Finite Group Databasebeta

← #30610818

Subgroups — #30610818

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#21462627C211yesmaximal, characteristic
2#21462519C211yesmaximal, characteristic
3#21461182C212yesmaximal, complemented
4#21461168C211yesmaximal, characteristic, complemented
5#21461162C212yesmaximal, complemented
6#21461148C211yesmaximal, characteristic, complemented
7#21461142C212yesmaximal, complemented
8#21461128C211yesmaximal, characteristic, complemented
9#21461122C212yesmaximal, complemented
10D61⋊(Q8⋊C8)C211yesmaximal, characteristic, complemented
11#21459401C211yesmaximal, characteristic
12#16840813—21no—
13#16840796—21no—
14#16840792—22no—
15#16840787—22no—
16#16840784C2212yes—
17#16840781C2212yes—
18#16840781C2211yescharacteristic
19#16840780C2212yes—
20#16840766C2211yescharacteristic
21D61⋊(C4⋊C8)C2211yescharacteristic
22#16840737—21no—
23#16840727—21no—
24C4×#11758213C2212yes—
25C4×D61⋊C8C2211yescharacteristic
26#16840612C2211yescharacteristic
27#16840612C2211yescharacteristic
28#16840612C2212yes—
29Q8⋊#92420C2211yescharacteristic
30Q8⋊#92420C2211yescharacteristic
31Q8⋊#92420C2212yes—
32Q8⋊#92420C2212yes—
33Q8⋊#92420C2212yes—
34Q8⋊#92420C2212yes—
35Q8⋊#92420C2212yes—
36Q8×Dic122C411yescharacteristic
37#16840488—21no—
38#16840472C411yescharacteristic
39#16840472C412yes—
40#16840468—22no—
41#16840468—21no—
42#16840427C412yes—
43#16840427C411yescharacteristic
44#16840318—21no—
45#16840318—22no—
46#16840317C2211yescharacteristic
47#16840317C2212yes—
48C4×#11758226—21no—
49C4×#11758226C2211yescharacteristic
50C4×Q8×D61C2211yescharacteristic
51C4×Q8×D61C2212yes—
52C61×#557C411yescharacteristic
53C2×#11758213—22no—
54C2×#11758213—21no—
55C2×D61⋊C8—21no—
56C2×D61⋊C8—21no—
57#14519844—21no—
58#14519844—22no—
59C4⋊#92420—21no—
60C4⋊#92420—21no—
61C4⋊#92420C2311yescharacteristic
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2312yes—
64#14519834—22no—
65#14519834—22no—
66#14519834—22no—
67#14519834—21no—
68#14519832—22no—
69#14519832—22no—
70C4×#92420—21no—
71C4×#92420—22no—
72C4×#92420C2311yescharacteristic
73C4×#92420C2311yescharacteristic
74C4×#92420C2312yes—
75C2×#11758226D411yescharacteristic
76C2×Q8×D61C2×C411yescharacteristic
77C2×Q8×D61C2×C412yes—
78#14519777C2×C411yescharacteristic
79#14519777C2×C412yes—
80#14519776—22no—
81#14519776—21no—
82#14519776—22no—
83#14519776—21no—
84#14519769—22no—
85#14519769—21no—
86#14519768—21no—
87#14519768—22no—
88#14519757—21no—
89#14519757—22no—
90#14519755—22no—
91#14519755—22no—
92#14519755—22no—
93#14519755—22no—
94#14519755—22no—
95#14519755—22no—
96#14519753C2×C412yes—
97#14519753C2×C412yes—
98#14519753C2×C412yes—
99Q8×Dic61—21no—
100Q8×Dic61D411yescharacteristic
101Q8×Dic61C2311yescharacteristic
102Q8×Dic61C2312yes—
103C4×Dic122—21no—
104C4×Dic122—22no—
105C4×Dic122C2×C411yescharacteristic
106C4×Dic122C2×C412yes—
107#14519742—22no—
108#14519742—21no—
109#14519742—22no—
110#14519742—21no—
111#14519742C2311yescharacteristic
112C4×D244—22no—
113C4×D244—21no—
114C4×D244C2×C412yes—
115C4×D244C2×C411yescharacteristic
116#14519739—22no—
117#14519739—21no—
118#14519739D412yes—
119#14519739D411yescharacteristic
120#14519735—22no—
121#14519735—21no—
122#14519735D411yescharacteristic
123#14519735D412yes—
124C4⋊C4×D61—21no—
125C4⋊C4×D61—22no—
126C4⋊C4×D61C2×C411yescharacteristic
127C4⋊C4×D61C2×C412yes—
128C42×D61—22no—
129C42×D61—21no—
130C42×D61C2311yescharacteristic
131C42×D61C2312yes—
132C4⋊Q8×C61C2×C411yescharacteristic
133C4⋊Q8×C61C2×C412yes—
134#125×C61—22no—
135#125×C61—21no—
136C4×Q8×C61—21no—
137C4×Q8×C61C2×C411yescharacteristic
138C4×Q8×C61C2×C412yes—
139C4×Q8×C61C2×C411yescharacteristic
140C2×Q8×C61C22⋊C411yescharacteristic
141C2×Q8×C61C22×C411yescharacteristic
142C2×Q8×C61C22×C412yes—
