Finite Group Databasebeta

← #30610817

Subgroups — #30610817

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462631C211yesmaximal, characteristic
2#21462515C211yesmaximal, characteristic
3#21461181C212yesmaximal, complemented
4#21461169C211yesmaximal, characteristic, complemented
5#21461163C212yesmaximal, complemented
6#21461148C211yesmaximal, characteristic, complemented
7#21461141C212yesmaximal, complemented
8#21461129C211yesmaximal, characteristic, complemented
9#21461123C212yesmaximal, complemented
10D61⋊(Q8⋊C8)C211yesmaximal, characteristic, complemented
11#21459398C211yesmaximal, characteristic
12#16840812—21no—
13#16840797—21no—
14#16840794—22no—
15#16840786—22no—
16#16840783C2212yes—
17#16840781C2211yescharacteristic
18#16840778C2212yes—
19#16840778C2212yes—
20#16840766C2211yescharacteristic
21D61⋊(C4⋊C8)C2211yescharacteristic
22#16840736—21no—
23#16840727—21no—
24C4×#11758217C2212yes—
25C4×D61⋊C8C2211yescharacteristic
26#16840612C2211yescharacteristic
27#16840612C2211yescharacteristic
28#16840612C2212yes—
29#16840612C2212yes—
30#16840612C2212yes—
31#16840612C2212yes—
32#16840612C2212yes—
33Q8⋊#92420C2211yescharacteristic
34Q8⋊#92420C2211yescharacteristic
35Q8⋊#92420C2212yes—
36#16840492C411yescharacteristic
37#16840490C411yescharacteristic
38#16840489—21no—
39#16840471—21no—
40#16840467C412yes—
41#16840466—22no—
42D4×Dic122C411yescharacteristic
43#16840425C412yes—
44#16840318—21no—
45#16840317C2211yescharacteristic
46#16840314—22no—
47#16840314C2212yes—
48C4×#11758224—21no—
49C4×#11758224C2211yescharacteristic
50C2×#14519742C2212yes—
51C4×Q8×D61C2211yescharacteristic
52C61×#554C411yescharacteristic
53C2×#11758213—21no—
54C2×#11758217—22no—
55C2×D61⋊C8—21no—
56C2×D61⋊C8—21no—
57#14519844—21no—
58C4⋊#92420—21no—
59C4⋊#92420—22no—
60C4⋊#92420—21no—
61C4⋊#92420C2311yescharacteristic
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2312yes—
64#14519834—21no—
65C22⋊#92420—22no—
66C22⋊#92420—22no—
67C22⋊#92420—22no—
68#14519832—22no—
69#14519832—22no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C2×#11758224D411yescharacteristic
76C22×Dic122C2×C412yes—
77C2×Q8×D61C2×C411yescharacteristic
78#14519777C2×C411yescharacteristic
79#14519776—21no—
80#14519776—21no—
81#14519776—22no—
82#14519776C2×C412yes—
83#14519775—22no—
84#14519775C2×C412yes—
85#14519769—21no—
86#14519768—21no—
87#14519767—22no—
88#14519767—22no—
89#14519767—22no—
90#14519757C2×C412yes—
91#14519756—22no—
92#14519756C2×C412yes—
93#14519755—22no—
94#14519755—22no—
95#14519754—22no—
96#14519754—22no—
97#14519754—22no—
98#14519753—21no—
99Q8×Dic61—21no—
100Q8×Dic61C2311yescharacteristic
101C4×Dic122—21no—
102C4×Dic122—21no—
103C4×Dic122C2×C411yescharacteristic
104C4×Dic122C2×C411yescharacteristic
105#14519742—21no—
106#14519742—22no—
107#14519742—21no—
108#14519742—22no—
109#14519742—22no—
110#14519742C2×C412yes—
111#14519742C2×C412yes—
112#14519742D411yescharacteristic
113#14519742C2311yescharacteristic
114#14519742C2312yes—
115D4×Dic61—21no—
116D4×Dic61D411yescharacteristic
117C4×C22⋊D61—22no—
118C4×C22⋊D61C2×C412yes—
119#14519737—22no—
120#14519737D412yes—
121#14519735—21no—
122#14519735D411yescharacteristic
123#14519733—22no—
124#14519733D412yes—
125C2×#11758196—22no—
126C4⋊C4×D61—21no—
127C4⋊C4×D61C2×C411yescharacteristic
128C2×C4×Dic61—22no—
129C2×C4×Dic61C2312yes—
130C42×D61—21no—
131C42×D61C2311yescharacteristic
132C4⋊Q8×C61C2×C411yescharacteristic
133#125×C61—21no—
134C22⋊Q8×C61—22no—
135C22⋊Q8×C61C2×C412yes—
136C4×Q8×C61C2×C411yescharacteristic
137C4×D4×C61—21no—
138C4×D4×C61C2×C411yescharacteristic
139C2×C4⋊C4×C61C2×C412yes—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61C22⋊C411yescharacteristic
143C22×C4×C61—22no—
144C22×C4×C61C22×C412yes—
145C2×C22⋊D61C22⋊C412yes—
146C22×Dic61—22no—
147C22×Dic61C2×D412yes—
148C22×Dic61C2×D412yes—
149Q8×D61—22no—
150Q8×D61—41no—
151#11758224—22no—
152#11758224—22no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61C22×C411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×Dic122—24no—
159C2×Dic122—22no—
160C2×Dic122—21no—
