Finite Group Databasebeta

← #30625041

Subgroups — #30625041

Order 15616.

2340 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462362C214yesmaximal, complemented
2#21462352C211yesmaximal, characteristic, complemented
3#21462344C214yesmaximal, complemented
4D4×C4⋊#6923C212yesmaximal, complemented
5#21462329C214yesmaximal, complemented
6#21462309C214yesmaximal, complemented
7#21462307C214yesmaximal, complemented
8#21462300C214yesmaximal, complemented
9C4⋊Q8×#6923C211yesmaximal, characteristic, complemented
10#21462276C212yesmaximal, complemented
11#555×D61C211yesmaximal, characteristic
12C2×Q8×#6923C2212yes—
13C2×D4×#6923C2214yes—
14C2×D4×#6923C2214yes—
15C2×#14519856C2214yes—
16C22×C4⋊#6923C2214yes—
17(C2×#6923)⋊Q8C2212yes—
18(C2×#6923)⋊Q8C2214yes—
19#16840714C2212yes—
20#16840714C2212yes—
21#16840714C2214yes—
22#16840713C2212yes—
23#16840713C2212yes—
24#16840713C2214yes—
25#16840713C2214yes—
26#16840713C2214yes—
27#16840712C2214yes—
28#16840712C2214yes—
29#16840712C2218yes—
30#16840712C2218yes—
31#16840710C2218yes—
32#16840709C2218yes—
33#16840709C2218yes—
34#16840709C2218yes—
35#16840708C2211yescharacteristic
36#16840708C2214yes—
37#16840706C2218yes—
38#16840704C2214yes—
39C4⋊C4×#6923C2214yes—
40C4⋊C4×#6923C2214yes—
41C4×C4⋊#6923C2212yes—
42C4×C22⋊#6923C2214yes—
43#16840698C2214yes—
44C42×#6923C2211yescharacteristic
45#16840478C411yescharacteristic, complemented
46#16840477C412yescomplemented
47#16840476C414yescomplemented
48#16840475C414yescomplemented
49#16840474C414yescomplemented
50C4⋊Q8×D61C2211yescharacteristic
51#124×D61C2214yes—
52#121×D61C2214yes—
53C4×D4×D61C2212yes—
54#117×D61C2214yes—
55C61×#555C411yescharacteristic, complemented
56C23×#6923C2318yes—
57Q8×#6923—44no—
58D4×#6923—48no—
59D4×#6923—48no—
60#14519856—48no—
61C2×C4⋊#6923—48no—
62C2×C4⋊#6923C2312yes—
63C2×C4⋊#6923C2314yes—
64C2×C4⋊#6923C2312yes—
65C2×C4⋊#6923C2314yes—
66C2×C4⋊#6923C2318yes—
67C2×C4⋊#6923C2314yes—
68C2×C4⋊#6923C2312yes—
69C2×C4⋊#6923C2314yes—
70C2×C4⋊#6923C2314yes—
71C2×C22⋊#6923C2314yes—
72C2×C22⋊#6923C2318yes—
73C2×C22⋊#6923C2314yes—
74C2×C22⋊#6923C2314yes—
75C2×C22⋊#6923C2314yes—
76C2×C4×#6923C2311yescharacteristic
77C2×C4×#6923C2314yes—
78C2×C4×#6923C2312yes—
79C2×C4×#6923C2314yes—
80C2×C4×#6923C2312yes—
81C2×C4×#6923C2311yescharacteristic
82C2×C4×#6923C2314yes—
83(C2×#6923)⋊C4C2318yes—
84(C2×#6923)⋊C4C2314yes—
85(C2×#6923)⋊C4C2318yes—
86(C2×#6923)⋊C4C2318yes—
87(C2×#6923)⋊C4C2314yes—
88(C2×#6923)⋊C4C2314yes—
89C2×Q8×D61C2312yes—
90C2×D4×D61C2312yes—
91C2×D4×D61C2314yes—
92C22×C4×D61C2314yes—
93#14519777C2×C411yescharacteristic, complemented
94#14519776C2×C414yescomplemented
95#14519775C2×C412yescomplemented
96#14519766C2×C414yescomplemented
97#14519765C2×C414yescomplemented
98#14519764C2×C414yescomplemented
99#14519763C2×C418yescomplemented
100#14519763C2×C418yescomplemented
