Finite Group Databasebeta

← #484×D97 #27151638

Subgroups — #484×D97 #27151638

Order 12416.

460 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22⋊Q8×D97C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic
3#20261072C212yesmaximal, complemented
4#20261071C211yesmaximal, characteristic, complemented
5#20261070C211yesmaximal, characteristic, complemented
6#20261069C212yesmaximal, complemented
7#20261068C211yesmaximal, characteristic, complemented
8#107×D97C212yesmaximal, complemented
9#103×D97C212yesmaximal, complemented
10C22⋊C8×D97C211yesmaximal, characteristic
11#484×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2211yescharacteristic
13C22×C4×D97—21no—
14C22×C4×D97C2211yescharacteristic
15C2×C8×D97C2212yes—
16#15690790C2211yescharacteristic
17#15690789C2211yescharacteristic
18#15690788—21no—
19#15690787—21no—
20#15690786C2211yescharacteristic
21C2×C4⋊Dic97C2211yescharacteristic
22C2×#13095490—21no—
23C4⋊C4×D97—21no—
24C4⋊C4×D97—21no—
25C4⋊C4×D97—21no—
26C4⋊C4×D97C2212yes—
27C4⋊C4×D97C2212yes—
28C4⋊C4×D97C2211yescharacteristic
29C22⋊C4×D97—21no—
30#15690744C2212yescomplemented
31#15690743C2212yescomplemented
32#15690743C2212yescomplemented
33#15690735C2212yescomplemented
34#15690734C2212yescomplemented
35#15690733C2212yescomplemented
36C22⋊#11715895C2211yescharacteristic
37#15690722C2212yes—
38C22⋊Q8×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#107×C97C2212yescomplemented
41#103×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic
43C23×D97D411yescharacteristic
44C2×Dic194—21no—
45C2×Dic194C2311yescharacteristic
46Q8×D97—24no—
47C22×Dic97—21no—
48C22×Dic97D411yescharacteristic
49C2×C4×D97—24no—
50C2×C4×D97—21no—
51C2×C4×D97—21no—
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97—44no—
55C2×C4×D97D411yescharacteristic
56C2×C4×D97D411yescharacteristic
57C2×C4×D97C2311yescharacteristic
58C2×C4×D97C2311yescharacteristic
59C2×#11715895C2312yes—
60C8×D97—24no—
61C4⋊Dic97—21no—
62C4⋊Dic97—21no—
63C4⋊Dic97C2312yes—
64C4⋊Dic97C2311yescharacteristic
65#13095490—21no—
66#13095490—22no—
67#13095490—22no—
68#13095490—21no—
69#13095490—21no—
70C22⋊Dic97—21no—
71#13095488—21no—
72#13095488—21no—
73C2×Q8×C97C2311yescharacteristic
74C22×C4×C97—21no—
75C22×C4×C97C2311yescharacteristic
76C2×C8×C97C2312yes—
77C4⋊C4×C97—21no—
78C4⋊C4×C97—21no—
79C4⋊C4×C97C2312yes—
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C23×C97C2×D411yescharacteristic
83C22×D97—22no—
84C22×D97—22no—
85C22×D97Q1614yes—
86C22×D97C2×D411yescharacteristic
87C22×D97C2×D411yescharacteristic
88Q8×C97—22no—
89C2×C4×C97—22no—
90C2×C4×C97—21no—
91C2×C4×C97—21no—
92C2×C4×C97—22no—
93C2×C4×C97—42no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2411yescharacteristic
96C2×Dic97—22no—
97C2×Dic97—21no—
98C2×Dic97—21no—
99C2×Dic97—22no—
100C2×Dic97—42no—
101C2×Dic97C2×D411yescharacteristic
102C2×Dic97C2×D411yescharacteristic
103C4×D97—22no—
104C4×D97—22no—
105C4×D97—22no—
106C4×D97—44no—
107C4×D97—42no—
108C4×D97—42no—
109C4×D97—44no—
110C4×D97#4114yes—
111Dic194—22no—
112Dic194—42no—
113C8×C97—22no—
114#11715895—22no—
115C22×C97—21no—
116C22×C97—21no—
117C22×C97C2×Q1612yes—
118C22×C97C22×D411yescharacteristic
119C2×D97—21no—
120C2×D97—21no—
121C2×D97—28no—
122C2×D97#12312yes—
