61C2×D411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—44no—
156M16×C61—42no—
157C2×C8×C61—22no—
158C2×C8×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61C22×C414yes—
163C42×C61C22×C411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2411yescharacteristic
167#11758198—24no—
168#11758198—24no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—24no—
173C4⋊Dic61—24no—
174C4⋊Dic61C22×C414yes—
175#11758196—24no—
176#11758196—24no—
177C4×Dic61—24no—
178C4×Dic61—24no—
179C4×Dic61C22⋊C411yescharacteristic
180C4×Dic61C22⋊C411yescharacteristic
181C4×Dic61C22⋊C411yescharacteristic
182C4×Dic61C22⋊C411yescharacteristic
183C4×Dic61C22⋊C411yescharacteristic
184C4×Dic61C22⋊C411yescharacteristic
185C4×Dic61C22×C411yescharacteristic
186C4×Dic61C22×C411yescharacteristic
187C4×Dic61C2×D411yescharacteristic
188C4×Dic61C2×D411yescharacteristic
189C4×Dic61C2×D411yescharacteristic
190C4×Dic61C2×D411yescharacteristic
191#11758194—42no—
192C2×#92418—22no—
193C2×#92418—22no—
194C8×D61—44no—
195C23×C61C2×C22⋊C411yescharacteristic
196C22×D61—22no—
197C22×D61—22no—
198C22×D61—41no—
199C22×D61C2×C22⋊C411yescharacteristic
200C22×D61C2×C22⋊C411yescharacteristic
201Q8×C61—42no—
202Q8×C61—42no—
203Q8×C61—44no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—24no—
207C2×C4×C61—44no—
208C2×C4×C61C2×C22⋊C411yescharacteristic
209C2×C4×C61C2×C22⋊C411yescharacteristic
210C2×C4×C61C2×C22⋊C411yescharacteristic
211C2×C4×C61#13512yes—
212C2×C4×C61#13512yes—
213C2×C4×C61C23×C411yescharacteristic
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×Dic61—22no—
217C2×Dic61—22no—
218C2×Dic61—24no—
219C2×Dic61—41no—
220C2×Dic61—44no—
221C2×Dic61C2×C22⋊C411yescharacteristic
222C2×Dic61C2×C22⋊C411yescharacteristic
223C2×Dic61C2×C22⋊C411yescharacteristic
224C2×Dic61C2×C22⋊C411yescharacteristic
225C2×Dic61C2×C22⋊C411yescharacteristic
226C2×Dic61C2×C22⋊C411yescharacteristic
227C4×D61—22no—
228C4×D61—21no—
229C4×D61—21no—
230C4×D61—21no—
231C4×D61—21no—
232C4×D61—22no—
233C4×D61—22no—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61#10114yes—
241Dic122—44no—
242Dic122—44no—
243C8×C61—42no—
244C8×C61—42no—
245#92418—42no—
246#92418—42no—
247#17715—611nomaximal, Sylow
248C22×C61—21no—
249C22×C61—21no—
250C22×C61#41612yes—
251C22×C61C22×C22⋊C411yescharacteristic
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—44no—
258C2×D61C2×#10112yes—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—44no—
262C4×C61C2×#10112yes—
263C4×C61#41612yes—
264C4×C61C2×#13511yescharacteristic
265C4×C61C2×#13511yescharacteristic
266C4×C61#57512yes—
267Dic61—21no—
268Dic61—21no—
269Dic61—21no—
270Dic61—21no—
271Dic61—21no—
272Dic61—44no—
273Dic61C2×#10112yes—
274C2×#533—611no—
275#2920—611no—
276C2×#431—611no—
277#1552—614no—
278#1549—611no—
279#1549—611no—
280#1546—611no—
281#1546D6111yescharacteristic, complemented
282#1544—614no—
283C2×C61—22no—
284C2×C61C22×#10111yescharacteristic
285C2×C61C2×#41611yescharacteristic
286C2×C61#289611yescharacteristic
287D61—22no—
288D61—41no—
289#533—612no—
290#533—612no—
291#533—612no—
292#533—612no—
293#533—1221no—
294#533—1221no—
295#533—1221no—
296#533—1221no—
297C2×C4⋊Q8—611no—
298C2×C4⋊Q8C2×D6111yescharacteristic
299C2×#125—612no—
300C2×C22⋊Q8—1221no—
301C2×C22⋊Q8—1221no—
302C2×C4×Q8—1221no—
303C2×C4×Q8—1221no—
304C2×#117—612no—
305C22×C42—611no—
306#432—612no—
307#432—1221no—
308#431—611no—
309#431—612no—
310#431—612no—
311#431—1221no—
312#431—1221no—
313#431—1221no—
314#431C2×D6111yescharacteristic
315#423—611no—
316#423C2×D6111yescharacteristic
317C2×C4⋊C8—612no—
318#407—1221no—
319C2×C22⋊C8—1221no—
320C4×M16—1221no—
321C2×C4×C8—1221no—
322#332—614no—
323#332—614no—
324#332—614no—
325#332C2×D6114yescomplemented
326C61#1771511yescharacteristic, complemented, Sylow
327C22×Q8—21no—
328C22×Q8—1221no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—611no—
332C2×M16—21no—
333C2×M16—1221no—
334C22×C8—1222no—
335C4⋊Q8—21no—
336C4⋊Q8—21no—
337C4⋊Q8—612no—
338C4⋊Q8—612no—
339C4⋊Q8—1221no—
340C4⋊Q8—1221no—
341C4⋊Q8C4×D6112yes—
342C4⋊Q8C22×D6112yes—
343#125—614no—
344#125—614no—
345#125—1222no—
346#125—1222no—
347C22⋊Q8—1222no—
348C22⋊Q8—1222no—
349C22⋊Q8—1222no—
350C22⋊Q8—1222no—
351C22⋊Q8—1222no—
352C22⋊Q8—1222no—
353C22⋊Q8—1222no—
354C22⋊Q8—1222no—
