Finite Group Databasebeta

← #30615513

Subgroups — #30615513

Order 15616.

1446 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×#6923C211yesmaximal, characteristic
2C2×#16840773C211yesmaximal, characteristic
3C2×#16840755C211yesmaximal, characteristic
4#21462279C211yesmaximal, characteristic
5#21461785C214yesmaximal, complemented
6#21461783C214yesmaximal, complemented
7#21461508C211yesmaximal, characteristic
8#21461481C211yesmaximal, characteristic
9#419×D61C211yesmaximal, characteristic
10C2×D61⋊M16C2211yescharacteristic
11C2×Q8×#6923—24no—
12C2×Q8×#6923—22no—
13C2×Q8×#6923—24no—
14C2×Q8×#6923C2214yes—
15C2×#14519856C2211yescharacteristic
16C22×C4⋊#6923—21no—
17C22×C4⋊#6923C2211yescharacteristic
18C22×C4×#6923—21no—
19C22×C4×#6923C2211yescharacteristic
20#16840773—24no—
21C2×#14519846C2212yes—
22C2×#14519845C2212yes—
23#16840755—24no—
24C2×#14519840C2212yes—
25C2×#14519840C2212yes—
26#16840713C2212yes—
27#16840713C2212yes—
28#16840709—21no—
29#16840709—21no—
30C4×C4⋊#6923—22no—
31#16840635C2214yes—
32(C2×#6923)⋊C8C2214yes—
33C22×Q8×D61C2211yescharacteristic
34C2×M16×D61C2211yescharacteristic
35#117×D61C2211yescharacteristic
36#16839772C411yescharacteristic, complemented
37#16839771C411yescharacteristic, complemented
38#16839770C411yescharacteristic, complemented
39#16839769C414yescomplemented
40#103×D61C2214yes—
41C61×#419C411yescharacteristic, complemented
42C23×#6923—21no—
43C23×#6923D411yescharacteristic
44C23×#6923D411yescharacteristic
45C2×#11758213—21no—
46D61⋊M16—22no—
47D61⋊M16—22no—
48C2×#11758217—21no—
49C2×D61⋊C8C2312yes—
50Q8×#6923—216no—
51Q8×#6923—28no—
52Q8×#6923—28no—
53#14519856—22no—
54#14519856—22no—
55C2×C4⋊#6923—28no—
56C2×C4⋊#6923—24no—
57C2×C4⋊#6923—21no—
58C2×C4⋊#6923—22no—
59C2×C4⋊#6923—22no—
60C2×C4⋊#6923—22no—
61C2×C4⋊#6923D414yes—
62C2×C4⋊#6923C2311yescharacteristic
63C2×C4⋊#6923C2311yescharacteristic
64C2×C4⋊#6923C2312yes—
65C2×C22⋊#6923—21no—
66C2×C4×#6923—28no—
67C2×C4×#6923—24no—
68C2×C4×#6923—22no—
69C2×C4×#6923—22no—
70C2×C4×#6923—21no—
71C2×C4×#6923D412yes—
72C2×C4×#6923C2312yes—
73#14519846—24no—
74#14519845—24no—
75#14519840—24no—
76#14519840—24no—
77(C2×#6923)⋊C4—22no—
78(C2×#6923)⋊C4—22no—
79C22×Dic122—21no—
80C22×Dic122C2×C411yescharacteristic
81C2×Q8×D61—21no—
82C2×Q8×D61—22no—
83C2×Q8×D61—22no—
84C2×Q8×D61—21no—
85C2×Q8×D61C2×C412yes—
86C2×Q8×D61D414yes—
87C2×Q8×D61Q814yes—
88C2×Q8×D61C2312yes—
89C22×C4×D61—21no—
90C22×C4×D61C2311yescharacteristic
91C2×#11758194C2×C411yescharacteristic
92C2×#11758189C2×C412yes—
93M16×D61—22no—
94M16×D61—22no—
95C2×C8×D61C2312yes—
96#14519736C2×C411yescharacteristic
97#14519735C2×C412yes—
98#14519734—21no—
99#14519733—21no—
100C4⋊C4×D61C2312yes—
101C22⋊C4×D61—21no—
102C42×D61—21no—
103#14519702C2×C414yescomplemented
104#14519701C2×C414yescomplemented
105#14519700C2×C414yescomplemented
106C22×Q8×C61C2×C411yescharacteristic
107C2×M16×C61C2×C411yescharacteristic
108#117×C61C2×C411yescharacteristic
109#103×C61C2×C414yescomplemented
110C23×D61C2×D411yescharacteristic
111C22×#6923—22no—
112C22×#6923—24no—
113C22×#6923—22no—
114C22×#6923—24no—
115C22×#6923—22no—
116C22×#6923—21no—
117C22×#6923—21no—
118C22×#6923—42no—
119C22×#6923—44no—
120C22×#6923C2×D412yes—
