Finite Group Databasebeta

← #30625938

Subgroups — #30625938

Order 15616.

1702 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462730C213yesmaximal, complemented
2#21462728C213yesmaximal, complemented
3#21462724C213yesmaximal, complemented
4#21462723C213yesmaximal, complemented
5#21462722C211yesmaximal, characteristic, complemented
6#21462721C211yesmaximal, characteristic, complemented
7#21462713C211yesmaximal, characteristic, complemented
8#21462712C211yesmaximal, characteristic, complemented
9Q8×#11758217C211yesmaximal, characteristic, complemented
10#21462614C213yesmaximal, complemented
11#21462607C213yesmaximal, complemented
12C2×Q8×#92420C211yesmaximal, characteristic, complemented
13#21462514C213yesmaximal
14#21462502C213yesmaximal
15Q8×#11758224C211yesmaximal, characteristic
16#16840814C2216yes—
17#16840814C2216yes—
18#16840813C2213yes—
19#16840813C2213yes—
20#16840813C2213yes—
21#16840813C2213yes—
22#16840812C2213yes—
23#16840812C2213yes—
24#16840812C2216yes—
25#16840811C2213yes—
26#16840811C2216yes—
27#16840811C2213yes—
28Q8×#92420C2211yescharacteristic
29Q8×#92420C2211yescharacteristic
30Q8×#92420C2213yes—
31Q8×#92420C2213yes—
32Q8×#92420C2213yes—
33Q8×#92420C2213yes—
34Q8×#92420C2211yescharacteristic
35Q8×#92420C2211yescharacteristic
36#16840787C2213yes—
37#16840787C2216yes—
38#16840787C2213yes—
39#16840786C2213yes—
40#16840786C2216yes—
41#16840786C2213yes—
42#16840784C2213yes—
43#16840783C2213yes—
44#16840781C2213yes—
45#16840781C2213yes—
46#16840780C2213yes—
47#16840779C2213yes—
48#16840778C2213yes—
49#16840778C2213yes—
50#16840768C2213yes—
51#16840768C2213yes—
52#16840765C2213yes—
53C2×C4⋊#92420C2213yes—
54#16840735C2211yescharacteristic
55#16840735C2213yes—
56#16840726C2216yes—
57C4×#11758217C2213yes—
58C2×C4×#92420C2213yes—
59Q8×Dic122C411yescharacteristic
60#16840487C413yes—
61#16840470C413yes—
62#16840470C413yes—
63#16840469C413yes—
64Q8×C22⋊D61C411yescharacteristic
65Q8×C22⋊D61C411yescharacteristic
66#16840317C2213yes—
67#16840314C2213yes—
68#16840314C2213yes—
69C4×#11758224C2213yes—
70C2×Q8×Dic61C2211yescharacteristic
71C2×Q8×Dic61C2211yescharacteristic
72C4×Q8×D61C2211yescharacteristic
73D4×Q8×C61C411yescharacteristic
74C2×#11758217C2311yescharacteristic
75C2×#11758217C2313yes—
76C22×#92420C2311yescharacteristic
77C22×#92420C2313yes—
78#14519844C2316yes—
79#14519844C2313yes—
80#14519844C2313yes—
81#14519844C2313yes—
82#14519844C2313yes—
83C4⋊#92420C2313yes—
84C4⋊#92420C2313yes—
85C4⋊#92420C2316yes—
86C4⋊#92420C2313yes—
87C4⋊#92420C2313yes—
88C4⋊#92420C2313yes—
89C4⋊#92420C2313yes—
90C4⋊#92420C2313yes—
91C4⋊#92420C2313yes—
92#14519834C2311yescharacteristic
93#14519834C2313yes—
94#14519834C2311yescharacteristic
95#14519834C2313yes—