161C2×Dic122—22no—
162C2×Dic122—42no—
163C2×Dic122C22⋊C411yescharacteristic
164C2×Dic122C22×C412yes—
165C2×Dic122C22×C411yescharacteristic
166#11758217—44no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—21no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420—21no—
173#11758213—42no—
174D61⋊C8—42no—
175D61⋊C8—42no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61C22⋊C411yescharacteristic
181C4⋊C4×C61C22×C411yescharacteristic
182C4⋊C4×C61C22×C412yes—
183C22⋊C4×C61—22no—
184C22⋊C4×C61C22⋊C412yes—
185C42×C61—21no—
186C42×C61C22×C411yescharacteristic
187C22⋊Dic61—22no—
188C22⋊Dic61C22⋊C412yes—
189#11758198—22no—
190#11758198C22⋊C412yes—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61C22⋊C411yescharacteristic
196C4⋊Dic61C22×C411yescharacteristic
197#11758196—21no—
198#11758196—22no—
199#11758196—21no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—21no—
206#11758196—22no—
207#11758196C22⋊C412yes—
208#11758196C22⋊C412yes—
209C4×Dic61—21no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—22no—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2411yescharacteristic
221C23×C61C2×C22⋊C412yes—
222C22×D61C2×C22⋊C411yescharacteristic
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—81no—
226D4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—82no—
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C23×C411yescharacteristic
237C22⋊D61—24no—
238C2×Dic61—21no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—42no—
244C2×Dic61—44no—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C22×D411yescharacteristic
249C2×Dic61C22×D411yescharacteristic
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—41no—
253C4×D61—42no—
254C4×D61—81no—
255C4×D61#13712yes—
256C4×D61#13712yes—
257Dic122—21no—
258Dic122—22no—
259Dic122—22no—
260Dic122—22no—
261Dic122—42no—
262Dic122—44no—
263Dic122—82no—
264Dic122#13112yes—
265#92420—22no—
266#92420—41no—
267#92420—42no—
268#92420—41no—
269#92420—41no—
270#92420—42no—
271#17514—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61—22no—
275C22×C61C22×C22⋊C411yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61#41912yes—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61#41912yes—
285C4×C61C2×#13111yescharacteristic
286C4×C61#56811yescharacteristic
287Dic61—22no—
288Dic61—21no—
289Dic61—22no—
290Dic61—21no—
291Dic61—22no—
292Dic61—42no—
293Dic61—41no—
294Dic61C2×#13711yescharacteristic
295Dic61C2×#13711yescharacteristic
296Q8×#41—611no—
297#2965—611no—
298#2925—611no—
299#1497—612no—
300#1493—611no—
301#1491—612no—
302#1487—611no—
303#1485—612no—
304#1481—611no—
305#1479—612no—
306C2×Q8⋊C8—611no—
307C2×C61—42no—
308C2×C61#288011yescharacteristic
309C2×C61C2×#41911yescharacteristic
310C2×C61#290111yescharacteristic
311D61—22no—
312Q82—611no—
313#557—611no—
314#557—1221no—
315#554—612no—
316#554—1221no—
317#554—1222no—
318#554#692311yescharacteristic
319D4×Q8—612no—
320D4×Q8—611no—
321#533—611no—
322#533—612no—
323#533—1222no—
324#533—1221no—
325C4×#41—611no—
326C4×#41—1221no—
327C2×C4×Q8—611no—
328C2×C4×Q8—612no—
329#446—1221no—
330#436—1221no—
331#435—1222no—
332#433—1222no—
333#432—612no—
334#431—612no—
335#431—611no—
336#431—612no—
337#423—611no—
338C2×C4⋊C8—611no—
339#408—1221no—
340#405—1221no—
341C4×M16—612no—
342C2×C4×C8—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—612no—
346Q8⋊C8—611no—
347Q8⋊C8—611no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—612no—
351Q8⋊C8—612no—
352Q8⋊C8—612no—
353C61#1751411yescharacteristic, complemented, Sylow
354C2×#41—611no—
355C22×Q8—611no—
356C22×Q8—612no—
357C2×M16—1222no—
358C2×M16—1221no—
359C22×C8—1221no—
360C22×C8—1221no—
361C4⋊Q8—612no—
362C4⋊Q8—612no—
363C4⋊Q8—611no—
364C4⋊Q8—1222no—
365C4⋊Q8—1221no—
366C4⋊Q8—1222no—
367C4⋊Q8—1221no—
368C4⋊Q8C2×#692311yescharacteristic
369#125—21no—
370#125—1221no—
371#125—1222no—
372#125—1221no—
373#125—1222no—
374#125—1222no—
375C22⋊Q8—22no—
376C22⋊Q8—612no—
377C22⋊Q8—612no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1221no—
384C22⋊Q8—1222no—
385C22⋊Q8C2×#692312yes—
386C4×Q8—611no—