143#11758226—22no—
144#11758226—22no—
145Q8×D61—22no—
146Q8×D61—24no—
147Q8×D61—41no—
148Q8×D61—42no—
149C2×D244C22⋊C412yes—
150C2×D244C22⋊C411yescharacteristic
151C2×C4×D61—22no—
152C2×C4×D61—21no—
153C2×C4×D61—22no—
154C2×C4×D61—21no—
155C2×C4×D61C22×C412yes—
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D411yescharacteristic
161C2×Dic122—22no—
162C2×Dic122—21no—
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122C22×C411yescharacteristic
166C2×#92420—21no—
167C2×#92420—21no—
168C2×#92420—21no—
169C2×#92420—22no—
170C2×#92420—22no—
171C2×#92420—21no—
172#11758213—44no—
173#11758213—42no—
174D61⋊C8—42no—
175D61⋊C8—42no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61C22⋊C412yes—
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22×C411yescharacteristic
185C42×C61—22no—
186C42×C61—21no—
187C42×C61C22×C412yes—
188C42×C61C22×C411yescharacteristic
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198C22⋊C412yes—
193#11758198C22⋊C412yes—
194#11758198C22⋊C412yes—
195C4⋊Dic61—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22⋊C412yes—
200C4⋊Dic61C22⋊C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204#11758196—21no—
205#11758196—22no—
206#11758196—21no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—21no—
211C4×Dic61—22no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—22no—
215C4×Dic61—21no—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2411yescharacteristic
221C22×D61C2×C22⋊C412yes—
222C22×D61C2×C22⋊C411yescharacteristic
223Q8×C61—22no—
224Q8×C61—21no—
225Q8×C61—22no—
226Q8×C61—21no—
227Q8×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61C2×C22⋊C412yes—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C23×C411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—22no—
238C2×Dic61—21no—
239C2×Dic61C2×C22⋊C412yes—
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C22×D411yescharacteristic
242C2×Dic61C22×D411yescharacteristic
243D244—24no—
244D244—22no—
245C4×D61—21no—
246C4×D61—22no—
247C4×D61—21no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—41no—
252C4×D61—42no—
253C4×D61—42no—
254C4×D61—44no—
255C4×D61—81no—
256C4×D61—82no—
257C4×D61#13712yes—
258C4×D61#13712yes—
259Dic122—21no—
260Dic122—42no—
261Dic122—44no—
262Dic122—81no—
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×C61C22×C22⋊C411yescharacteristic
273C2×D61—21no—
274C2×D61—21no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61#41912yes—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—22no—
282C4×C61—41no—
283C4×C61—42no—
284C4×C61—41no—
285C4×C61#41912yes—
286C4×C61C2×#13111yescharacteristic
287C4×C61#56811yescharacteristic
288Dic61—22no—
289Dic61—21no—
290Dic61—22no—
291Dic61—21no—
292Dic61—41no—
293Dic61—42no—
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#288011yescharacteristic
308C2×C61C2×#41911yescharacteristic
309C2×C61#290111yescharacteristic
310D61—22no—
311D61—42no—
312Q82—611no—
313#557—1221no—
314#557#692311yescharacteristic
315#554—612no—
316#554—611no—
317#554—1222no—
318#554—1221no—
319D4×Q8—611no—
320D4×Q8—612no—
321#533—611no—
322#533—612no—
323#533—1221no—
324#533—1222no—
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—612no—
347Q8⋊C8—612no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—611no—
351Q8⋊C8—611no—
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—611no—
362C4⋊Q8—612no—
363C4⋊Q8—1222no—
364C4⋊Q8—1221no—
365C4⋊Q8—1222no—
366C4⋊Q8—1221no—
367C4⋊Q8C2×#692311yescharacteristic
368C4⋊Q8C2×#692312yes—
369#125—22no—
370#125—21no—
371#125—1222no—
372#125—1221no—
373#125—1221no—
374#125—1222no—
375C22⋊Q8—612no—
376C22⋊Q8—612no—
377C22⋊Q8—612no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1221no—
385C22⋊Q8—1222no—
386C4×Q8—21no—
387C4×Q8—611no—
388C4×Q8—611no—
389C4×Q8—612no—
