Finite Group Databasebeta

← #29670201

Subgroups — #29670201

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#21458718C211yesmaximal, characteristic
2Q8×#11758199C211yesmaximal, characteristic
3Q16×D244C211yesmaximal, characteristic, complemented
4#21445498C211yesmaximal, characteristic, complemented
5#21445335C211yesmaximal, characteristic, complemented
6#21445267C211yesmaximal, characteristic, complemented
7#21445203C211yesmaximal, characteristic, complemented
8#21445184C211yesmaximal, characteristic, complemented
9#21445068C211yesmaximal, characteristic, complemented
10#21445066C211yesmaximal, characteristic, complemented
11#21445033C211yesmaximal, characteristic, complemented
12#21444982C211yesmaximal, characteristic, complemented
13#21444670C211yesmaximal, characteristic, complemented
14#21444641C211yesmaximal, characteristic
15C61×#3299C211yesmaximal, characteristic, complemented
16C2×Q16×D61—21no—
17Q8×D244C2211yescharacteristic
18Q8×D244C2211yescharacteristic
19#16840172—21no—
20#16840165—21no—
21#16840165C2211yescharacteristic
22#16840165C2211yescharacteristic
23#16840158C2211yescharacteristic
24#16840142C2211yescharacteristic
25#16840112C2211yescharacteristic
26#16840059—21no—
27#16840052—21no—
28#16840049C2211yescharacteristic
29#16840042—21no—
30#16840042C2211yescharacteristic
31#16840042C2211yescharacteristic
32#16840039—21no—
33#16840039C2211yescharacteristic
34#16840004—21no—
35#16839972C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839919—21no—
38#16839919—21no—
39#16839893C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41#16839872—21no—
42C4×#11758199—21no—
43C4×#11758199C2211yescharacteristic
44C8×D244C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56C61×#554C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#487C2211yescharacteristic
59C61×#484—21no—
60C61×#477C2211yescharacteristic
61C61×#467—21no—
62C61×#462C2211yescharacteristic
63C61×#451—21no—
64C4×Q16×C61C2211yescharacteristic
65C8×D4×C61C2211yescharacteristic
66C2×#103×C61—21no—
67C2×Q8×D61—21no—
68C2×Q8×D61—21no—
69C2×Dic244—21no—
70C2×Dic244—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73Q16×D61—44no—
74C2×#11758199D411yescharacteristic
75C2×C8×D61—21no—
76#14519777—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519757—21no—
81#14519757—21no—
82#14519757—41no—
83#14519757—41no—
84C4×D244—21no—
85C4×D244—21no—
86C4×D244C2311yescharacteristic
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—41no—
93#14519710D411yescharacteristic
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97C4⋊#92418—21no—
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—41no—
103#14519700—21no—
104#14519700—21no—
105#14519698—21no—
106#14519698—21no—
107#14519698D411yescharacteristic
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111#14519689—21no—
112#14519688—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×Q16×C61—21no—
118C2×Q16×C61D411yescharacteristic
119C22×C8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126C22⋊Q8×C61—41no—
127C22⋊Q8×C61—41no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61—21no—
134#107×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
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—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×C4×C61—41no—
156Q8×D61—42no—
157Q8×D61—42no—
158Q8×D61—81no—
159Q8×D61—81no—
160C2×D244C2×D411yescharacteristic
161C2×C4×D61—21no—
162C2×C4×D61—41no—
163C2×C4×D61—41no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—41no—
167C2×Dic122—41no—
168Q16×C61—22no—
169Q16×C61—81no—
170C2×C8×C61—21no—
171C2×C8×C61—81no—
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—41no—
183C4⋊C4×C61C2×D411yescharacteristic
184C4⋊C4×C61C2×D411yescharacteristic
185C4⋊C4×C61C2×D411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—41no—
188C22⋊C4×C61—41no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C2411yescharacteristic
192#11758202—42no—
193#11758202—42no—
194#11758199Q1614yescomplemented
195#11758198—21no—
196#11758198—41no—
197#11758198—41no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61—41no—
204C4⋊Dic61C2×D411yescharacteristic
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418C2×D411yescharacteristic