96C22⋊#92420C2313yes—
97C22⋊#92420C2311yescharacteristic
98C22⋊#92420C2311yescharacteristic
99C22⋊#92420C2313yes—
100#14519832C2313yes—
101#14519832C2313yes—
102#14519832C2316yes—
103#14519832C2316yes—
104C4×#92420C2313yes—
105C4×#92420C2313yes—
106C4×#92420C2313yes—
107C4×#92420C2313yes—
108C4×#92420C2313yes—
109C4×#92420C2313yes—
110C4×#92420C2316yes—
111C4×#92420C2316yes—
112C2×#11758224C2311yescharacteristic
113C2×Q8×D61C2×C411yescharacteristic
114#14519777C2×C413yes—
115#14519776C2×C413yes—
116#14519776C2×C413yes—
117#14519775C2×C413yes—
118#14519775C2×C413yes—
119#14519768C2×C416yes—
120#14519767C2×C416yes—
121#14519767C2×C416yes—
122#14519757C2×C416yes—
123#14519756C2×C413yes—
124#14519756C2×C413yes—
125#14519754C2×C416yes—
126#14519754C2×C416yes—
127#14519753C2×C413yes—
128#14519753C2×C413yes—
129Q8×Dic61C2×C413yes—
130Q8×Dic61C2×C413yes—
131Q8×Dic61C2×C411yescharacteristic
132Q8×Dic61C2×C411yescharacteristic
133Q8×Dic61C2311yescharacteristic
134Q8×Dic61C2313yes—
135Q8×Dic61C2313yes—
136C4×Dic122C2×C413yes—
137C4×Dic122C2×C413yes—
138#14519742C2×C413yes—
139#14519742C2313yes—
140#14519742C2316yes—
141D4×Dic61C2×C413yes—
142C4×C22⋊D61C2×C413yes—
143C4×C22⋊D61C2×C413yes—
144#14519737C2×C413yes—
145#14519737C2×C413yes—
146#14519735C2313yes—
147#14519733C2313yes—
148#14519733C2313yes—
149C2×C4⋊Dic61C2×C413yes—
150C2×C4⋊Dic61C2×C413yes—
151C4⋊C4×D61C2×C413yes—
152C2×C4×Dic61C2313yes—
153C2×C4×Dic61C2313yes—
154C42×D61C2313yes—
155C22×Q8×C61C2×C411yescharacteristic
156C22×Q8×C61C2×C411yescharacteristic
157C4⋊Q8×C61C2×C413yes—
158C22⋊Q8×C61C2×C413yes—
159C22⋊Q8×C61C2×C413yes—
160C4×Q8×C61C2×C411yescharacteristic
161C4×D4×C61C2×C413yes—
162C2×Q8×C61—24no—
163C2×Q8×C61—24no—
164C2×Q8×C61C22×C413yes—
165C2×Q8×C61C22×C413yes—
166C2×Q8×C61C22×C411yescharacteristic
167C2×D4×C61C22×C411yescharacteristic
168C22×C4×C61C22×C413yes—
169C22×C4×C61C22×C413yes—
170C2×C22⋊D61C22×C411yescharacteristic
171C2×C22⋊D61C22×C411yescharacteristic
172C22×Dic61C22×C413yes—
173C22×Dic61C22×C413yes—
174C22×Dic61C2411yescharacteristic
175C22×Dic61C2411yescharacteristic
176Q8×D61—24no—
177#11758224—22no—
178#11758224—22no—
179C2×C4×D61C22×C413yes—
180C2×C4×D61C22×C413yes—
181C2×C4×D61C2411yescharacteristic
182C2×Dic122C22×C413yes—
183C2×Dic122C22×C411yescharacteristic
184C2×Dic122C22×C416yes—
185#11758217—22no—
186#11758217—26no—
187C2×#92420—26no—
188C2×#92420—22no—
189C2×#92420C2413yes—
190C2×#92420C2411yescharacteristic
191C2×#92420C2411yescharacteristic
192C2×#92420C2411yescharacteristic
193C2×#92420C2413yes—
194C2×#92420C2413yes—
195C2×#92420C2411yescharacteristic
196C2×#92420C2413yes—
197C4⋊C4×C61C22×C413yes—
198C4⋊C4×C61C22×C413yes—
199C4⋊C4×C61C22×C416yes—
200C22⋊C4×C61C22×C413yes—