101#14519752C2×C414yescomplemented
102#14519751C2×C414yescomplemented
103#14519750C2×C418yescomplemented
104#14519749C2×C418yescomplemented
105#14519748C2×C414yescomplemented
106C4×D244C2×C412yescomplemented
107D4×Dic61C2×C412yescomplemented
108#14519739C2×C412yescomplemented
109C4×C22⋊D61C2×C414yescomplemented
110#14519737C2×C414yescomplemented
111#14519736C2×C414yescomplemented
112#14519735C2×C414yescomplemented
113#14519735C2×C414yescomplemented
114#14519734C2×C418yescomplemented
115#14519733C2×C418yescomplemented
116C4⋊C4×D61C2312yes—
117C4⋊C4×D61C2314yes—
118C22⋊C4×D61C2318yes—
119C22⋊C4×D61C2314yes—
120C42×D61C2314yes—
121C42×D61C2311yescharacteristic
122C4⋊Q8×C61C2×C411yescharacteristic, complemented
123#124×C61C2×C414yescomplemented
124#121×C61C2×C414yescomplemented
125C4×D4×C61C2×C412yescomplemented
126#117×C61C2×C414yescomplemented
127C23×D61C2414yes—
128C22×#6923—48no—
129C22×#6923—48no—
130C22×#6923—48no—
131C22×#6923C2412yes—
132C22×#6923C2414yes—
133C22×#6923C2414yes—
134C22×#6923C2414yes—
135C22×#6923C2411yescharacteristic
136C22×#6923C2411yescharacteristic
137C2×Q8×C61C22×C412yes—
138C2×D4×C61C22×C412yes—
139C2×D4×C61C22×C414yes—
140C22×C4×C61C22×C414yes—
141C2×C22⋊D61C22×C414yes—
142C2×C22⋊D61C22×C418yes—
143C22×Dic61C22×C414yes—
144Q8×D61—42no—
145Q8×D61—42no—
146D4×D61—24no—
147D4×D61—24no—
148D4×D61—44no—
149D4×D61—44no—
150C2×D244C22×C412yes—
151C2×D244C22×C414yes—
152C2×C4×D61—28no—
153C2×C4×D61—28no—
154C2×C4×D61C22×C414yes—
155C2×C4×D61C22×C414yes—
156C2×C4×D61C2414yes—
157C2×C4×D61C2414yes—
158C2×C4×D61C2411yescharacteristic
159C2×C4×D61C2412yes—
160C2×Dic122C22×C412yes—
161C2×Dic122C22×C414yes—
162C22⋊#6923—28no—
163C22⋊#6923—44no—
164C22⋊#6923—44no—
165C22⋊#6923—44no—
166C22⋊#6923—48no—
167C4⋊#6923—24no—
168C4⋊#6923—28no—
169C4⋊#6923—24no—
170C4⋊#6923—48no—
171C4⋊#6923—44no—
172C4⋊#6923—44no—
173C4⋊#6923—44no—
174C4⋊#6923—44no—
175C4⋊#6923—42no—
176C4×#6923—28no—
177C4×#6923—22no—
178C4×#6923—24no—
179C4×#6923—28no—
180C4×#6923—24no—
181C4×#6923—44no—
182C4×#6923—41no—
183C4⋊C4×C61C22×C412yes—
184C4⋊C4×C61C22×C414yes—
185C22⋊C4×C61C22×C418yes—
186C22⋊C4×C61C22×C414yes—
187C42×C61C22×C414yes—
188C42×C61C22×C411yescharacteristic
189C22⋊Dic61C22×C418yes—
190C22⋊Dic61C22×C414yes—
191#11758198C22×C418yes—
192#11758198C22×C418yes—
193#11758198C22×C414yes—
194#11758198C22×C414yes—
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C414yes—
197#11758196C22×C414yes—
198#11758196C22×C418yes—
199C4×Dic61C22×C414yes—
200C4×Dic61C22×C418yes—
201C4×Dic61C22×C411yescharacteristic
202C4×Dic61C22×C412yes—
203C23×C61C23×C414yes—
204C22×D61—24no—
205C22×D61—24no—
206C22×D61—24no—
207C22×D61—24no—
208C22×D61—24no—
209C22×D61—24no—
210C22×D61C23×C414yes—
211C22×D61C2511yescharacteristic
212C2×#6923—22no—
213C2×#6923—21no—
214C2×#6923—22no—
