Finite Group Databasebeta

← #29670196

Subgroups — #29670196

Order 15616.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458714C211yesmaximal, characteristic
2Q8×#11758199C211yesmaximal, characteristic
3#21457697C211yesmaximal, characteristic, complemented
4#21445494C211yesmaximal, characteristic, complemented
5#21445335C211yesmaximal, characteristic, complemented
6#21445270C211yesmaximal, characteristic, complemented
7#21445199C211yesmaximal, characteristic, complemented
8#21445181C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445066C211yesmaximal, characteristic, complemented
11#21445031C211yesmaximal, characteristic, complemented
12#21444984C211yesmaximal, characteristic, complemented
13#21444670C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic
15D4×Q16×C61C211yesmaximal, characteristic, complemented
16#16840472C2211yescharacteristic
17Q8×D244C2211yescharacteristic
18#16840168—21no—
19#16840165—21no—
20#16840165C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840158C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840113C2211yescharacteristic
25#16840099—21no—
26#16840062—21no—
27#16840057C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840049—21no—
30#16840042—21no—
31#16840042C2211yescharacteristic
32#16840039—21no—
33#16840039C2211yescharacteristic
34#16840039C2211yescharacteristic
35#16840002—21no—
36#16839972C2211yescharacteristic
37#16839916—21no—
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839872—21no—
41C4×#11758199—21no—
42C4×#11758199C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#103×D61—21no—
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839510C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56C22×Q16×C61—21no—
57D4×Q8×C61C2211yescharacteristic
58D4×Q8×C61C2211yescharacteristic
59C61×#494C2211yescharacteristic
60C61×#467—21no—
61C61×#462C2211yescharacteristic
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C61×#451—21no—
65C4×Q16×C61C2211yescharacteristic
66C8×D4×C61C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×#11758202—21no—
69C2×#11758199D411yescharacteristic
70C2×C8×D61—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519769—21no—
75#14519757—21no—
76#14519757—41no—
77#14519755—41no—
78#14519753—21no—
79C4×D244—21no—
80C4×D244—21no—
81C4×D244C2311yescharacteristic
82C4⋊C4×D61—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—41no—
90#14519710D411yescharacteristic
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519700—21no—
103#14519700—41no—
104#14519698—21no—
105#14519698—21no—
106#14519698D411yescharacteristic
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111#14519688—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C22×Q8×C61—21no—
117C2×Q16×C61—21no—
118C2×Q16×C61—21no—
119C2×Q16×C61—21no—
120C2×Q16×C61—21no—
121C2×Q16×C61—44no—
122C2×Q16×C61D411yescharacteristic
123C22×C8×C61—21no—
124C4⋊Q8×C61—21no—
125C4⋊Q8×C61—21no—
126C4⋊Q8×C61C2311yescharacteristic
127C4⋊Q8×C61C2311yescharacteristic
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61—21no—
130C22⋊Q8×C61—41no—
131C22⋊Q8×C61—41no—
132C4×Q8×C61C2311yescharacteristic
133C4×Q8×C61C2311yescharacteristic
134C4×D4×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61C2311yescharacteristic
137#107×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#103×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—41no—
149C2×Q8×C61—42no—
150C2×Q8×C61—42no—
151C2×Q8×C61—81no—
152C2×Q8×C61—81no—
153C2×Q8×C61C2×D411yescharacteristic
154C2×Q8×C61C2×D411yescharacteristic
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—41no—
158C22×C4×C61—41no—
159Q8×D61—42no—
160Q8×D61—81no—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168Q16×C61—22no—
169Q16×C61—42no—
170Q16×C61—42no—
171Q16×C61—42no—
172Q16×C61—81no—
173C2×C8×C61—21no—
174C2×C8×C61—42no—
175C2×C8×C61C2×D411yescharacteristic
176C2×C8×C61C2×D411yescharacteristic
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—41no—