387C4×Q8—611no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—611no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×Q8—1222no—
397C4×Q8—1221no—
398C4×Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8C2×#692311yescharacteristic
403C4×D4—21no—
404C4×D4—611no—
405C4×D4—612no—
406C4×D4—612no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1221no—
410C4×D4C2×#692311yescharacteristic
411#117—611no—
412#117—612no—
413#117—1222no—
414#117—1221no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4C2×#692312yes—
419C2×C42—612no—
420C2×C42—611no—
421C2×C42—1221no—
422C2×C42—1222no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—612no—
426C4⋊C8—1221no—
427C4⋊C8—1222no—
428C4⋊C8—1221no—
429C4⋊C8—1221no—
430C22⋊C8—1222no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1221no—
434#97—1222no—
435#97—1222no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8—612no—
439C4×C8—1221no—
440C4×C8—1222no—
441#41—1222no—
442#41—1222no—
443C2×Q8—21no—
444C2×Q8—611no—
445C2×Q8—611no—
446C2×Q8—612no—
447C2×Q8—1222no—
448C2×Q8—1221no—
449C2×Q8—1224no—
450C2×Q8—1222no—
451C2×Q8—1222no—
452C2×Q8—2442no—
453C2×Q8—2441no—
454C2×Q8C22×#692311yescharacteristic
455C2×D4—612no—
456C2×D4C22⋊#692311yescharacteristic
457C22×C4—22no—
458C22×C4—611no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1222no—
466C22×C4C22×#692312yes—
467M16—2444no—
468M16—2442no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1222no—
472C2×C8—1221no—
473C2×C8—1222no—
474C2×C8—1221no—
475C2×C8—2442no—
476C2×C8—2442no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—612no—
482C4⋊C4—612no—
483C4⋊C4—611no—
484C4⋊C4—611no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4C22⋊#692311yescharacteristic
500C4⋊C4C22×#692311yescharacteristic
501C4⋊C4C22×#692312yes—
502C22⋊C4—22no—
503C22⋊C4—612no—
504C22⋊C4—612no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4C22⋊#692312yes—
508C42—21no—
509C42—612no—
510C42—611no—
511C42—612no—
512C42—611no—
513C42—611no—
514C42—612no—
515C42—1221no—
516C42—1221no—
517C42—1222no—
518C42—1222no—
519C42—1222no—
520C42—1221no—
521C42C22×#692311yescharacteristic
522C23—611no—
523C23C2×C22⋊#692312yes—
524Q8—21no—
525Q8—21no—
526Q8—81no—
527Q8—612no—
528Q8—1222no—
529Q8—1221no—
530Q8—1222no—
531Q8—1222no—
532Q8—4882no—
533Q8—2444no—
534Q8—2442no—
535D4—22no—
536D4—1224no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—82no—
543C2×C4—611no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—611no—
548C2×C4—612no—
549C2×C4—611no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—4881no—
558C2×C4—2441no—
559C2×C4—2442no—
560C2×C4—2442no—
561C2×C4—2444no—
562C2×C4C2×C22⋊#692312yes—
563C2×C4C2×C22⋊#692311yescharacteristic
564C2×C4C2×C22⋊#692311yescharacteristic
565C2×C4C23×#692311yescharacteristic
566C8—1222no—
567C8—2441no—
568C8—2441no—
569C8—2442no—
570C8—2441no—
571C8—2442no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—612no—
576C22—1221no—
577C22—1221no—
578C22C22×C22⋊#692311yescharacteristic
579C4—22no—
580C4—21no—
581C4—41no—
582C4—41no—
583C4—42no—
584C4—611no—
585C4—611no—
586C4—1221no—
587C4—1222no—
588C4—1222no—
589C4—1221no—
590C4—1222no—
591C4—2442no—
592C4—2441no—
593C4#1684075512yes—
594C4D61⋊#13111yescharacteristic
595C4#1684084111yescharacteristic
596C2—42no—
597C2—1222no—
598C2#2146240911yescharacteristic
599C2C2×#1684075511yescharacteristic
600C2#2146244911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#554C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758224D4
cosocleC4×Dic61C24
hypercenter#554#6923
hypocenterC61#17514
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
3. #21461181C2semidirect
4. #21461169C2semidirect
5. #21461163C2semidirect
6. #21461148C2semidirect
7. #21461141C2semidirect
8. #21461129C2semidirect
9. #21461123C2semidirect
10. D61⋊(Q8⋊C8)C2semidirect
353. C61#17514semidirect

Maximal subgroups

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

subgroupindex
#214626312
#214625152
#214611812
#214611692
#214611632
#214611482
#214611412
#214611292
#214611232
D61⋊(Q8⋊C8)2
#214593982
#1751461