Finite Group Databasebeta

← #27153247

Subgroups — #27153247

Order 12416.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690841C211yesmaximal, characteristic, complemented
2C22×#13095498C211yesmaximal, characteristic, complemented
3#20261381C211yesmaximal, characteristic
4#20261127C211yesmaximal, characteristic, complemented
5#20260983C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260957C211yesmaximal, characteristic, complemented
8#20260937C211yesmaximal, characteristic, complemented
9#20260895C211yesmaximal, characteristic, complemented
10#20260891C211yesmaximal, characteristic, complemented
11#20260885C211yesmaximal, characteristic, complemented
12#20260874C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14#20260807C211yesmaximal, characteristic, complemented
15D4×Q8×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×D388C2211yescharacteristic
18#15690841—24no—
19C2×#13095501C2211yescharacteristic, complemented
20C2×#13095501C2211yescharacteristic, complemented
21C2×#13095501C2211yescharacteristic, complemented
22C2×#13095498—24no—
23C2×#13095498C2211yescharacteristic
24C2×#13095498C2211yescharacteristic
25C2×#13095498C2211yescharacteristic
26C2×#13095498C2211yescharacteristic
27C2×#13095498C2211yescharacteristic
28C2×#13095498C2211yescharacteristic
29C2×#13095495C2211yescharacteristic, complemented
30C22×#11715895C2211yescharacteristic
31#15690798C2211yescharacteristic
32#15690795—21no—
33#15690789C2211yescharacteristic
34#15690784C2211yescharacteristic
35#15690783—21no—
36#15690778—21no—
37C4×D388C2211yescharacteristic
38C22⋊C4×D97—21no—
39#15690743C2211yescharacteristic, complemented
40C4⋊#11715895C2211yescharacteristic, complemented
41#15690734C2211yescharacteristic, complemented
42#15690734C2211yescharacteristic, complemented
43#15690734C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690730C2211yescharacteristic
46#15690730C2211yescharacteristic
47C22⋊#11715895C2211yescharacteristic
48C22⋊#11715895C2211yescharacteristic, complemented
49#15690721C2211yescharacteristic, complemented
50C22×Q8×C97C2211yescharacteristic
51C22×Q8×C97C2211yescharacteristic, complemented
52C4⋊Q8×C97—21no—
53C4⋊Q8×C97C2211yescharacteristic, complemented
54C22⋊Q8×C97—21no—
55C22⋊Q8×C97—21no—
56C22⋊Q8×C97C2211yescharacteristic
57C22⋊Q8×C97C2211yescharacteristic, complemented
58C4×Q8×C97C2211yescharacteristic, complemented
59C4×D4×C97—21no—
60C4×D4×C97C2211yescharacteristic
61C23×D97—21no—
62#13095511—22no—
63#13095511—41no—
64C2×Dic194C2311yescharacteristic
65C2×D388—22no—
66C2×D388—21no—
67C2×D388—41no—
68C2×D388C2311yescharacteristic
69C2×D388C2311yescharacteristic
70C2×C22⋊D97—21no—
71C2×C22⋊D97—21no—
72C2×C4×D97—21no—
73C2×C4×D97—21no—
74#13095501—22no—
75#13095501—22no—
76#13095501—41no—
77#13095498—22no—
78#13095498—22no—
79#13095498—22no—
80#13095498—22no—
81#13095498—41no—
82#13095498D414yescomplemented
83#13095495—22no—
84C2×#11715895—22no—
85C2×#11715895C2311yescharacteristic
86C2×#11715895C2311yescharacteristic
87C2×#11715895C2311yescharacteristic
88C2×#11715895C2311yescharacteristic
89C4⋊Dic97C2311yescharacteristic
90#13095490—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95#13095488—21no—
96C2×Q8×C97—21no—
97C2×Q8×C97—21no—
98C2×Q8×C97—22no—
99C2×Q8×C97—22no—
100C2×Q8×C97—41no—
101C2×Q8×C97—41no—
102C2×Q8×C97C2311yescharacteristic
103C2×Q8×C97C2311yescharacteristic
104C2×Q8×C97C2311yescharacteristic
105C2×D4×C97D411yescharacteristic
106C22×C4×C97—21no—
107C22×C4×C97—21no—
