Finite Group Databasebeta

← #30613129

Subgroups — #30613129

Order 15616.

680 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#21462614C211yesmaximal, characteristic, complemented
2D4×#11758217C211yesmaximal, characteristic
3#21461456C211yesmaximal, characteristic, complemented
4#21461455C211yesmaximal, characteristic, complemented
5#21461429C211yesmaximal, characteristic, complemented
6#21461428C211yesmaximal, characteristic, complemented
7#21461424C211yesmaximal, characteristic, complemented
8#21461422C211yesmaximal, characteristic, complemented
9#21461402C211yesmaximal, characteristic, complemented
10SD16×#92420C211yesmaximal, characteristic, complemented
11#21461386C211yesmaximal, characteristic, complemented
12#21461167C211yesmaximal, characteristic, complemented
13C2×D4⋊#92420C211yesmaximal, characteristic
14#21461020C211yesmaximal, characteristic, complemented
15#21455822C211yesmaximal, characteristic
16C22×#11758217—21no—
17#16840814—21no—
18#16840813C2211yescharacteristic, complemented
19#16840812—21no—
20#16840812C2211yescharacteristic, complemented
21#16840811C2211yescharacteristic, complemented
22Q8×#92420C2211yescharacteristic, complemented
23#16840797C2211yescharacteristic
24#16840797C2211yescharacteristic
25#16840793—21no—
26#16840793—21no—
27D4×#92420C2211yescharacteristic
28D4×#92420C2211yescharacteristic
29#16840768—21no—
30#16840765C2211yescharacteristic
31C2×C4⋊#92420C2211yescharacteristic
32#16840732—21no—
33#16840726—21no—
34C4×#11758217C2211yescharacteristic
35C2×C4×#92420C2211yescharacteristic
36#16840624C2211yescharacteristic, complemented
37#16840624C2211yescharacteristic, complemented
38#16840623C2211yescharacteristic, complemented
39#16840623C2211yescharacteristic, complemented
40#16840612C2211yescharacteristic, complemented
41#16840612C2211yescharacteristic, complemented
42Q8⋊#92420C2211yescharacteristic, complemented
43Q8⋊#92420C2211yescharacteristic, complemented
44D4⋊#92420C2211yescharacteristic
45D4⋊#92420C2211yescharacteristic
46D4⋊#92420C2211yescharacteristic
47D4⋊#92420C2211yescharacteristic
48#92420⋊C8C2211yescharacteristic, complemented
49#92420⋊C8C2211yescharacteristic, complemented
50C8×#92420C2211yescharacteristic, complemented
51C8×#92420C2211yescharacteristic, complemented
52#16840314C2211yescharacteristic
53C2×D4×Dic61C2211yescharacteristic
54#16840093C411yescharacteristic
55#16840090C411yescharacteristic
56#16840050C411yescharacteristic
57#16840050C411yescharacteristic
58#16839965C411yescharacteristic
59#16839965C411yescharacteristic
60#16839913C411yescharacteristic
61#16839853C2211yescharacteristic
62#16839853C2211yescharacteristic
63SD16×Dic61C2211yescharacteristic
64SD16×Dic61C2211yescharacteristic
65#16839812C2211yescharacteristic
66C61×#450C411yescharacteristic
67C2×#11758217—21no—
68C2×#11758217—21no—
69C2×#11758217—21no—
70C2×#11758217—44no—
71C2×#11758217D411yescharacteristic
72C2×#11758217D411yescharacteristic
73C22×#92420—21no—
74C22×#92420—21no—
75C22×#92420—21no—
76C22×#92420—21no—
77#14519844—21no—
78#14519844—21no—
79#14519844—21no—
80C4⋊#92420—21no—
81C4⋊#92420—21no—
82C4⋊#92420—21no—
83C4⋊#92420C2311yescharacteristic
84C4⋊#92420C2311yescharacteristic
85C4⋊#92420C2311yescharacteristic
86C4⋊#92420C2311yescharacteristic
87C22⋊#92420—21no—
88C22⋊#92420—21no—
89C22⋊#92420—21no—
90C22⋊#92420—21no—
91#14519832—21no—
92#14519832—21no—
93#14519832D411yescharacteristic
94#14519832D411yescharacteristic
95C4×#92420—21no—
96C4×#92420—21no—
97C4×#92420—21no—
98C4×#92420C2311yescharacteristic
99C4×#92420C2311yescharacteristic
100C4×#92420C2311yescharacteristic