201C22⋊C4×C61C22×C413yes—
202C42×C61C22×C413yes—
203C22⋊Dic61C22×C413yes—
204C22⋊Dic61C22×C413yes—
205#11758198C22×C413yes—
206#11758198C22×C413yes—
207C4⋊Dic61C22×C413yes—
208C4⋊Dic61C22×C413yes—
209C4⋊Dic61C22×C413yes—
210C4⋊Dic61C22×C416yes—
211C4⋊Dic61C22×C416yes—
212C4⋊Dic61C22×C413yes—
213#11758196C22×C416yes—
214#11758196C22×C413yes—
215#11758196C22×C413yes—
216#11758196C22×C413yes—
217#11758196C22×C413yes—
218#11758196C22×C416yes—
219#11758196C22×C416yes—
220C4×Dic61C22×C413yes—
221C4×Dic61C22×C413yes—
222C4×Dic61C22×C413yes—
223C4×Dic61C22×C413yes—
224C4×Dic61C22×C413yes—
225C4×Dic61C2413yes—
226C4×Dic61C2413yes—
227C4×Dic61C2413yes—
228C4×Dic61C2413yes—
229C23×C61C23×C411yescharacteristic
230C23×C61C23×C411yescharacteristic
231C22×D61C23×C411yescharacteristic
232Q8×C61—26no—
233Q8×C61—26no—
234Q8×C61#13114yes—
235D4×C61—22no—
236C2×C4×C61—26no—
237C2×C4×C61—26no—
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C23×C413yes—
240C2×C4×C61C23×C413yes—
241C2×C4×C61C23×C413yes—
242C2×C4×C61C23×C413yes—
243C22⋊D61—22no—
244C22⋊D61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—43no—
248C2×Dic61—43no—
249C2×Dic61C23×C411yescharacteristic
250C2×Dic61C23×C413yes—
251C2×Dic61C23×C413yes—
252C2×Dic61C23×C413yes—
253C2×Dic61C23×C413yes—
254C2×Dic61C23×C411yescharacteristic
255C2×Dic612−1+412yes—
256C2×Dic612−1+412yes—
257C2×Dic61C2511yescharacteristic
258C4×D61—26no—
259C4×D61—21no—
260C4×D61—21no—
261C4×D61—43no—
262Dic122—26no—
263Dic122—22no—
264Dic122—46no—
265#92420—23no—
266#92420—21no—
267#92420—21no—
268#92420—23no—
269#92420—21no—
270#92420—23no—
271#92420—23no—
272#92420—21no—
273#60253—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61#51912yes—
280C22×C61C24×C411yescharacteristic
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C4×C61—21no—
285C4×C61—23no—
286C4×C61—23no—
287C4×C61—23no—
288C4×C61C2×#13116yes—
289Dic61—21no—
290Dic61—23no—
291Dic61—23no—
292Dic61—23no—
293Dic61—23no—
294Dic61#51912yes—
295Dic61C2×2−1+411yescharacteristic
296Dic61C2×2−1+411yescharacteristic
297Q8×#41—611no—
298#2986—613no—
299#2986—613no—
300#2985—613no—
301#2985—613no—
302#2984—611no—
303#2984—611no—
304#2982—611no—
305#2982—611no—
306Q8×M16—611no—
307#2962—613no—
308#2960—613no—
309C2×C8×Q8—611no—
310#2925—613no—
311#2922—613no—
312C2×C61—22no—
313C2×C61—41no—
314C2×C61C2×#51911yescharacteristic
315C2×C61C22×#13111yescharacteristic
316C2×C61#357311yescharacteristic
317D61—41no—
318Q82—611no—
319#557—613no—
320#554—613no—
321#554—613no—
322#554—613no—
323D4×Q8—611no—
324D4×Q8—611no—
325D4×Q8#692311yescharacteristic
326#533—613no—
327#533—613no—
328#533—613no—
329C4×#41—613no—
330C2×C4×Q8—611no—
331C2×C4×Q8—611no—
332C2×C4×Q8—611no—
333#446—613no—