121C22×#6923C2×D412yes—
122C2×Q8×C61—21no—
123C2×Q8×C61—22no—
124C2×Q8×C61C4⋊C414yescomplemented
125C2×Q8×C61C22×C412yes—
126C22×C4×C61—21no—
127C22×C4×C61C22×C411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61#4111yescharacteristic
130Q8×D61—24no—
131Q8×D61—24no—
132Q8×D61—24no—
133Q8×D61—24no—
134Q8×D61—24no—
135Q8×D61—22no—
136Q8×D61—22no—
137Q8×D61C4⋊C418yescomplemented
138Q8×D61C2×D414yes—
139Q8×D61C2×Q814yes—
140C2×C4×D61—22no—
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61C22×C411yescharacteristic
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D411yescharacteristic
149C2×C4×D61C2×Q812yes—
150C2×C4×D61#4112yes—
151C2×C4×D61#4111yescharacteristic
152C2×C4×D61C2411yescharacteristic
153C2×Dic122—22no—
154C2×Dic122—21no—
155C2×Dic122—22no—
156C2×Dic122—24no—
157C2×Dic122—24no—
158C2×Dic122—22no—
159C2×Dic122C4⋊C414yescomplemented
160C2×Dic122C22×C412yes—
161C22⋊#6923—42no—
162#11758217—42no—
163C2×#92420—22no—
164C4⋊#6923—216no—
165C4⋊#6923—24no—
166C4⋊#6923—22no—
167C4⋊#6923—21no—
168C4⋊#6923—21no—
169C4⋊#6923—21no—
170C4⋊#6923—21no—
171C4⋊#6923—44no—
172C4⋊#6923—44no—
173C4⋊#6923C2×D414yes—
174C4×#6923—216no—
175C4×#6923—24no—
176C4×#6923—22no—
177C4×#6923—22no—
178C4×#6923—42no—
179#11758213—42no—
180D61⋊C8—22no—
181D61⋊C8—22no—
182M16×C61—22no—
183C2×C8×C61C22×C412yes—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—21no—
186C42×C61—21no—
187C22⋊Dic61—21no—
188#11758198—22no—
189C4⋊Dic61C22×C412yes—
190#11758196—22no—
191C4×Dic61—21no—
192C4×Dic61—22no—
193#11758194—22no—
194C2×#92418C22×C412yes—
195#11758189—24no—
196C8×D61—22no—
197C8×D61—22no—
198C23×C61C4×D411yescharacteristic
199C22×D61—22no—
200C22×D61—22no—
201C22×D61C4×D411yescharacteristic
202C22×D61#12012yes—
203C22×D61C22⋊Q812yes—
204C22×D61C22×D411yescharacteristic
205C2×#6923—28no—
206C2×#6923—22no—
207C2×#6923—22no—
208C2×#6923—22no—
209C2×#6923—24no—
210C2×#6923—22no—
211C2×#6923—22no—
212C2×#6923—44no—
213C2×#6923—48no—
214C2×#6923—44no—
215C2×#6923—41no—
216C2×#6923—41no—
217C2×#6923—42no—
218C2×#6923—42no—
219C2×#6923#13714yes—
220Q8×C61—22no—
221Q8×C61—24no—
222Q8×C61C2×C4⋊C414yes—
223C2×C4×C61—22no—
224C2×C4×C61—24no—
225C2×C4×C61—22no—
226C2×C4×C61C2×C4⋊C412yes—
227C2×C4×C61C4×D412yes—
228C2×C4×C61C4×D411yescharacteristic
229C2×C4×C61C23×C411yescharacteristic
230C2×Dic61—22no—
231C2×Dic61—24no—
232C2×Dic61—22no—
233C2×Dic61C4×D411yescharacteristic
234C2×Dic61C22⋊Q812yes—
235C2×Dic61#12312yes—
236C2×Dic61C2×#4111yescharacteristic
237C4×D61—24no—
238C4×D61—24no—
239C4×D61—28no—
240C4×D61—42no—
241C4×D61—42no—
242C4×D61C2×C4⋊C412yes—
243C4×D61C4×D412yes—
244C4×D61#12012yes—
245C4×D61C22⋊Q812yes—
246C4×D61C22⋊Q812yes—
247C4×D61#12312yes—
248C4×D61C22×D411yescharacteristic
249C4×D61C22×D411yescharacteristic
250C4×D61C22×Q811yescharacteristic
251C4×D61C2×#4111yescharacteristic
252Dic122—22no—
253Dic122—24no—
254Dic122—24no—
255Dic122—24no—
256Dic122—28no—
257Dic122C2×C4⋊C414yes—
258#92420—42no—
259C8×C61—22no—
260#92418—22no—
261#20874—611nomaximal, Sylow
262C22×C61—22no—
263C22×C61#38512yes—
264C22×C61C2×C4×D411yescharacteristic
265C2×D61—22no—
266C2×D61—24no—
267C2×D61—22no—
268C2×D61—22no—
269C2×D61#38512yes—
270C2×D61C2×#12011yescharacteristic