184C4⋊C4×C61—41no—
185C4⋊C4×C61C2×D411yescharacteristic
186C4⋊C4×C61C2×D411yescharacteristic
187C4⋊C4×C61C2×D411yescharacteristic
188C22⋊C4×C61—21no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61C2411yescharacteristic
194#11758202—81no—
195#11758199Q1614yescomplemented
196#11758198—21no—
197#11758198—41no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—41no—
204C4⋊Dic61C2×D411yescharacteristic
205#11758196—41no—
206#11758196—41no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213C8×D61—81no—
214C23×C61—21no—
215C22×D61—21no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—82no—
224Q8×C61—82no—
225Q8×C61AGL(1,ℤ/8)12yes—
226D4×C61—21no—
227D4×C61C2×Q1612yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61—82no—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245D244—21no—
246D244C2×Q1612yescomplemented
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—82no—
250C4×D61—161no—
251Dic122—41no—
252Dic122—41no—
253Dic122—82no—
254Dic122—81no—
255C8×C61—21no—
256C8×C61—21no—
257C8×C61—41no—
258#92418—21no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418—41no—
263#92418C2×Q1612yescomplemented
264#19811—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61D4211yescharacteristic
269C2×D61—22no—
270C2×D61—41no—
271C2×D61—41no—
272C4×C61—22no—
273C4×C61—21no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61—81no—
280C4×C61C22×Q1611yescharacteristic
281C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
282C4×C61#57611yescharacteristic
283C4×C61#57811yescharacteristic
284Dic61—41no—
285Dic61—81no—
286Dic61—81no—
287#3384—611no—
288Q8×D8—611no—
289#3299—611no—
290D4×Q16D6111yescharacteristic, complemented
291#1723—611no—
292#1694—611no—
293#1683—611no—
294#1672—611no—
295#1669—611no—
296#1649—611no—
297#1649—611no—
298#1644—611no—
299#1637—611no—
300#1584—611no—
301#1580—611no—
302C2×C61—42no—
303C2×C61#326411yescharacteristic
304C2×C61D4×Q1611yescharacteristic
305C2×C61#328911yescharacteristic
306D61—42no—
307C22×Q16—21no—
308#554—611no—
309D4×Q8—611no—
310D4×Q8C2×D6111yescharacteristic
311D4×Q8C2×D6111yescharacteristic
312#501—1221no—
313#500—611no—
314#500—611no—
315#500—1221no—
316#498—611no—
317#494C2×D6111yescharacteristic
318#487—611no—
319#487—611no—
320#484—1221no—
321#478—1221no—
322#477—611no—
323#476—611no—
324#476—1221no—
325#474—611no—
326#474—611no—
327#474—611no—
328#474—1221no—
329#474—1221no—
330#467—21no—
331#467—1221no—
332#462—611no—
333#462C2×D6111yescharacteristic
334#462C2×D6111yescharacteristic
335#451—21no—
336#451—21no—
337#451—1221no—
338#446—611no—
339C8×Q8—611no—
340#442—1221no—
341C4×Q16C2×D6111yescharacteristic
342C4×D8—611no—
343C4×D8—1221no—
344C8×D4—611no—
345C8×D4C2×D6111yescharacteristic
346C2×#103—1221no—
347#335—611no—
348#334—611no—
349#333—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353Q8⋊C8—611no—
354Q8⋊C8—611no—
355D4⋊C8—611no—
356D4⋊C8—611no—
357#322—611no—
358C61#1981111yescharacteristic, complemented, Sylow
359C22×Q8—21no—
360C22×Q8—21no—
361C22×Q8—1221no—
362C2×Q16—21no—
363C2×Q16—21no—
364C2×Q16—21no—
365C2×Q16—21no—
366C2×Q16—44no—
367C2×Q16—1221no—
368C2×Q16C22⋊D6111yescharacteristic
369C2×D8—611no—
370C22×C8—21no—
371C22×C8—1221no—
372C4⋊Q8—21no—
373C4⋊Q8—21no—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376C4⋊Q8—1221no—
377C4⋊Q8C22×D6111yescharacteristic
378C4⋊Q8C22×D6111yescharacteristic
379#125—1221no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—41no—
383C22⋊Q8—41no—
384C22⋊Q8—1221no—
385C22⋊Q8—1221no—
386C22⋊Q8—2441no—
387C22⋊Q8—2441no—
388C4×Q8C22×D6111yescharacteristic
389C4×Q8C22×D6111yescharacteristic
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—611no—
393C4×D4—1221no—
394C4×D4—1221no—
395C4×D4C22×D6111yescharacteristic
396C2×C4⋊C4—1221no—
397#107—611no—
398#107—1221no—
399#107—1221no—
400#107—1221no—
401#107—1221no—
402#107—1221no—
403#107—1221no—
404#107—2441no—
405#107C22⋊D6111yescharacteristic
