Finite Group Databasebeta

← #27151211

Subgroups — #27151211

Order 12416.

584 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690834C212yesmaximal, complemented
2#127×D97C211yesmaximal, characteristic, complemented
3C4×#13095510C211yesmaximal, characteristic, complemented
4#20260936C211yesmaximal, characteristic, complemented
5#20260934C212yesmaximal, complemented
6#20260932C211yesmaximal, characteristic, complemented
7#20260930C211yesmaximal, characteristic, complemented
8#20260928C212yesmaximal, complemented
9#20260757C211yesmaximal, characteristic, complemented
10#20260738C212yesmaximal, complemented
11#459×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2211yescharacteristic
13C2×D4×D97—21no—
14C2×D4×D97—21no—
15C2×D4×D97C2212yes—
16#15690834—28no—
17C2×#13095500C2212yescomplemented
18C2×#13095499C2212yescomplemented
19C2×#13095496C2212yescomplemented
20C2×#13095494C2212yes—
21#15690807C2211yescharacteristic
22#15690806—21no—
23#15690806—21no—
24#15690806C2212yes—
25C4×Dic194C2211yescharacteristic, complemented
26#15690775C2211yescharacteristic, complemented
27D4×Dic97C2211yescharacteristic, complemented
28C4×C22⋊D97—21no—
29#15690770—21no—
30#15690768C2211yescharacteristic
31#15690766—21no—
32C2×C4×Dic97—21no—
33C42×D97C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic, complemented
35#15690739C2212yescomplemented
36#15690732C2212yescomplemented
37#15690731C2212yescomplemented
38#15690730C2212yescomplemented
39C2×D8×C97C2212yescomplemented
40#127×C97C2211yescharacteristic
41C4×D4×C97C2211yescharacteristic, complemented
42C4⋊C8×C97C2211yescharacteristic, complemented
43#102×C97C2212yescomplemented
44C23×D97—22no—
45#13095510—22no—
46#13095510D414yescomplemented
47C2×Dic194C2311yescharacteristic
48C2×D388—21no—
49C2×D388—21no—
50C2×D388C2312yes—
51C2×C22⋊D97—21no—
52C2×C22⋊D97—22no—
53C2×C22⋊D97—22no—
54C2×C22⋊D97—22no—
55D4×D97—24no—
56D4×D97—24no—
57D4×D97—44no—
58D4×D97—44no—
59C22×Dic97—21no—
60C22×Dic97—21no—
61C2×C4×D97D411yescharacteristic
62C2×C4×D97D411yescharacteristic
63C2×C4×D97C2311yescharacteristic
64#13095500—24no—
65#13095499—24no—
66#13095496—24no—
67#13095494—24no—
68C2×#11715895C2312yes—
69C4⋊Dic97C2311yescharacteristic
70#13095490—21no—
71#13095490—21no—
72C22⋊Dic97—21no—
73#13095488—21no—
74C4×Dic97—21no—
75C4×Dic97—21no—
76C4×Dic97—21no—
77C4×Dic97D411yescharacteristic
78C4×Dic97D411yescharacteristic
79C4×Dic97C2311yescharacteristic
80C2×D4×C97—21no—
81C2×D4×C97—21no—
82C2×D4×C97C2312yes—
83C2×D4×C97C2311yescharacteristic
84C22×C4×C97—21no—
85D8×C97—24no—
86C2×C8×C97C2312yes—
87C4⋊C4×C97C2311yescharacteristic
88C22⋊C4×C97—21no—
89C42×C97C2311yescharacteristic
90C23×C97—21no—
91C23×C97—22no—
92C22×D97—22no—
93C22×D97—44no—
94C22×D97—44no—
95C22×D97—44no—
96C22×D97C2×D411yescharacteristic
97D4×C97—22no—
98D4×C97—21no—
99D4×C97—22no—