101C4×#92420C2311yescharacteristic
102C23×Dic61—21no—
103C2×#11758200C2×C411yescharacteristic
104C2×#11758200C2×C411yescharacteristic
105#14519776C2×C411yescharacteristic
106#14519775C2×C411yescharacteristic
107#14519767—21no—
108#14519756C2×C411yescharacteristic
109#14519754—21no—
110Q8×Dic61C2311yescharacteristic
111#14519742—21no—
112#14519742C2311yescharacteristic
113D4×Dic61—21no—
114D4×Dic61—21no—
115D4×Dic61C2×C411yescharacteristic
116D4×Dic61C2×C411yescharacteristic
117D4×Dic61C2311yescharacteristic
118D4×Dic61C2311yescharacteristic
119#14519733—21no—
120C2×C4⋊Dic61C2×C411yescharacteristic
121C2×C22⋊Dic61—21no—
122C2×C4×Dic61C2311yescharacteristic
123#14519709C2×C411yescharacteristic
124#14519709C2×C411yescharacteristic
125#14519708C2×C411yescharacteristic
126#14519708C2×C411yescharacteristic
127#14519706C2×C411yescharacteristic
128#14519706C2×C411yescharacteristic
129#14519702C2×C411yescharacteristic
130#14519702C2×C411yescharacteristic
131#14519701C2×C411yescharacteristic
132#14519701C2×C411yescharacteristic
133#14519698C2×C411yescharacteristic
134#14519698C2×C411yescharacteristic
135C22⋊#92418C2×C411yescharacteristic
136C8×Dic61C2311yescharacteristic
137C8×Dic61C2311yescharacteristic
138C22×D4×C61C2×C411yescharacteristic
139C2×SD16×C61C2×C411yescharacteristic
140C2×SD16×C61C2×C411yescharacteristic
141C22⋊Q8×C61C2×C411yescharacteristic
142#102×C61C2×C411yescharacteristic
143#102×C61C2×C411yescharacteristic
144C22⋊C8×C61C2×C411yescharacteristic
145C24×C61—21no—
146C2×Q8×C61C22×C411yescharacteristic
147C2×D4×C61—22no—
148C2×D4×C61—21no—
149C2×D4×C61—41no—
150C2×D4×C61C22×C411yescharacteristic
151C2×D4×C61C22×C411yescharacteristic
152C22×C4×C61C22×C411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61#4111yescharacteristic
162C2×Dic122—21no—
163C2×Dic122C22×C411yescharacteristic
164#11758217—22no—
165#11758217—42no—
166#11758217—42no—
167#11758217—42no—
168#11758217SD1614yescomplemented
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—21no—
173C2×#92420—21no—
174C2×#92420—21no—
175C2×#92420—42no—
176C2×#92420—42no—
177C2×#92420—42no—
178C2×#92420—42no—
179C2×#92420C2×D411yescharacteristic
180C2×#92420C2×D411yescharacteristic
181C2×#92420C2×D411yescharacteristic
182C2×#92420C2×D411yescharacteristic
183SD16×C61—22no—
184SD16×C61—41no—
185C2×C8×C61C22×C411yescharacteristic
186C2×C8×C61C22×C411yescharacteristic
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22×C411yescharacteristic
189C22⋊C4×C61—21no—
190C22⋊Dic61—21no—
191C22⋊Dic61—21no—
192C22⋊Dic61—21no—
193C22⋊Dic61—21no—
194C22⋊Dic61—21no—
195#11758200—22no—
196#11758200—41no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61C22×C411yescharacteristic
200C4⋊Dic61C22×C411yescharacteristic
201C4⋊Dic61C22×C411yescharacteristic
202#11758196—21no—
203#11758196—21no—
204#11758196—21no—
205#11758196—21no—
206#11758196—21no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61C22×C411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61#4111yescharacteristic
212C4×Dic61C2411yescharacteristic
213C2×#92418C22×C411yescharacteristic
214C2×#92418C22×C411yescharacteristic
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—81no—
220C23×C61C4×D411yescharacteristic
221Q8×C61—21no—
222Q8×C61—21no—
223D4×C61—21no—
224D4×C61—21no—
225D4×C61—21no—
226D4×C61—42no—
227D4×C61#13112yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61C4×D411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—41no—
239C2×Dic61—42no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243C2×Dic61—41no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61C4×D411yescharacteristic