355C4×Q8—1222no—
356C4×Q8—1222no—
357C4×Q8—1222no—
358C4×Q8—1222no—
359C4×Q8—1222no—
360C4×Q8—1222no—
361C4×Q8—1222no—
362C4×Q8—1222no—
363#117—614no—
364#117—614no—
365#117—1222no—
366#117—1222no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C4×D6112yes—
373C2×C22⋊C4—1222no—
374C2×C42—611no—
375C2×C42—611no—
376C2×C42—611no—
377C2×C42—611no—
378C2×C42—611no—
379C2×C42—611no—
380C2×C42—611no—
381C2×C42—611no—
382C2×C42—611no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—1222no—
386C2×C42C22×D6111yescharacteristic
387C4⋊C8—21no—
388C4⋊C8—612no—
389C4⋊C8—612no—
390C4⋊C8—1222no—
391C4⋊C8—1221no—
392C4⋊C8C22×D6112yes—
393C4⋊C8C22×D6112yes—
394C22⋊C8—21no—
395C22⋊C8—1222no—
396C22⋊C8—1222no—
397C22⋊C8—1222no—
398C22⋊C8—1221no—
399#97—1222no—
400C4×C8—21no—
401C4×C8—1222no—
402C4×C8—1222no—
403C4×C8—1221no—
404C24—611no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—1222no—
410C2×Q8—1222no—
411C2×Q8—2444no—
412C22×C4—22no—
413C22×C4—611no—
414C22×C4—611no—
415C22×C4—611no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—611no—
420C22×C4—611no—
421C22×C4—611no—
422C22×C4—611no—
423C22×C4—611no—
424C22×C4—611no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—611no—
428C22×C4—1224no—
429C22×C4—1222no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4—2442no—
433C22×C4C2×C4×D6111yescharacteristic
434C22×C4D4×D6111yescharacteristic
435C22×C4D4×D6111yescharacteristic
436M16—42no—
437M16—2442no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—1222no—
441C2×C8—1222no—
442C2×C8—2444no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—614no—
447C4⋊C4—1224no—
448C4⋊C4—1224no—
449C4⋊C4—1224no—
450C4⋊C4—1224no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4C2×C4×D6114yes—
455C22⋊C4—1224no—
456C22⋊C4—1224no—
457C42—611no—
458C42—611no—
459C42—611no—
460C42—611no—
461C42—611no—
462C42—611no—
463C42—611no—
464C42—611no—
465C42—611no—
466C42—611no—
467C42—611no—
468C42—611no—
469C42—1224no—
470C42—1224no—
471C42C2×C4×D6111yescharacteristic
472C42D4×D6111yescharacteristic
473C42D4×D6111yescharacteristic
474C42C23×D6111yescharacteristic
475C23—611no—
476C23—611no—
477C23—1222no—
478C23—1222no—
479C23—2441no—
480C23C22⋊C4×D6111yescharacteristic
481Q8—44no—
482Q8—42no—
483Q8—42no—
484Q8—2444no—
485Q8—2444no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—44no—
490C2×C4—611no—
491C2×C4—614no—
492C2×C4—611no—
493C2×C4—611no—
494C2×C4—611no—
495C2×C4—611no—
496C2×C4—611no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1221no—
505C2×C4—1221no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—1222no—
509C2×C4—1222no—
510C2×C4—1224no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2444no—
515C2×C4C22⋊C4×D6111yescharacteristic
516C2×C4C22⋊C4×D6111yescharacteristic
517C2×C4C22⋊C4×D6111yescharacteristic
518C2×C4#1451979612yes—
519C2×C4#1451979812yes—
520C2×C4C22×C4×D6111yescharacteristic
521C2×C4C2×D4×D6111yescharacteristic
522C2×C4C2×D4×D6111yescharacteristic
523C8—42no—
524C8—42no—
525C8—2442no—
526C8—2442no—
527C22—21no—
528C22—21no—
529C22—612no—
530C22—1221no—
531C22—1221no—
532C22—1221no—
533C22—1221no—
534C22—1221no—
535C22—2444no—
536C22#1683975512yes—
537C22C2×C22⋊C4×D6111yescharacteristic
538C4—21no—
539C4—21no—
540C4—44no—
541C4—612no—
542C4—1221no—
543C4—1221no—
544C4—1221no—
545C4—1221no—
546C4—1221no—
547C4—2444no—
548C4#101×D6112yes—
549C4#1683975512yes—
550C4C2×#1451979611yescharacteristic
551C4C2×#1451979811yescharacteristic
552C4#1684056112yes—
553C2—22no—
554C2—1222no—
555C2—2441no—
556C2C2×#101×D6111yescharacteristic
557C2C2×#1683975511yescharacteristic
558C2#2145433711yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1546C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1546D61
hypocenterC61#17715
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. #21444516C2semidirect
5. #21444505C2semidirect
6. #21444503C2semidirect
7. #21444491C2semidirect
8. #21444482C2semidirect
9. C61×#1546C2semidirect
40. #16839509C22semidirect
41. #16839508C22semidirect
42. #16839507C22semidirect
46. C61×#332C22semidirect
281. #1546D61semidirect
325. #332C2×D61semidirect
326. C61#17715semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214544512
#431×D612
#214445162
#214445052
#214445032
#214444912
#214444822
C61×#15462
#1771561