Finite Group Databasebeta

← #27151379

Subgroups — #27151379

Order 12416.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D97C211yesmaximal, characteristic, complemented
2C2×#15690816C211yesmaximal, characteristic, complemented
3#20261301C212yesmaximal, complemented
4#20260987C212yesmaximal, complemented
5#20260986C212yesmaximal, complemented
6#20260972C211yesmaximal, characteristic, complemented
7#20260971C211yesmaximal, characteristic, complemented
8#20260968C211yesmaximal, characteristic, complemented
9#20260772C211yesmaximal, characteristic, complemented
10#20260738C212yesmaximal, complemented
11#466×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C2×D4×D97C2212yes—
14C2×D776C2211yescharacteristic
15C2×#13095496C2212yes—
16D8×D97—24no—
17#15690816—24no—
18C2×#13095495C2211yescharacteristic, complemented
19C2×#13095494C2211yescharacteristic
20C2×#13095494C2211yescharacteristic, complemented
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic, complemented
23#15690806—22no—
24#15690796—22no—
25#15690785C2212yescomplemented
26#15690784C2212yescomplemented
27#15690781C2212yescomplemented
28#15690780—22no—
29#15690778—22no—
30#15690768C2212yes—
31#15690744C2211yescharacteristic, complemented
32#15690741C2212yescomplemented
33#15690732C2212yescomplemented
34#15690731C2212yescomplemented
35#15690730C2212yescomplemented
36C2×D8×C97C2211yescharacteristic
37C22×C8×C97C2211yescharacteristic, complemented
38#121×C97C2212yescomplemented
39#107×C97C2211yescharacteristic, complemented
40#102×C97C2212yescomplemented
41C23×D97—22no—
42#13095511—22no—
43#13095511—22no—
44C2×Dic194D411yescharacteristic
45C2×D388—22no—
46C2×D388D411yescharacteristic
47C2×D388C2312yes—
48C2×C22⋊D97—22no—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—22no—
51C2×C22⋊D97D411yescharacteristic
52C2×C22⋊D97D411yescharacteristic
53D4×D97—24no—
54D4×D97—24no—
55C2×C4×D97C2311yescharacteristic
56C2×C4×D97C2311yescharacteristic
57D776—22no—
58#13095496—24no—
59#13095495—22no—
60#13095494—22no—
61#13095494—22no—
62C2×#11715895C2311yescharacteristic
63C2×#11715895C2311yescharacteristic
64C8×D97—22no—
65C8×D97D414yescomplemented
66C4⋊Dic97C2312yes—
67C22⋊Dic97—22no—
68#13095488—22no—
69#13095488—22no—
70#13095488—22no—
71C4×Dic97—22no—
72C2×D4×C97—22no—
73C2×D4×C97C2312yes—
74C22×C4×C97C2311yescharacteristic
75D8×C97—22no—
76C2×C8×C97—22no—
77C2×C8×C97C2311yescharacteristic
78C2×C8×C97C2311yescharacteristic
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—22no—
81C23×C97—22no—
82C23×C97C2×D411yescharacteristic
83C22×D97—22no—
84C22×D97—44no—
85C22×D97—82no—
86C22×D97—82no—
87C22×D97C2×D411yescharacteristic
88C22×D97C2×D411yescharacteristic
89D4×C97—22no—
90D4×C97—22no—
91D4×C97—82no—
92C2×C4×C97—21no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—22no—
98C22⋊D97—22no—
99C22⋊D97—44no—
100C22⋊D97—82no—
101C22⋊D97—82no—
102C2×Dic97—22no—
103C2×Dic97C2×D411yescharacteristic
104C2×Dic97C2×D411yescharacteristic
105D388—22no—
106D388—22no—
107D388—22no—
108D388—82no—
109C4×D97—21no—
110C4×D97—21no—
111C4×D97C2×D412yes—
112C4×D97#4112yes—
113Dic194—22no—
114C8×C97—21no—
115C8×C97C2×D412yes—
116#11715895—21no—
117#11715895C2×D412yes—
118C22×C97—21no—
119C22×C97—21no—
120C22×C97—21no—
121C22×C97—42no—
122C22×C97—42no—
123C22×C97—42no—
124C22×C97C22×D411yescharacteristic
125C2×D97—21no—
126C2×D97—21no—
127C2×D97—21no—
128C2×D97—21no—
129C2×D97—21no—
130C2×D97—42no—
131C2×D97—42no—
132C2×D97—42no—
133C2×D97—88no—
134C2×D97#12112yes—
135C4×C97—21no—
136C4×C97—42no—
137C4×C97C22×D411yescharacteristic
138C4×C97C2×#4111yescharacteristic
139Dic97—21no—
140Dic97—42no—
141Dic97#12112yes—
142C2×C97—41no—
143C2×C97—82no—
144C2×C97C2×#12111yescharacteristic
145C2×C97#57611yescharacteristic
146C2×C97#57611yescharacteristic
147D97—22no—
148D97—41no—
149D97—82no—
150#3058—971nomaximal, Sylow