247C2×Dic61C4×D411yescharacteristic
248C2×Dic61C2×SD1612yes—
249C2×Dic61#13512yes—
250C2×Dic61C22×D411yescharacteristic
251C2×Dic61C2×#4111yescharacteristic
252Dic122—21no—
253Dic122—21no—
254Dic122—81no—
255#92420—22no—
256#92420—21no—
257#92420—21no—
258#92420—41no—
259#92420—41no—
260#92420—41no—
261#92420—41no—
262#92420—41no—
263#92420C2×SD1612yes—
264#92420C2×SD1612yes—
265C8×C61—21no—
266C8×C61—21no—
267#92418—21no—
268#92418—21no—
269#17995—611nomaximal, Sylow
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—22no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61—41no—
278C22×C61—84no—
279C22×C61C4×SD1612yes—
280C22×C61C2×C4×D411yescharacteristic
281C4×C61—21no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61C2×#13111yescharacteristic
285C4×C61#56811yescharacteristic
286Dic61—21no—
287Dic61—21no—
288Dic61—41no—
289Dic61—41no—
290Dic61—41no—
291Dic61—41no—
292Dic61C4×SD1612yes—
293Dic61C22×SD1611yescharacteristic
294Dic61C2×#13511yescharacteristic
295#3101—611no—
296#2962—611no—
297D4×M16—611no—
298#1590—611no—
299#1590—611no—
300#1583—611no—
301#1583—611no—
302#1582—611no—
303#1582—611no—
304C8×SD16—611no—
305C8×SD16—611no—
306#1570—611no—
307#1493—611no—
308C2×D4⋊C8—611no—
309C8×M16—611no—
310C2×C61—22no—
311C2×C61—42no—
312C2×C61—81no—
313C2×C61#293311yescharacteristic
314C2×C61C2×C4×SD1611yescharacteristic
315C2×C61#295611yescharacteristic
316C22×M16—1221no—
317#533—611no—
318C2×C4×D4—611no—
319#483—611no—
320#483—611no—
321#476—611no—
322#476—611no—
323#461—611no—
324#461—611no—
325#450—611no—
326#450#692311yescharacteristic
327#446—611no—
328#446—611no—
329#446—1221no—
330#446—1221no—
331C8×Q8—611no—
332C8×Q8—611no—
333C4×SD16—611no—
334C4×SD16—611no—
335C4×SD16—611no—
336C4×SD16—611no—
337#436—611no—
338#436—611no—
339#435—1221no—
340#435—1221no—
341C8×D4—611no—
342C8×D4—611no—
343#431—611no—
344#424—1221no—
345#423—611no—
346C2×C4⋊C8—611no—
347#407—1221no—
348#405—1221no—
349C4×M16—611no—
350C2×C4×C8—611no—
351#334—611no—
352#334—611no—
353#334—611no—
354#334—611no—
355Q8⋊C8—611no—
356Q8⋊C8—611no—
357Q8⋊C8—611no—
358Q8⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363#322—611no—
364#322—611no—
365C82—611no—
366C82—611no—
367C61#1799511yescharacteristic, complemented, Sylow
368C22×D4C2×#692311yescharacteristic
369C23×C4—1221no—
370C2×SD16—611no—
371C2×SD16—611no—
372C2×SD16C2×#692311yescharacteristic
373C2×SD16C2×#692311yescharacteristic
374C2×M16—611no—
375C2×M16—611no—
376C2×M16—1221no—
377C2×M16—1221no—
378C2×M16—1221no—
379C2×M16—2444no—
380C22×C8—1221no—
381C22×C8—1221no—
382C22×C8—1221no—
383C22×C8—1221no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386#125—1221no—
387C22⋊Q8—611no—
388C22⋊Q8—1221no—
389C22⋊Q8C2×#692311yescharacteristic
390C4×Q8—611no—
391C4×Q8—611no—
392C4×Q8—1221no—
393C4×D4—611no—
394C4×D4—611no—
395C4×D4—611no—
396C4×D4—611no—
397C4×D4—1221no—
398C4×D4—1221no—
399#117—1221no—
400C2×C4⋊C4—611no—
401C2×C22⋊C4—1221no—
402C2×C42—611no—
403#106—611no—
404#106—611no—
405#106—611no—
406#106—611no—
407C4⋊C8—611no—
408C4⋊C8—611no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—611no—
413C4⋊C8—1221no—
414C4⋊C8—1221no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417C4⋊C8—1221no—
418C4⋊C8—1221no—
419#103—611no—
420#103—611no—
421#103—611no—
422#103—611no—
423#102—611no—
424#102—611no—
425#102C2×#692311yescharacteristic
426#102C2×#692311yescharacteristic
427C22⋊C8—611no—
428C22⋊C8—1221no—
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431C22⋊C8—1221no—
432C22⋊C8C2×#692311yescharacteristic
433#97—611no—
434#97—611no—