100D4×C97—42no—
101D4×C97—42no—
102D4×C97C2×D412yescomplemented
103C2×C4×C97—21no—
104C2×C4×C97—42no—
105C2×C4×C97C2×D411yescharacteristic
106C2×C4×C97C2×D411yescharacteristic
107C2×C4×C97C2411yescharacteristic
108C22⋊D97—24no—
109C22⋊D97—44no—
110C22⋊D97—44no—
111C22⋊D97—44no—
112C2×Dic97—21no—
113C2×Dic97—24no—
114C2×Dic97—21no—
115C2×Dic97—42no—
116C2×Dic97C2×D411yescharacteristic
117C2×Dic97C2×D411yescharacteristic
118C2×Dic97C2×D411yescharacteristic
119D388—22no—
120D388—22no—
121D388—42no—
122D388—42no—
123C4×D97—22no—
124C4×D97—22no—
125C4×D97C2×D412yes—
126C4×D97#4112yes—
127Dic194—21no—
128Dic194C2×D412yescomplemented
129C8×C97—22no—
130#11715895—22no—
131C22×C97—22no—
132C22×C97—41no—
133C22×C97—41no—
134C22×C97—42no—
135C22×C97—42no—
136C22×C97—42no—
137C22×C97C22×D411yescharacteristic
138C2×D97—21no—
139C2×D97—21no—
140C2×D97—42no—
141C2×D97—42no—
142C2×D97—42no—
143C2×D97—88no—
144C2×D97#12112yes—
145C4×C97—21no—
146C4×C97—41no—
147C4×C97AGL(1,ℤ/8)12yescomplemented
148C4×C97C22×D411yescharacteristic
149C4×C97C2×#4111yescharacteristic
150Dic97—22no—
151Dic97—21no—
152Dic97—41no—
153Dic97#12112yes—
154Dic97AGL(1,ℤ/8)12yescomplemented
155C2×C97—42no—
156C2×C97—82no—
157C2×C97C2×#12111yescharacteristic
158C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
160D97—22no—
161D97—82no—
162#3038—971nomaximal, Sylow
163C97#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—972no—
165C2×#127—971no—
166C4×#41—971no—
167#460—971no—
168#460—971no—
169#460—972no—
170#459—972no—
171#459—971no—
172#459D9711yescharacteristic, complemented
173#423—971no—
174#418—972no—
175C2×#41—971no—
176C22×D4—972no—
177C22×D4—1941no—
178C22×D4—1941no—
179AGL(1,ℤ/8)—1948no—
180C2×SD16—972no—
181C2×SD16—972no—
182C2×D8—972no—
183C2×D8C2×D9712yescomplemented
184C2×M16—972no—
185#127—972no—
186#127—971no—
187#127—1941no—
188#127—1941no—
189#127C2×D9711yescharacteristic
190C4×Q8—971no—
191C4×Q8—971no—
192C4×D4—971no—
193C4×D4—1941no—
194C4×D4—1941no—
195C4×D4C2×D9711yescharacteristic, complemented
196#117—971no—
197#117—1941no—
198C2×C42—971no—
199C2×C42—1941no—
200C4⋊C8—972no—
201C4⋊C8—971no—
202C4⋊C8C2×D9711yescharacteristic, complemented
203#102—972no—
204#102—972no—
205#102—972no—
206#102C2×D9712yescomplemented
207C24—1942no—
208#41—974no—
209#41—1942no—
210C2×Q8—971no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—972no—
214C2×D4—3884no—
215C2×D4—3884no—
216C2×D4—1944no—
217C2×D4—1944no—
218C2×D4—1942no—
219C2×D4—1942no—
220C2×D4—1942no—
221C2×D4—1941no—
222C2×D4—1941no—
223C2×D4—1941no—
224C2×D4C22×D9712yes—
225C2×D4C22×D9711yescharacteristic
226C22×C4—21no—
227C22×C4—971no—
228C22×C4—971no—
229C22×C4—971no—
230C22×C4—1941no—
231C22×C4—1941no—
232SD16—1944no—
233SD16—1944no—
234D8—24no—
235D8—1944no—
236M16—1944no—
237C2×C8—972no—
238C2×C8C22×D9712yes—
239C4⋊C4—971no—
240C4⋊C4—1941no—
241C4⋊C4—1941no—