406#106—1221no—
407#106—1221no—
408#106—1221no—
409C4⋊C8—611no—
410C4⋊C8—611no—
411C4⋊C8—611no—
412C4⋊C8—1221no—
413C4⋊C8—1221no—
414C4⋊C8C22×D6111yescharacteristic
415#103—21no—
416#103—21no—
417#103—1221no—
418#103—1221no—
419#103—1221no—
420#103—1221no—
421#103—2441no—
422#103C22⋊D6111yescharacteristic
423#103C22⋊D6111yescharacteristic
424#102—611no—
425#102—611no—
426#102—1221no—
427#102—1221no—
428#102—1221no—
429#102—1221no—
430C22⋊C8—21no—
431C22⋊C8—1221no—
432#97—1221no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8—1221no—
436C4×C8C22×D6111yescharacteristic
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—41no—
440C2×Q8—41no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—81no—
444C2×Q8—81no—
445C2×Q8—1221no—
446C2×Q8—1221no—
447C2×Q8—4881no—
448C2×Q8—2441no—
449C2×Q8—2442no—
450C2×Q8C2×C22⋊D6111yescharacteristic
451C2×Q8C2×C22⋊D6111yescharacteristic
452C2×D4—611no—
453C2×D4D4×D6111yescharacteristic
454C22×C4—21no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—1221no—
458C22×C4—2441no—
459C22×C4—2441no—
460Q16—22no—
461Q16—42no—
462Q16—42no—
463Q16—42no—
464Q16—81no—
465Q16—4881no—
466D8—614no—
467C2×C8—21no—
468C2×C8—42no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—4881no—
476C2×C8C2×C22⋊D6111yescharacteristic
477C2×C8C2×C22⋊D6111yescharacteristic
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—611no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—2441no—
492C4⋊C4—2441no—
493C4⋊C4—2441no—
494C4⋊C4D4×D6111yescharacteristic
495C4⋊C4C2×C22⋊D6111yescharacteristic
496C4⋊C4C2×C22⋊D6111yescharacteristic
497C22⋊C4—21no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—1221no—
501C22⋊C4—2441no—
502C22⋊C4—2441no—
503C42—21no—
504C42—21no—
505C42C23×D6111yescharacteristic
506C23—21no—
507C23—1221no—
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—41no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—82no—
516Q8—82no—
517Q8—4881no—
518Q8—4882no—
519Q8—2441no—
520Q8—2441no—
521Q8#1451980312yes—
522D4—21no—
523D4—612no—
524D4—1221no—
525D4Q16×D6112yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—1221no—
538C2×C4—9761no—
539C2×C4—4882no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4C2×D4×D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4#1451982211yescharacteristic
547C8—21no—
548C8—21no—
549C8—41no—
550C8—612no—
551C8—1221no—
552C8—2441no—
553C8—2441no—
554C8—2441no—
555C8—2441no—
556C22—22no—
557C22—41no—
558C22—41no—
559C22—1222no—
560C22—2441no—
561C22—2441no—
562C22D4×C22⋊D6111yescharacteristic
563C4—22no—
564C4—21no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—81no—
570C4—81no—
571C4—4881no—
572C4—4881no—
573C4—2441no—
574C4C2×Q16×D6111yescharacteristic
575C4C2×#1451980311yescharacteristic
576C4#1684056511yescharacteristic
577C4#1684057511yescharacteristic
578C2—42no—
579C2—2442no—
580C2#2145729511yescharacteristic
581C2Q16×C22⋊D6111yescharacteristic
582C2#2145759011yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingD4×Q16×C61C2
socleC22×C61D42
prime radicalC2×#11758199D4
hypercenterD4×Q16D61
hypocenterC61#19811
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ D4×Q16

Complements

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

NH ≅ G/Ntype
3. #21457697C2semidirect
4. #21445494C2semidirect
5. #21445335C2semidirect
6. #21445270C2semidirect
7. #21445199C2semidirect
8. #21445181C2semidirect
9. #21445068C2semidirect
10. #21445066C2semidirect
11. #21445031C2semidirect
12. #21444984C2semidirect
13. #21444670C2semidirect
15. D4×Q16×C61C2semidirect
195. #11758199Q16semidirect
246. D244C2×Q16semidirect
263. #92418C2×Q16semidirect
290. D4×Q16D61semidirect
358. C61#19811semidirect

Maximal subgroups

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

subgroupindex
#214587142
Q8×#117581992
#214576972
#214454942
#214453352
#214452702
#214451992
#214451812
#214450682
#214450662
#214450312
#214449842
#214446702
#214446412
D4×Q16×C612
#1981161