334#446—613no—
335#446—616no—
336#446—613no—
337#446—613no—
338#446—613no—
339#446—613no—
340#446—616no—
341#446—616no—
342C8×Q8—611no—
343C8×Q8—611no—
344C8×Q8—613no—
345C8×Q8—613no—
346C8×Q8—613no—
347C8×Q8—613no—
348C8×Q8—611no—
349C8×Q8—611no—
350C8×Q8—613no—
351C8×Q8—616no—
352C8×Q8—613no—
353#433—613no—
354#433—613no—
355#433—616no—
356#433—613no—
357#433—613no—
358#433—616no—
359#432—613no—
360#432—613no—
361#431—613no—
362#431—613no—
363#431—613no—
364#431—613no—
365#431—613no—
366#431—613no—
367#424—613no—
368#424—613no—
369#423—613no—
370C2×C4⋊C8—613no—
371#408—611no—
372#408—613no—
373#405—616no—
374C4×M16—613no—
375C2×C4×C8—613no—
376C61#6025311yescharacteristic, complemented, Sylow
377C2×#41—611no—
378C22×Q8—611no—
379C22×Q8C2×#692311yescharacteristic
380C22×Q8C2×#692311yescharacteristic
381C2×M16—611no—
382C2×M16—613no—
383C22×C8—611no—
384C22×C8—613no—
385C4⋊Q8—613no—
386C4⋊Q8—613no—
387C4⋊Q8—613no—
388C4⋊Q8—613no—
389C4⋊Q8—613no—
390C4⋊Q8C2×#692313yes—
391#125—616no—
392#125—616no—
393#125—616no—
394C22⋊Q8—613no—
395C22⋊Q8—613no—
396C22⋊Q8—613no—
397C22⋊Q8—613no—
398C22⋊Q8—616no—
399C22⋊Q8—616no—
400C22⋊Q8—616no—
401C22⋊Q8C2×#692313yes—
402C22⋊Q8C2×#692313yes—
403C4×Q8—611no—
404C4×Q8—613no—
405C4×Q8—613no—
406C4×Q8—611no—
407C4×Q8—611no—
408C4×Q8—613no—
409C4×Q8—613no—
410C4×Q8—613no—
411C4×Q8—613no—
412C4×Q8—616no—
413C4×Q8—613no—
414C4×Q8—613no—
415C4×Q8C2×#692311yescharacteristic
416C4×D4—613no—
417C4×D4—613no—
418C4×D4—613no—
419C4×D4—613no—
420C4×D4—613no—
421C4×D4C2×#692313yes—
422#117—613no—
423#117—613no—
424#117—613no—
425C2×C4⋊C4—613no—
426C2×C4⋊C4—613no—
427C2×C4⋊C4—613no—
428C2×C42—613no—
429C2×C42—613no—
430C2×C42—613no—
431C4⋊C8—613no—
432C4⋊C8—613no—
433C4⋊C8—616no—
434C4⋊C8—613no—
435C4⋊C8—613no—
436C4⋊C8—613no—
437C4⋊C8—613no—
438C4⋊C8—613no—
439C4⋊C8—613no—
440C4⋊C8—616no—
441C4⋊C8—613no—
442C4⋊C8—613no—
443C4⋊C8—613no—
444C4⋊C8—613no—
445C22⋊C8—611no—
446C22⋊C8—613no—
447C22⋊C8—611no—
448C22⋊C8—613no—
449C22⋊C8—611no—
450C22⋊C8—613no—
451C22⋊C8—611no—
452C22⋊C8—613no—
453#97—613no—
454#97—613no—
455#97—616no—
456#97—616no—
457C4×C8—613no—
458C4×C8—613no—
459C4×C8—613no—
460C4×C8—613no—
461C4×C8—613no—
462C4×C8—613no—
463C4×C8—616no—
464C4×C8—616no—
465#41—1222no—
466#41—1222no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—611no—
470C2×Q8—613no—
471C2×Q8—616no—
472C2×Q8—1224no—
473C2×Q8C22×#692313yes—
474C2×Q8C22×#692313yes—
475C2×Q8C22×#692311yescharacteristic
476C2×D4—611no—
477C2×D4—611no—
478C2×D4C22×#692311yescharacteristic
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—613no—
482C22×C4—613no—
483C22×C4—611no—
484C22×C4—613no—
485C22×C4—613no—
486C22×C4C22×#692313yes—
487C22×C4C22×#692313yes—
488M16—1222no—