108C22×C4×C97C2311yescharacteristic
109C22×C4×C97C2311yescharacteristic
110C4⋊C4×C97—21no—
111C4⋊C4×C97—21no—
112C4⋊C4×C97—21no—
113C4⋊C4×C97—21no—
114C4⋊C4×C97—21no—
115C4⋊C4×C97D411yescharacteristic
116C4⋊C4×C97C2311yescharacteristic
117C22⋊C4×C97—21no—
118C22⋊C4×C97—21no—
119C22⋊C4×C97D411yescharacteristic
120C22⋊C4×C97D411yescharacteristic
121C42×C97—21no—
122C42×C97C2311yescharacteristic
123C23×C97C2×D411yescharacteristic
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—42no—
128C22×D97—42no—
129C22×D97—81no—
130Q8×C97—21no—
131Q8×C97—21no—
132Q8×C97—21no—
133Q8×C97—21no—
134Q8×C97—21no—
135Q8×C97—42no—
136Q8×C97—42no—
137Q8×C97C2×D412yes—
138D4×C97—22no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—21no—
145C2×C4×C97—21no—
146C2×C4×C97—21no—
147C2×C4×C97—42no—
148C2×C4×C97—42no—
149C2×C4×C97C2×D411yescharacteristic
150C2×C4×C97C2×D411yescharacteristic
151C2×C4×C97C2×D411yescharacteristic
152C2×C4×C97C2×D411yescharacteristic
153C2×C4×C97C2411yescharacteristic
154C22⋊D97—42no—
155C22⋊D97—81no—
156C2×Dic97—21no—
157C2×Dic97—21no—
158D388—21no—
159D388—21no—
160D388—21no—
161D388—42no—
162D388C2×D412yes—
163C4×D97—42no—
164C4×D97—81no—
165Dic194—21no—
166Dic194—21no—
167#11715895—21no—
168#11715895—21no—
169#11715895—21no—
170#11715895C2×D412yes—
171C22×C97—21no—
172C22×C97—21no—
173C22×C97—21no—
174C22×C97—21no—
175C22×C97—21no—
176C22×C97#13712yescomplemented
177C22×C97C22×D411yescharacteristic
178C2×D97—22no—
179C2×D97—41no—
180C2×D97—41no—
181C2×D97—41no—
182C2×D97—41no—
183C2×D97—41no—
184C2×D97—84no—
185C4×C97—21no—
186C4×C97—21no—
187C4×C97—22no—
188C4×C97—21no—
189C4×C97—21no—
190C4×C97—41no—
191C4×C97—41no—
192C4×C97—41no—
193C4×C97C22×D411yescharacteristic
194C4×C97H(5,𝔽2)11yescharacteristic
195Dic97—41no—
196Dic97—41no—
197C2×C97—22no—
198C2×C97—41no—
199C2×C97D4211yescharacteristic
200C2×C97C2×#13711yescharacteristic
201C2×C97#57711yescharacteristic
202D97—42no—
203D97—81no—
204#3266—971nomaximal, Sylow
205C97#326611yescharacteristic, complemented, Sylow
206C2×#137—971no—
207C22×SD16—971no—
208D4×Q8D9711yescharacteristic, complemented
209#546—971no—
210#497—971no—
211#468—971no—
212#467—971no—
213#464—971no—
214#461—971no—
215#452—971no—
216#451—971no—
217#450—971no—
218#448—971no—
219#440—971no—
220#435—971no—
221C2×#41—971no—
222C22×Q8C2×D9711yescharacteristic
223C22×Q8C2×D9711yescharacteristic, complemented
224C22×D4—971no—
225#137—1944no—
226C2×Q16—971no—
227C2×Q16—971no—
228C2×Q16—971no—
229C2×SD16—971no—
230C2×SD16—971no—
231C2×SD16—971no—
232C2×SD16—971no—
233C2×SD16—971no—
234C2×SD16—971no—
235C2×SD16—1944no—
236C2×M16—971no—
237C22×C8—971no—
238C4⋊Q8—21no—
239C4⋊Q8C2×D9711yescharacteristic, complemented
240#124—971no—
241#123—1941no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—971no—
245C22⋊Q8C2×D9711yescharacteristic
246C22⋊Q8C2×D9711yescharacteristic, complemented
247#121—971no—
248#121—1941no—
249#120—1941no—
250C4×Q8C2×D9711yescharacteristic, complemented
251C4×D4—21no—
252C4×D4—971no—
253C4×D4C2×D9711yescharacteristic
254C2×C22⋊C4—1941no—
255#107—971no—
256C4⋊C8—971no—
257#103—971no—
258#103—971no—
259#103—971no—
260#103—971no—
261#102—971no—
262#102—971no—
263C22⋊C8—971no—
264C22⋊C8—971no—
265#97—971no—
266C24—1941no—
267#41—3881no—
268#41—1942no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—971no—
276C2×Q8C22×D9711yescharacteristic
277C2×Q8C22×D9711yescharacteristic
278C2×Q8C22×D9711yescharacteristic
279C2×D4—971no—
280C2×D4—971no—
281C2×D4—3881no—
282C2×D4—1942no—
283C2×D4—1941no—