215C2×#6923—22no—
216C2×#6923—24no—
217C2×#6923—24no—
218C2×#6923—24no—
219C2×#6923—24no—
220C2×#6923—28no—
221C2×#6923—24no—
222C2×#6923—22no—
223C2×#6923—24no—
224C2×#6923—24no—
225C2×#6923—22no—
226C2×#6923—816no—
227C2×#6923H(5,𝔽2)12yes—
228Q8×C61—42no—
229D4×C61—24no—
230D4×C61—44no—
231C2×C4×C61—28no—
232C2×C4×C61C23×C414yes—
233C2×C4×C61C23×C414yes—
234C2×C4×C61C23×C411yescharacteristic
235C2×C4×C61C23×C412yes—
236C22⋊D61—28no—
237C22⋊D61—48no—
238C2×Dic61—28no—
239C2×Dic61C23×C414yes—
240C2×Dic61C23×C414yes—
241C2×Dic61C23×C411yescharacteristic
242C2×Dic61C23×C412yes—
243D244—24no—
244D244—44no—
245C4×D61—24no—
246C4×D61—21no—
247C4×D61—24no—
248C4×D61—24no—
249C4×D61—24no—
250C4×D61—21no—
251C4×D61—28no—
252C4×D61—28no—
253C4×D61H(5,𝔽2)14yes—
254C4×D612−1+414yes—
255Dic122—44no—
256Dic122—42no—
257#34485—611nomaximal, Sylow
258C22×C61—24no—
259C22×C61—24no—
260C22×C61—24no—
261C22×C61C24×C411yescharacteristic
262C2×D61—24no—
263C2×D61—24no—
264C2×D61—24no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61—44no—
269C2×D61C2×H(5,𝔽2)11yescharacteristic
270C2×D61C2×H(5,𝔽2)12yes—
271C2×D61C2×2−1+412yes—
272C2×D61#58511yescharacteristic
273#6923—22no—
274#6923—42no—
275#6923—44no—
276#6923—41no—
277#6923—44no—
278#6923—44no—
279C4×C61—24no—
280C4×C61—21no—
281C4×C61—24no—
282C4×C61#52014yes—
283Dic61—24no—
284Dic61—21no—
285Dic61—24no—
286Dic61#52014yes—
287C2×#555—611no—
288#2379—614no—
289#2369—611no—
290#2361—614no—
291D4×C4⋊C4—612no—
292#2346—614no—
293#2326—614no—
294#2324—614no—
295#2317—614no—
296C4×C4⋊Q8—611no—
297#2293—612no—
298C2×C61—44no—
299C2×C61C2×#52012yes—
300C2×C61C4×H(5,𝔽2)11yescharacteristic
301D61—44no—
302D61#354411yescharacteristic
303D61#354712yes—
304D61#354911yescharacteristic
305#555—611no—
306#555—614no—
307#555—614no—
308#555—612no—
309#555—614no—
310#555#692311yescharacteristic, complemented
311C2×C4⋊Q8—611no—
312C2×#124—614no—
313C2×#121—614no—
314C2×C4×Q8—612no—
315C2×C4×D4—612no—
316C2×C4×D4—614no—
317C2×C4×D4—614no—
318C2×#117—614no—
319C2×#117—614no—
320C22×C4⋊C4—614no—
321#391—612no—
322#391—614no—
323#390—614no—
324#390—612no—
325#390—612no—
326#389—614no—
327#389—612no—
328#389—614no—
329#389—614no—
330#389—612no—
331#388—618no—
332#388—614no—
333#388—618no—
334#388—614no—
335#386—618no—
336#385—618no—
337#385—618no—
338#385—618no—
339#384—611no—
340#384—614no—
341#382—618no—
342#380—614no—
343C4×C4⋊C4—614no—
344C4×C4⋊C4—614no—
345C4×C4⋊C4—612no—
346C4×C22⋊C4—614no—
347#376—614no—
348C43—611no—
349C61#3448511yescharacteristic, complemented, Sylow
350C22×Q8—612no—
351C22×D4—612no—
352C22×D4—614no—
353C23×C4—614no—
354C23×C4—618no—
355C4⋊Q8—612no—
356C4⋊Q8—614no—
357C4⋊Q8—611no—
358C4⋊Q8C2×#692311yescharacteristic, complemented
359#124—614no—
360#124—614no—
361#124—618no—
362#124—618no—
363#124—614no—
364#124C2×#692314yescomplemented