489M16—1226no—
490C2×C8—611no—
491C2×C8—613no—
492C2×C8—611no—
493C2×C8—611no—
494C2×C8—611no—
495C2×C8—613no—
496C2×C8—613no—
497C2×C8—613no—
498C2×C8—1226no—
499C2×C8—1222no—
500C4⋊C4—613no—
501C4⋊C4—616no—
502C4⋊C4—616no—
503C4⋊C4—616no—
504C4⋊C4—613no—
505C4⋊C4—613no—
506C4⋊C4—613no—
507C4⋊C4—613no—
508C4⋊C4—613no—
509C4⋊C4—613no—
510C4⋊C4—616no—
511C4⋊C4—616no—
512C4⋊C4—613no—
513C4⋊C4C22×#692313yes—
514C4⋊C4C22×#692313yes—
515C4⋊C4C22×#692316yes—
516C22⋊C4—613no—
517C22⋊C4—613no—
518C22⋊C4—613no—
519C22⋊C4—613no—
520C22⋊C4C22×#692313yes—
521C22⋊C4C22×#692313yes—
522C42—613no—
523C42—613no—
524C42—613no—
525C42—613no—
526C42—613no—
527C42—613no—
528C42—613no—
529C42—613no—
530C42—613no—
531C42C22×#692313yes—
532C23—611no—
533C23C23×#692311yescharacteristic
534C23C23×#692311yescharacteristic
535Q8—26no—
536Q8—26no—
537Q8—1222no—
538Q8—1226no—
539Q8—2446no—
540Q8#1451986414yes—
541D4—22no—
542D4—1222no—
543D4—1222no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—611no—
551C2×C4—613no—
552C2×C4—613no—
553C2×C4—613no—
554C2×C4—613no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1226no—
560C2×C4—2443no—
561C2×C4—2443no—
562C2×C4—2443no—
563C2×C4C23×#692311yescharacteristic
564C2×C4C23×#692313yes—
565C2×C4C23×#692313yes—
566C2×C4C23×#692313yes—
567C2×C4C23×#692313yes—
568C8—1223no—
569C8—1221no—
570C8—1221no—
571C8—1223no—
572C8—1221no—
573C8—1221no—
574C8—1223no—
575C8—1223no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—1221no—
582C22—1221no—
583C22—1221no—
584C22#1684083012yes—
585C22C24×#692311yescharacteristic
586C4—21no—
587C4—23no—
588C4—23no—
589C4—23no—
590C4—612no—
591C4—611no—
592C4—611no—
593C4—1221no—
594C4—1223no—
595C4—1223no—
596C4—1223no—
597C4—1223no—
598C4C2×#1451986416yes—
599C2—22no—
600C2—41no—
601C2—2441no—
602C2C2×#1684083011yescharacteristic
603C2C22×#1451986411yescharacteristic
604C2#2146281511yescharacteristic
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingD4×Q8×C61C4
prime radicalC2×#11758224C23
cosocleC2×Dic61C25
hypercenterD4×Q8#6923
hypocenterC61#60253
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. #21462730C2semidirect
2. #21462728C2semidirect
3. #21462724C2semidirect
4. #21462723C2semidirect
5. #21462722C2semidirect
6. #21462721C2semidirect
7. #21462713C2semidirect
8. #21462712C2semidirect
9. Q8×#11758217C2semidirect
10. #21462614C2semidirect
11. #21462607C2semidirect
12. C2×Q8×#92420C2semidirect
376. C61#60253semidirect

Maximal subgroups

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

subgroupindex
#214627302
#214627282
#214627242
#214627232
#214627222
#214627212
#214627132
#214627122
Q8×#117582172
#214626142
#214626072
C2×Q8×#924202
#214625142
#214625022
Q8×#117582242
#6025361