242C4⋊C4C22×D9711yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—1941no—
245C22⋊C4—1941no—
246C42—971no—
247C42—971no—
248C42—971no—
249C42—1941no—
250C42—1941no—
251C42—1941no—
252C42C22×D9711yescharacteristic
253C23—21no—
254C23—22no—
255C23—971no—
256C23—3884no—
257C23—3884no—
258C23—3884no—
259C23—1942no—
260Q8—972no—
261Q8—1941no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—3882no—
268D4—3884no—
269D4—3884no—
270D4—3884no—
271D4—3882no—
272D4—1944no—
273D4—1942no—
274D4—1942no—
275D4D4×D9712yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—971no—
279C2×C4—971no—
280C2×C4—972no—
281C2×C4—972no—
282C2×C4—971no—
283C2×C4—3882no—
284C2×C4—1942no—
285C2×C4—1944no—
286C2×C4—1941no—
287C2×C4—1941no—
288C2×C4—1942no—
289C2×C4D4×D9711yescharacteristic
290C2×C4D4×D9711yescharacteristic
291C2×C4C23×D9711yescharacteristic
292C8—22no—
293C8—1942no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—7768no—
301C22—972no—
302C22—3882no—
303C22—3882no—
304C22—3882no—
305C22—1941no—
306C22—1941no—
307C22C2×D4×D9711yescharacteristic
308C4—21no—
309C4—41no—
310C4—972no—
311C4—972no—
312C4—3881no—
313C4—1942no—
314C4—1941no—
315C4#1569083412yescomplemented
316C4C2×D4×D9711yescharacteristic
317C4#41×D9711yescharacteristic
318C2—42no—
319C2—82no—
320C2—7762no—
321C2—1942no—
322C2#121×D9711yescharacteristic
323C2AGL(1,ℤ/8)×D9711yescharacteristic
324C2C2×#1569083411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#459×C97C2
socleC22×C97C22×D4
hypercenter#459D97
hypocenterC97#3038
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#15690834C2semidirect
2. #127×D97C2semidirect
3. C4×#13095510C2semidirect
4. #20260936C2semidirect
5. #20260934C2semidirect
6. #20260932C2semidirect
7. #20260930C2semidirect
8. #20260928C2semidirect
9. #20260757C2semidirect
10. #20260738C2semidirect
11. #459×C97C2semidirect
17. C2×#13095500C22semidirect
18. C2×#13095499C22semidirect
19. C2×#13095496C22semidirect
25. C4×Dic194C22semidirect
26. #15690775C22semidirect
27. D4×Dic97C22semidirect
34. C4⋊#11715895C22semidirect
35. #15690739C22semidirect
36. #15690732C22semidirect
37. #15690731C22semidirect
38. #15690730C22semidirect
39. C2×D8×C97C22semidirect
41. C4×D4×C97C22semidirect
42. C4⋊C8×C97C22semidirect
43. #102×C97C22semidirect
46. #13095510D4semidirect
102. D4×C97C2×D4semidirect
128. Dic194C2×D4semidirect
147. C4×C97AGL(1,ℤ/8)semidirect
154. Dic97AGL(1,ℤ/8)semidirect
163. C97#3038semidirect
172. #459D97semidirect
183. C2×D8C2×D97semidirect
195. C4×D4C2×D97semidirect
202. C4⋊C8C2×D97semidirect
206. #102C2×D97semidirect
275. D4D4×D97semidirect
315. C4#15690834semidirect

Maximal subgroups

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

subgroupindex
C2×#156908342
#127×D972
C4×#130955102
#202609362
#202609342
#202609322
#202609302
#202609282
#202607572
#202607382
#459×C972
#303897