271C2×D61C2×C22⋊Q811yescharacteristic
272C2×D61C2×#13712yes—
273C2×D61#57512yes—
274#6923—24no—
275#6923—42no—
276#6923—44no—
277#6923—81no—
278#6923—81no—
279C4×C61—24no—
280C4×C61—42no—
281C4×C61#38512yes—
282C4×C61C22×C4⋊C411yescharacteristic
283C4×C61C2×C4×D411yescharacteristic
284Dic61—24no—
285Dic61—42no—
286Dic61#38512yes—
287Dic61C2×C22⋊Q811yescharacteristic
288Dic61C2×#12311yescharacteristic
289C22×C4×Q8—611no—
290C2×#428—611no—
291C2×#419—611no—
292C2×#419—611no—
293#2296—611no—
294#1865—614no—
295#1863—614no—
296#1736—611no—
297#1728—611no—
298C2×C61—22no—
299C2×C61C2×#38511yescharacteristic
300C2×C61C4×#13712yes—
301D61—22no—
302D61#300411yescharacteristic
303D61#300511yescharacteristic
304D61#307812yes—
305C23×Q8—611no—
306C22×M16—611no—
307C22×M16—611no—
308C2×C4×Q8—614no—
309C2×C4×Q8—1222no—
310C2×C4×Q8—1224no—
311C2×C4×Q8—1224no—
312C2×#117—611no—
313C2×#117—611no—
314C22×C4⋊C4—611no—
315C22×C4⋊C4—1221no—
316C22×C42—611no—
317C22×C42—1221no—
318#428—1224no—
319C2×#107—612no—
320C2×#106—612no—
321#419—611no—
322#419—611no—
323#419—611no—
324#419—614no—
325#419—1224no—
326#419#692311yescharacteristic, complemented
327C2×#103—614no—
328C2×#103—612no—
329C2×#103—612no—
330#389—612no—
331#389—612no—
332#385—1221no—
333#385—1221no—
334C4×C4⋊C4—1222no—
335#340—614no—
336#336—614no—
337C61#2087411yescharacteristic, complemented, Sylow
338C22×Q8—614no—
339C22×Q8—614no—
340C22×Q8—612no—
341C22×Q8—612no—
342C22×Q8—611no—
343C22×Q8—1221no—
344C22×Q8—1221no—
345C22×Q8—1222no—
346C22×Q8—1221no—
347C22×Q8—1222no—
348C22×Q8C2×#692311yescharacteristic
349C23×C4—611no—
350C23×C4—611no—
351C23×C4—611no—
352C23×C4—1221no—
353C23×C4—1221no—
354C2×M16—611no—
355C2×M16—612no—
356C2×M16—1222no—
357C2×M16—1222no—
358C2×M16—1221no—
359C2×M16—1222no—
360C2×M16—1222no—
361C2×M16—1221no—
362C2×M16C2×#692311yescharacteristic
363C22×C8—612no—
364C22×C8—612no—
365C4×Q8—1228no—
366C4×Q8—1228no—
367C4×Q8—12216no—
368#117—611no—
369#117—612no—
370#117—1221no—
371#117—1221no—
372#117—1222no—
373#117—1222no—
374#117C2×#692311yescharacteristic
375C2×C4⋊C4—612no—
376C2×C4⋊C4—614no—
377C2×C4⋊C4—612no—
378C2×C4⋊C4—611no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—1222no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1224no—
383C2×C4⋊C4—1228no—
384C2×C4⋊C4—1222no—
385C2×C4⋊C4—1222no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4—1221no—
388C2×C42—612no—
389C2×C42—612no—
390C2×C42—1221no—
391C2×C42—1224no—
392C2×C42—1222no—
393C2×C42—1222no—
394C2×C42—1228no—
395C2×C42—1221no—
396#107—1224no—
397#106—1224no—
398#103—614no—
399#103—614no—
400#103—614no—
401#103—1224no—
402#103—1224no—
403#103C2×#692314yescomplemented
404#95—1222no—
405#95—1222no—
406C24—611no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—614no—
410C2×Q8—618no—
411C2×Q8—614no—
412C2×Q8—614no—
413C2×Q8—612no—
414C2×Q8—1224no—
415C2×Q8—1224no—
416C2×Q8—1222no—
417C2×Q8—1222no—
418C2×Q8—1224no—
419C2×Q8—1222no—
420C2×Q8—1224no—
421C2×Q8—1224no—
422C2×Q8—1224no—
423C2×Q8—1224no—
424C2×Q8—1222no—
425C2×Q8—1222no—
426C2×Q8—1221no—
427C2×Q8C4⋊#692314yescomplemented
428C2×Q8C22×#692312yes—
429C22×C4—21no—
430C22×C4—611no—
431C22×C4—612no—
432C22×C4—612no—
433C22×C4—611no—
434C22×C4—612no—