390C4×Q8—611no—
391C4×Q8—611no—
392C4×Q8—612no—
393C4×Q8—1221no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1221no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8C2×#692311yescharacteristic
401C4×Q8C2×#692312yes—
402C4×Q8C2×#692311yescharacteristic
403C4×D4—612no—
404C4×D4—611no—
405C4×D4—612no—
406C4×D4—611no—
407C4×D4—1222no—
408C4×D4—1221no—
409C4×D4—1222no—
410C4×D4—1221no—
411#117—611no—
412#117—612no—
413#117—1221no—
414#117—1222no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1222no—
419C2×C42—611no—
420C2×C42—612no—
421C2×C42—1221no—
422C2×C42—1222no—
423C4⋊C8—611no—
424C4⋊C8—611no—
425C4⋊C8—612no—
426C4⋊C8—1221no—
427C4⋊C8—1221no—
428C4⋊C8—1221no—
429C4⋊C8—1222no—
430C22⋊C8—1222no—
431C22⋊C8—1222no—
432C22⋊C8—1221no—
433C22⋊C8—1222no—
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—611no—
444C2×Q8—1222no—
445C2×Q8—1221no—
446C2×Q8—1222no—
447C2×Q8—1221no—
448C2×Q8—1222no—
449C2×Q8—1224no—
450C2×Q8—2441no—
451C2×Q8—2442no—
452C2×Q8C22⋊#692311yescharacteristic
453C2×Q8C22×#692311yescharacteristic
454C2×Q8C22×#692312yes—
455C2×D4—612no—
456C2×D4—611no—
457C22×C4—611no—
458C22×C4—612no—
459C22×C4—611no—
460C22×C4—612no—
461C22×C4—611no—
462C22×C4—612no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1221no—
466C22×C4—1222no—
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—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—612no—
484C4⋊C4—611no—
485C4⋊C4—611no—
486C4⋊C4—612no—
487C4⋊C4—1222no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1222no—
498C4⋊C4—1221no—
499C4⋊C4C22⋊#692312yes—
500C4⋊C4C22⋊#692311yescharacteristic
501C4⋊C4C22×#692311yescharacteristic
502C22⋊C4—612no—
503C22⋊C4—612no—
504C22⋊C4—612no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C42—22no—
509C42—21no—
510C42—612no—
511C42—611no—
512C42—612no—
513C42—611no—
514C42—611no—
515C42—1221no—
516C42—1221no—
517C42—1222no—
518C42—1222no—
519C42—1221no—
520C42C22×#692312yes—
521C42C22×#692311yescharacteristic
522C23—611no—
523C23—612no—
524Q8—22no—
525Q8—21no—
526Q8—22no—
527Q8—21no—
528Q8—22no—
529Q8—612no—
530Q8—1221no—
531Q8—4881no—
532Q8—4882no—
533Q8—2442no—
534Q8—2444no—
535D4—1224no—
536D4—1222no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—611no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—611no—
545C2×C4—611no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1222no—
551C2×C4—1221no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1221no—
555C2×C4—4881no—
556C2×C4—4882no—
557C2×C4—2441no—
558C2×C4—2442no—
559C2×C4—2442no—
560C2×C4—2444no—
561C2×C4C2×C22⋊#692312yes—
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—612no—
573C22—1221no—
574C22—1221no—
575C22—1222no—
576C22—1222no—
577C22—1222no—
578C22C22×C22⋊#692311yescharacteristic
579C4—22no—
580C4—21no—
581C4—22no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—611no—
586C4—611no—
587C4—1222no—
588C4—1221no—
589C4—1222no—
590C4—1221no—
591C4—2441no—
592C4—2442no—
593C4#1684075512yes—
594C4D61⋊#13111yescharacteristic
595C4#1684084111yescharacteristic
596C2—1222no—
597C2—2442no—
598C2#2146241011yescharacteristic
599C2C2×#1684075511yescharacteristic
600C2#2146245011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#557C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758226D4
cosocleC4×Dic61C24
hypercenter#557#6923
hypocenterC61#17514
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
3. #21461182C2semidirect
4. #21461168C2semidirect
5. #21461162C2semidirect
6. #21461148C2semidirect
7. #21461142C2semidirect
8. #21461128C2semidirect
9. #21461122C2semidirect
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
#214626272
#214625192
#214611822
#214611682
#214611622
#214611482
#214611422
#214611282
#214611222
D61⋊(Q8⋊C8)2
#214594012
#1751461