365#121—614no—
366#121—618no—
367#121—614no—
368#121—614no—
369#121—618no—
370#121C2×#692314yescomplemented
371C4×Q8—2444no—
372C4×D4—612no—
373C4×D4—614no—
374C4×D4—612no—
375C4×D4—614no—
376C4×D4—612no—
377C4×D4—2448no—
378C4×D4—2448no—
379C4×D4C2×#692312yescomplemented
380#117—614no—
381#117—614no—
382#117—614no—
383#117—618no—
384#117—618no—
385#117—2448no—
386#117C2×#692314yescomplemented
387C2×C4⋊C4—612no—
388C2×C4⋊C4—614no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—614no—
391C2×C4⋊C4—612no—
392C2×C4⋊C4—618no—
393C2×C4⋊C4—614no—
394C2×C4⋊C4—614no—
395C2×C4⋊C4—614no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—614no—
398C2×C4⋊C4—2448no—
399C2×C22⋊C4—618no—
400C2×C22⋊C4—614no—
401C2×C22⋊C4—614no—
402C2×C22⋊C4—614no—
403C2×C22⋊C4—614no—
404C2×C22⋊C4—618no—
405C2×C22⋊C4—614no—
406C2×C42—614no—
407C2×C42—611no—
408C2×C42—612no—
409C2×C42—614no—
410C2×C42—614no—
411C2×C42—611no—
412C2×C42—612no—
413C2×C42—611no—
414C2×C42—614no—
415#95—614no—
416#95—618no—
417#95—614no—
418#95—618no—
419#95—618no—
420#95—614no—
421C24—614no—
422C2×Q8—614no—
423C2×Q8—612no—
424C2×Q8—2442no—
425C2×Q8—2442no—
426C2×Q8C22×#692312yes—
427C2×D4—614no—
428C2×D4—618no—
429C2×D4—612no—
430C2×D4—614no—
431C2×D4—1224no—
432C2×D4—1224no—
433C2×D4—2444no—
434C2×D4—2444no—
435C2×D4C22×#692312yes—
436C2×D4C22×#692314yes—
437C22×C4—612no—
438C22×C4—611no—
439C22×C4—614no—
440C22×C4—611no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—614no—
446C22×C4—612no—
447C22×C4—614no—
448C22×C4—614no—
449C22×C4—611no—
450C22×C4—1228no—
451C22×C4—1228no—
452C22×C4—2448no—
453C22×C4—2448no—
454C22×C4—2448no—
455C22×C4C22×#692314yes—
456C4⋊C4—614no—
457C4⋊C4—618no—
458C4⋊C4—612no—
459C4⋊C4—614no—
460C4⋊C4—1224no—
461C4⋊C4—1228no—
462C4⋊C4—1224no—
463C4⋊C4—2448no—
464C4⋊C4—2444no—
465C4⋊C4—2444no—
466C4⋊C4—2444no—
467C4⋊C4—2444no—
468C4⋊C4—2442no—
469C4⋊C4C22×#692312yes—
470C4⋊C4C22×#692314yes—
471C22⋊C4—618no—
472C22⋊C4—614no—
473C22⋊C4—618no—
474C22⋊C4—618no—
475C22⋊C4—614no—
476C22⋊C4—614no—
477C22⋊C4—1228no—
478C22⋊C4—2444no—
479C22⋊C4—2444no—
480C22⋊C4—2448no—
481C22⋊C4—2444no—
482C22⋊C4C22×#692318yes—
483C22⋊C4C22×#692314yes—
484C42—614no—
485C42—618no—
486C42—611no—
487C42—612no—
488C42—1224no—
489C42—1228no—
490C42—1228no—
491C42—1222no—
492C42—1224no—
493C42—2444no—
494C42—2441no—
495C42C22×#692314yes—
496C42C22×#692311yescharacteristic
497C23—611no—
498C23—614no—
499C23—1224no—
500C23—1224no—
501C23—1224no—
502C23—1224no—
503C23—1224no—
504C23—1224no—
505C23C23×#692314yes—
506Q8—42no—
507Q8—2444no—
508Q8—2442no—
509D4—24no—
510D4—44no—
511D4—1228no—
512D4—1224no—
513D4—2444no—
514D4—2448no—
515C2×C4—28no—
516C2×C4—612no—
517C2×C4—614no—
518C2×C4—614no—
519C2×C4—612no—
520C2×C4—614no—
521C2×C4—611no—
522C2×C4—614no—
523C2×C4—1228no—
524C2×C4—1224no—
525C2×C4—1221no—