435C22×C4—611no—
436C22×C4—612no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—1224no—
442C22×C4—1222no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1224no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4—1222no—
450C22×C4—1224no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—2442no—
455C22×C4—2444no—
456C22×C4C22×#692311yescharacteristic
457M16—22no—
458M16—1224no—
459M16—1222no—
460M16—2442no—
461M16—2442no—
462C2×C8—612no—
463C2×C8—1222no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8—1222no—
468C2×C8C22×#692312yes—
469C4⋊C4—614no—
470C4⋊C4—612no—
471C4⋊C4—1222no—
472C4⋊C4—1224no—
473C4⋊C4—1222no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—12216no—
479C4⋊C4—2444no—
480C4⋊C4—2444no—
481C4⋊C4C22×#692312yes—
482C22⋊C4—21no—
483C22⋊C4—1222no—
484C22⋊C4—1221no—
485C22⋊C4—2442no—
486C42—21no—
487C42—1222no—
488C42—1224no—
489C42—1222no—
490C42—1222no—
491C42—12216no—
492C42—1221no—
493C42—2442no—
494C23—611no—
495C23—612no—
496C23—612no—
497C23—611no—
498C23—1222no—
499C23—1222no—
500C23D4×#692311yescharacteristic
501Q8—24no—
502Q8—22no—
503Q8—614no—
504Q8—1224no—
505Q8—1224no—
506Q8—1228no—
507Q8—1224no—
508Q8—1222no—
509Q8C2×C4⋊#692314yes—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—612no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—612no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—614no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1228no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1228no—
534C2×C4—1224no—
535C2×C4—1222no—
536C2×C4—1224no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2444no—
548C2×C4—2444no—
549C2×C4—2448no—
550C2×C4C2×C4⋊#692312yes—
551C2×C4D4×#692311yescharacteristic
552C2×C4D4×#692312yes—
553C2×C4C23×#692311yescharacteristic
554C8—22no—
555C8—1222no—
556C8—2442no—
557C22—22no—
558C22—612no—
559C22—612no—
560C22—611no—
561C22—611no—
562C22—612no—
563C22—1222no—
564C22—1222no—
565C22—1224no—
566C22—1222no—
567C22#1684070912yes—
568C22C2×D4×#692311yescharacteristic
569C4—24no—
570C4—42no—
571C4—611no—
572C4—611no—
573C4—612no—
574C4—1224no—
575C4—1224no—
576C4—4881no—
577C4—4881no—
578C4—2442no—
579C4—2444no—
580C4—2442no—
581C4#1684070912yes—
582C4C22×C4⋊#692311yescharacteristic
583C4C2×D4×#692311yescharacteristic
584C2—22no—
585C2—611no—
586C2—611no—
587C2—612no—
588C2—1222no—
589C2C2×#1684070911yescharacteristic
590C2#137×#692312yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#419C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#419#6923
hypocenterC61#20874
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
5. #21461785C2semidirect
6. #21461783C2semidirect
36. #16839772C4semidirect
37. #16839771C4semidirect
38. #16839770C4semidirect
39. #16839769C4semidirect
41. C61×#419C4semidirect
103. #14519702C2×C4semidirect
104. #14519701C2×C4semidirect
105. #14519700C2×C4semidirect
109. #103×C61C2×C4semidirect
124. C2×Q8×C61C4⋊C4semidirect
137. Q8×D61C4⋊C4semidirect
159. C2×Dic122C4⋊C4semidirect
326. #419#6923semidirect
337. C61#20874semidirect
403. #103C2×#6923semidirect
427. C2×Q8C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×#69232
C2×#168407732
C2×#168407552
#214622792
#214617852
#214617832
#214615082
#214614812
#419×D612
#2087461