435#97—1221no—
436#97—1221no—
437C4×C8—611no—
438C4×C8—611no—
439C4×C8—611no—
440C4×C8—611no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8—1221no—
444C4×C8—1221no—
445C4×C8—1221no—
446C24—21no—
447C2×Q8—611no—
448C2×Q8—1221no—
449C2×Q8C22×#692311yescharacteristic
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—41no—
453C2×D4C22×#692311yescharacteristic
454C2×D4C22×#692311yescharacteristic
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4C22×#692311yescharacteristic
465SD16—22no—
466SD16—41no—
467SD16—1222no—
468SD16—2441no—
469M16—614no—
470M16—1222no—
471M16—2442no—
472M16—2442no—
473M16—2442no—
474C2×C8—611no—
475C2×C8—611no—
476C2×C8—611no—
477C2×C8—611no—
478C2×C8—611no—
479C2×C8—611no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—2442no—
487C2×C8—2442no—
488C2×C8—2442no—
489C2×C8—2442no—
490C2×C8C22×#692311yescharacteristic
491C2×C8C22×#692311yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—611no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4C22×#692311yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—1221no—
506C22⋊C4—1221no—
507C22⋊C4—1221no—
508C22⋊C4—1221no—
509C22⋊C4—1221no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—1221no—
515C42—1221no—
516C23—21no—
517C23—21no—
518C23—42no—
519C23—42no—
520C23—81no—
521C23D4×#692311yescharacteristic
522Q8—21no—
523Q8—21no—
524Q8—1221no—
525Q8—1221no—
526Q8—4881no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—42no—
531D4#1451986412yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2441no—
553C2×C4—2442no—
554C2×C4D4×#692311yescharacteristic
555C2×C4C23×#692311yescharacteristic
556C8—21no—
557C8—21no—
558C8—612no—
559C8—612no—
560C8—1221no—
561C8—1221no—
562C8—1222no—
563C8—1221no—
564C8—1221no—
565C8—2441no—
566C8—2441no—
567C8—2441no—
568C8—2441no—
569C8—2441no—
570C22—21no—
571C22—21no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—84no—
579C22SD16×#692312yes—
580C22C2×D4×#692311yescharacteristic
581C4—21no—
582C4—41no—
583C4—41no—
584C4—611no—
585C4—611no—
586C4—612no—
587C4—1221no—
588C4—1221no—
589C4—2441no—
590C4—2441no—
591C4—2441no—
592C4—2441no—
593C4C2×#1451986411yescharacteristic
594C4#1684084011yescharacteristic
595C2—22no—
596C2—42no—
597C2—81no—
598C2#2146255111yescharacteristic
599C2C2×SD16×#692311yescharacteristic
600C2#2146259311yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#450C4
socleC22×C61C2×C4×D4
prime radicalC22×D4×C61C2×C4
cosocleC4×Dic61C24
hypercenter#450#6923
hypocenterC61#17995
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #21462614C2semidirect
3. #21461456C2semidirect
4. #21461455C2semidirect
5. #21461429C2semidirect
6. #21461428C2semidirect
7. #21461424C2semidirect
8. #21461422C2semidirect
9. #21461402C2semidirect
10. SD16×#92420C2semidirect
11. #21461386C2semidirect
12. #21461167C2semidirect
14. #21461020C2semidirect
18. #16840813C22semidirect
20. #16840812C22semidirect
21. #16840811C22semidirect
22. Q8×#92420C22semidirect
36. #16840624C22semidirect
37. #16840624C22semidirect
38. #16840623C22semidirect
39. #16840623C22semidirect
40. #16840612C22semidirect
41. #16840612C22semidirect
42. Q8⋊#92420C22semidirect
43. Q8⋊#92420C22semidirect
48. #92420⋊C8C22semidirect
49. #92420⋊C8C22semidirect
50. C8×#92420C22semidirect
51. C8×#92420C22semidirect
168. #11758217SD16semidirect
367. C61#17995semidirect

Maximal subgroups

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

subgroupindex
#214626142
D4×#117582172
#214614562
#214614552
#214614292
#214614282
#214614242
#214614222
#214614022
SD16×#924202
#214613862
#214611672
C2×D4⋊#924202
#214610202
#214558222
#1799561