526C2×C4—1224no—
527C2×C4—1221no—
528C2×C4—1224no—
529C2×C4—1224no—
530C2×C4—1228no—
531C2×C4—1222no—
532C2×C4—1228no—
533C2×C4—1224no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1224no—
537C2×C4—1221no—
538C2×C4—1224no—
539C2×C4—1222no—
540C2×C4—1224no—
541C2×C4—1222no—
542C2×C4—1224no—
543C2×C4—1228no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—48816no—
547C2×C4C23×#692314yes—
548C2×C4C23×#692311yescharacteristic
549C2×C4C23×#692314yes—
550C2×C4C23×#692312yes—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—612no—
555C22—612no—
556C22—611no—
557C22—611no—
558C22—1224no—
559C22—1224no—
560C22—1224no—
561C22—2444no—
562C22—2444no—
563C22—2444no—
564C22—2444no—
565C22C24×#692311yescharacteristic
566C4—24no—
567C4—24no—
568C4—21no—
569C4—614no—
570C4—1222no—
571C4—1224no—
572C4—1224no—
573C4—1221no—
574C4—2444no—
575C4—2444no—
576C4—2441no—
577C4—2444no—
578C4—2442no—
579C4#1684083114yes—
580C2—44no—
581C2—611no—
582C2—611no—
583C2—612no—
584C2—2444no—
585C2C2×#1684083112yes—
586C2H(5,𝔽2)×#692311yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#555C4
prime radical#555×D61C2
cosocleC22×D61C25
hypercenter#555#6923
hypocenterC61#34485
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #21462362C2semidirect
2. #21462352C2semidirect
3. #21462344C2semidirect
4. D4×C4⋊#6923C2semidirect
5. #21462329C2semidirect
6. #21462309C2semidirect
7. #21462307C2semidirect
8. #21462300C2semidirect
9. C4⋊Q8×#6923C2semidirect
10. #21462276C2semidirect
45. #16840478C4semidirect
46. #16840477C4semidirect
47. #16840476C4semidirect
48. #16840475C4semidirect
49. #16840474C4semidirect
55. C61×#555C4semidirect
93. #14519777C2×C4semidirect
94. #14519776C2×C4semidirect
95. #14519775C2×C4semidirect
96. #14519766C2×C4semidirect
97. #14519765C2×C4semidirect
98. #14519764C2×C4semidirect
99. #14519763C2×C4semidirect
100. #14519763C2×C4semidirect
101. #14519752C2×C4semidirect
102. #14519751C2×C4semidirect
103. #14519750C2×C4semidirect
104. #14519749C2×C4semidirect
105. #14519748C2×C4semidirect
106. C4×D244C2×C4semidirect
107. D4×Dic61C2×C4semidirect
108. #14519739C2×C4semidirect
109. C4×C22⋊D61C2×C4semidirect
110. #14519737C2×C4semidirect
111. #14519736C2×C4semidirect
112. #14519735C2×C4semidirect
113. #14519735C2×C4semidirect
114. #14519734C2×C4semidirect
115. #14519733C2×C4semidirect
122. C4⋊Q8×C61C2×C4semidirect
123. #124×C61C2×C4semidirect
124. #121×C61C2×C4semidirect
125. C4×D4×C61C2×C4semidirect
126. #117×C61C2×C4semidirect
310. #555#6923semidirect
349. C61#34485semidirect
358. C4⋊Q8C2×#6923semidirect
364. #124C2×#6923semidirect
370. #121C2×#6923semidirect
379. C4×D4C2×#6923semidirect
386. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623622
#214623522
#214623442
D4×C4⋊#69232
#214623292
#214623092
#214623072
#214623002
C4⋊Q8×#69232
#214622762
#555×D612
#3448561