151C97#305811yescharacteristic, complemented, Sylow
152C22×D8—971no—
153C2×#131—971no—
154#535—972no—
155#469—972no—
156#469—972no—
157#466—971no—
158#466—971no—
159#466—971no—
160#466D9711yescharacteristic, complemented
161#427—971no—
162#418—972no—
163C2×#41—971no—
164C22×D4—972no—
165C2×D8—972no—
166C2×D8—971no—
167C2×D8—1944no—
168C2×D8C2×D9711yescharacteristic
169#131—1944no—
170C2×M16—971no—
171C2×M16—971no—
172C2×M16—971no—
173C22×C8—971no—
174C22×C8—971no—
175C22×C8C2×D9711yescharacteristic, complemented
176#127—1942no—
177#124—1942no—
178#121—972no—
179#121—972no—
180#121—972no—
181#121C2×D9712yescomplemented
182#120—1942no—
183#120—1942no—
184#117—972no—
185#107—971no—
186#107C2×D9711yescharacteristic, complemented
187#106—972no—
188#102—972no—
189#102—972no—
190#102—972no—
191#102C2×D9712yescomplemented
192C24—1942no—
193#41—1942no—
194#41—1942no—
195C2×Q8—971no—
196C2×D4—22no—
197C2×D4—971no—
198C2×D4—971no—
199C2×D4—971no—
200C2×D4—972no—
201C2×D4—1942no—
202C2×D4—1944no—
203C2×D4—1944no—
204C2×D4—1942no—
205C2×D4—1942no—
206C2×D4—1942no—
207C2×D4C22×D9712yes—
208C22×C4—971no—
209C22×C4—971no—
210C22×C4C22×D9711yescharacteristic
211D8—22no—
212D8—1944no—
213D8—1942no—
214M16—1942no—
215M16—1942no—
216M16—1942no—
217C2×C8—22no—
218C2×C8—971no—
219C2×C8—974no—
220C2×C8—971no—
221C2×C8—1942no—
222C2×C8C22×D9711yescharacteristic
223C2×C8C22×D9711yescharacteristic
224C4⋊C4—972no—
225C4⋊C4C22×D9712yes—
226C22⋊C4—22no—
227C22⋊C4—1942no—
228C22⋊C4—1942no—
229C22⋊C4—1942no—
230C22⋊C4—1942no—
231C42—1942no—
232C23—22no—
233C23—7762no—
234C23—7762no—
235C23—971no—
236C23—971no—
237C23—3884no—
238C23—1942no—
239C23D4×D9711yescharacteristic
240Q8—1942no—
241D4—22no—
242D4—22no—
243D4—82no—
244D4—7762no—
245D4—7762no—
246D4—7762no—
247D4—3884no—
248D4—1942no—
249D4—1942no—
250D4—1942no—
251D4—1942no—
252D4—1942no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—971no—
257C2×C4—972no—
258C2×C4—972no—
259C2×C4—971no—
260C2×C4—1942no—
261C2×C4—1941no—
262C2×C4—1941no—
263C2×C4D4×D9711yescharacteristic
264C2×C4C23×D9711yescharacteristic
265C8—21no—
266C8—972no—
267C8—1941no—
268C8D4×D9712yes—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—7768no—
276C22—972no—
277C22—3882no—
278C22—3882no—
279C22—3882no—
280C22—1941no—
281C22—1941no—
282C22—1941no—
283C22—1941no—
284C22—1941no—
285C22C2×D4×D9711yescharacteristic
286C4—21no—
287C4—42no—
288C4—972no—
289C4—3882no—
290C4—1941no—
291C4C2×D4×D9711yescharacteristic
292C4#41×D9711yescharacteristic
293C2—41no—
294C2—82no—
295C2—7762no—
296C2—3881no—
297C2—1942no—
298C2#121×D9711yescharacteristic
299C2#2026153611yescharacteristic
300C2#2026153711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#466×C97C2
socleC22×C97C22×D4
hypercenter#466D97
hypocenterC97#3058
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D8×D97C2semidirect
2. C2×#15690816C2semidirect
3. #20261301C2semidirect
4. #20260987C2semidirect
5. #20260986C2semidirect
6. #20260972C2semidirect
7. #20260971C2semidirect
8. #20260968C2semidirect
9. #20260772C2semidirect
10. #20260738C2semidirect
11. #466×C97C2semidirect
18. C2×#13095495C22semidirect
20. C2×#13095494C22semidirect
22. C2×C8×D97C22semidirect
25. #15690785C22semidirect
26. #15690784C22semidirect
27. #15690781C22semidirect
31. #15690744C22semidirect
32. #15690741C22semidirect
33. #15690732C22semidirect
34. #15690731C22semidirect
35. #15690730C22semidirect
37. C22×C8×C97C22semidirect
38. #121×C97C22semidirect
39. #107×C97C22semidirect
40. #102×C97C22semidirect
65. C8×D97D4semidirect
151. C97#3058semidirect
160. #466D97semidirect
175. C22×C8C2×D97semidirect
181. #121C2×D97semidirect
186. #107C2×D97semidirect
191. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D972
C2×#156908162
#202613012
#202609872
#202609862
#202609722
#202609712
#202609682
#202607722
#202607382
#466×C972
#305897