210C2×#92418C2×D411yescharacteristic
211Dic244—42no—
212Dic244—81no—
213C8×D61—42no—
214C23×C61—21no—
215C22×D61—21no—
216Q8×C61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—41no—
220Q8×C61—41no—
221Q8×C61—82no—
222Q8×C61—81no—
223Q8×C61AGL(1,ℤ/8)12yes—
224D4×C61—21no—
225D4×C61C2×Q1612yescomplemented
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—82no—
236C2×C4×C61—161no—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61H(5,𝔽2)11yescharacteristic
240C2×Dic61—21no—
241C2×Dic61—41no—
242C2×Dic61—41no—
243D244—21no—
244D244C2×Q1612yes—
245C4×D61—41no—
246C4×D61—41no—
247C4×D61—82no—
248C4×D61—82no—
249Dic122—41no—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253Dic122—82no—
254Dic122—82no—
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#3299D6111yescharacteristic, complemented
290D4×Q16—611no—
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—1221no—
308#554C2×D6111yescharacteristic
309D4×Q8—611no—
310D4×Q8—611no—
311D4×Q8C2×D6111yescharacteristic
312#501—1221no—
313#500—611no—
314#500—611no—
315#500—1221no—
316#498—611no—
317#494—611no—
318#487—611no—
319#487C2×D6111yescharacteristic
320#484—21no—
321#478—1221no—
322#477C2×D6111yescharacteristic
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#462—611no—
334#462C2×D6111yescharacteristic
335#451—21no—
336#451—1221no—
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—21no—
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—1221no—
361C22×Q8—1221no—
362C2×Q16—21no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×Q16—1221no—
367C2×Q16—2444no—
368C2×Q16C22⋊D6111yescharacteristic
369C2×D8—611no—
370C22×C8—21no—
371C22×C8—1221no—
372C4⋊Q8—21no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377C4⋊Q8C22×D6111yescharacteristic
378C4⋊Q8C22×D6111yescharacteristic
379#125—21no—
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—21no—
397#107—21no—
398#107—611no—
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—21no—
418#103—21no—
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—42no—
441C2×Q8—81no—
442C2×Q8—1221no—
443C2×Q8—1221no—
444C2×Q8—4881no—
445C2×Q8—4881no—
446C2×Q8—2441no—
447C2×Q8—2441no—
448C2×Q8—2442no—
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—81no—
462Q16—4881no—
463Q16—2442no—
464Q16—2442no—
465Q16—2442no—
466D8—614no—
467C2×C8—21no—
468C2×C8—81no—
469C2×C8—611no—
470C2×C8—611no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—2442no—
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—41no—
487C4⋊C4—611no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
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—82no—
514Q8—81no—
515Q8—4882no—
516Q8—4882no—
517Q8—2441no—
518Q8—2441no—
519Q8—2441no—
520Q8—2441no—
521Q8#1451980212yes—
522D4—21no—
523D4—612no—
524D4—1221no—
525D4Q16×D6112yescomplemented
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—161no—
537C2×C4—1221no—
538C2×C4—4882no—
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×#1451980211yescharacteristic
576C4#1684056511yescharacteristic
577C4#1684057511yescharacteristic
578C2—42no—
579C2—2442no—
580C2#2145730011yescharacteristic
581C2Q16×C22⋊D6111yescharacteristic
582C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3299C2
socleC22×C61D42
prime radicalC2×#11758199D4
hypercenter#3299D61
hypocenterC61#19811
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3299

Complements

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

NH ≅ G/Ntype
3. Q16×D244C2semidirect
4. #21445498C2semidirect
5. #21445335C2semidirect
6. #21445267C2semidirect
7. #21445203C2semidirect
8. #21445184C2semidirect
9. #21445068C2semidirect
10. #21445066C2semidirect
11. #21445033C2semidirect
12. #21444982C2semidirect
13. #21444670C2semidirect
15. C61×#3299C2semidirect
194. #11758199Q16semidirect
225. D4×C61C2×Q16semidirect
263. #92418C2×Q16semidirect
289. #3299D61semidirect
358. C61#19811semidirect
525. D4Q16×D61semidirect

Maximal subgroups

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

subgroupindex
#214587182
Q8×#117581992
Q16×D2442
#214454982
#214453352
#214452672
#214452032
#214451842
#214450682
#214450662
#214450332
#214449822
#214446702
#214446412
C61×#32992
#1981161