284C2×D4—1941no—
285C2×D4—1941no—
286C2×D4C22⋊D9711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1941no—
290C22×C4—1941no—
291C22×C4C22×D9711yescharacteristic
292C22×C4C22×D9711yescharacteristic
293Q16—3881no—
294Q16—1942no—
295Q16—1942no—
296SD16—974no—
297SD16—3881no—
298SD16—1942no—
299SD16—1942no—
300SD16—1942no—
301SD16—1942no—
302M16—1942no—
303C2×C8—971no—
304C2×C8—971no—
305C2×C8—971no—
306C2×C8—971no—
307C2×C8—1942no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—971no—
314C4⋊C4—1941no—
315C4⋊C4C22⋊D9711yescharacteristic
316C4⋊C4C22×D9711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—21no—
319C22⋊C4—1941no—
320C22⋊C4—1941no—
321C22⋊C4—1941no—
322C22⋊C4—1941no—
323C22⋊C4—1941no—
324C22⋊C4C22⋊D9711yescharacteristic
325C22⋊C4C22⋊D9711yescharacteristic
326C42—21no—
327C42C22×D9711yescharacteristic
328C23—7761no—
329C23—3882no—
330C23—3882no—
331C23—1941no—
332C23—1941no—
333C23C2×C22⋊D9711yescharacteristic
334C23C2×C22⋊D9711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—42no—
342Q8—1941no—
343Q8—1941no—
344Q8D4×D9712yes—
345D4—22no—
346D4—7761no—
347D4—972no—
348D4—3882no—
349D4—3882no—
350D4—1941no—
351D4—1941no—
352D4—1941no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—7761no—
364C2×C4—3882no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4C2×C22⋊D9711yescharacteristic
368C2×C4C2×C22⋊D9711yescharacteristic
369C2×C4C2×C22⋊D9711yescharacteristic
370C2×C4C2×C22⋊D9711yescharacteristic
371C2×C4C23×D9711yescharacteristic
372C8—972no—
373C8—1941no—
374C8—1941no—
375C8—1941no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—7764no—
382C22—3881no—
383C22—3881no—
384C22—3881no—
385C22—3881no—
386C22—3881no—
387C22—1942no—
388C22#1569084112yescomplemented
389C22C22×C22⋊D9711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—3881no—
399C4—3881no—
400C4C2×D4×D9711yescharacteristic
401C4#1569085511yescharacteristic
402C2—22no—
403C2—41no—
404C2—7761no—
405C2—3882no—
406C2D4×C22⋊D9711yescharacteristic
407C2C2×#1569084111yescharacteristic
408C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingD4×Q8×C97C2
socleC22×C97C22×D4
prime radical#20261381C2
hypercenterD4×Q8D97
hypocenterC97#3266
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15690841C2semidirect
2. C22×#13095498C2semidirect
4. #20261127C2semidirect
5. #20260983C2semidirect
6. #20260974C2semidirect
7. #20260957C2semidirect
8. #20260937C2semidirect
9. #20260895C2semidirect
10. #20260891C2semidirect
11. #20260885C2semidirect
12. #20260874C2semidirect
13. #20260833C2semidirect
14. #20260807C2semidirect
15. D4×Q8×C97C2semidirect
19. C2×#13095501C22semidirect
20. C2×#13095501C22semidirect
21. C2×#13095501C22semidirect
29. C2×#13095495C22semidirect
39. #15690743C22semidirect
40. C4⋊#11715895C22semidirect
41. #15690734C22semidirect
42. #15690734C22semidirect
43. #15690734C22semidirect
44. #15690733C22semidirect
48. C22⋊#11715895C22semidirect
49. #15690721C22semidirect
51. C22×Q8×C97C22semidirect
53. C4⋊Q8×C97C22semidirect
57. C22⋊Q8×C97C22semidirect
58. C4×Q8×C97C22semidirect
82. #13095498D4semidirect
176. C22×C97#137semidirect
205. C97#3266semidirect
208. D4×Q8D97semidirect
223. C22×Q8C2×D97semidirect
239. C4⋊Q8C2×D97semidirect
246. C22⋊Q8C2×D97semidirect
250. C4×Q8C2×D97semidirect
388. C22#15690841semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C22×#130954982
#202613812
#202611272
#202609832
#202609742
#202609572
#202609372
#202608952
#202608912
#202608852
#202608742
#202608332
#202608072
D4×Q8×C972
#326697