123C2×D97C2×Q1612yes—
124C2×D97C2×Q1612yes—
125C2×D97AGL(1,ℤ/8)12yes—
126C4×C97—21no—
127C4×C97—41no—
128C4×C97—42no—
129C4×C97—41no—
130C4×C97C2×#4112yes—
131Dic97—21no—
132Dic97—41no—
133Dic97—42no—
134Dic97—41no—
135Dic97#12312yes—
136C2×C97—22no—
137C2×C97C2×#12311yescharacteristic
138C2×C97C22×Q1611yescharacteristic
139C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
140D97—22no—
141D97#48414yesdirect product factor
142C2×#484—971nomaximal, Sylow
143C97C2×#48411yescharacteristic, complemented, Sylow
144C2×C22⋊Q8—971no—
145C22×C4⋊C4—971no—
146#484—971no—
147#484—971no—
148#484—972no—
149#484—972no—
150#484—971no—
151#484D9711yescharacteristic, direct product factor
152C2×#107—972no—
153C2×#103—972no—
154C2×C22⋊C8—971no—
155C22×Q8—971no—
156C23×C4—971no—
157C23×C4—1941no—
158C22×C8—972no—
159C22⋊Q8—971no—
160C22⋊Q8—971no—
161C22⋊Q8—971no—
162C22⋊Q8—1941no—
163C22⋊Q8—1941no—
164C22⋊Q8C2×D9711yescharacteristic
165C2×C4⋊C4—971no—
166C2×C4⋊C4—972no—
167C2×C4⋊C4—972no—
168C2×C4⋊C4—971no—
169C2×C4⋊C4—1941no—
170C2×C4⋊C4—1941no—
171C2×C4⋊C4—1941no—
172C2×C4⋊C4—1941no—
173C2×C4⋊C4C2×D9711yescharacteristic
174C2×C22⋊C4—1941no—
175#107—972no—
176#107—972no—
177#107—972no—
178#107C2×D9712yescomplemented
179#103—972no—
180#103—972no—
181#103—972no—
182#103C2×D9712yescomplemented
183C22⋊C8—971no—
184C22⋊C8—972no—
185C22⋊C8C2×D9711yescharacteristic
186C24—971no—
187C2×Q8—971no—
188C2×Q8—1941no—
189C2×Q8—1944no—
190C2×Q8C22×D9711yescharacteristic
191C22×C4—21no—
192C22×C4—971no—
193C22×C4—971no—
194C22×C4—971no—
195C22×C4—971no—
196C22×C4—971no—
197C22×C4—3884no—
198C22×C4—1944no—
199C22×C4—1941no—
200C22×C4—1941no—
201C22×C4—1942no—
202C22×C4—1942no—
203C22×C4—1941no—
204C22×C4C22×D9711yescharacteristic
205C2×C8—972no—
206C2×C8—1944no—
207C2×C8C22×D9712yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—972no—
211C4⋊C4—971no—
212C4⋊C4—1942no—
213C4⋊C4—1942no—
214C4⋊C4—1941no—
215C4⋊C4—1941no—
216C4⋊C4—1941no—
217C4⋊C4—1941no—
218C4⋊C4—1941no—
219C4⋊C4C22×D9712yes—
220C4⋊C4C22×D9711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—1941no—
223C22⋊C4—1941no—
224C22⋊C4—1941no—
225C23—971no—
226C23—974no—
227C23—971no—
228C23—1942no—
229C23—1942no—
230C23D4×D9711yescharacteristic
231Q8—22no—
232Q8—3882no—
233Q8—1942no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—971no—
240C2×C4—974no—
241C2×C4—971no—
242C2×C4—3884no—
243C2×C4—3882no—
244C2×C4—3882no—
245C2×C4—3884no—
246C2×C4—3882no—
247C2×C4—1942no—
248C2×C4—1942no—
249C2×C4—1942no—
250C2×C4—1941no—
251C2×C4—1941no—
252C2×C4—1942no—
253C2×C4—1942no—
254C2×C4D4×D9711yescharacteristic
255C2×C4C23×D9711yescharacteristic
256C8—22no—
257C8—1942no—
258C22—21no—
259C22—21no—
260C22—972no—
261C22—972no—
262C22—972no—
263C22—972no—
264C22—1948no—
265C22—1941no—
266C22—1941no—
267C22Q16×D9712yes—
268C22C2×D4×D9711yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—972no—
274C4—3881no—
275C4—3882no—
276C4—3881no—
277C4—1941no—
278C4#41×D9712yes—
279C2—22no—
280C2—974no—
281C2—1942no—
282C2#123×D9711yescharacteristic
283C2C2×Q16×D9711yescharacteristic
284C2AGL(1,ℤ/8)×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#484×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
hypercenter#484D97
hypocenterC97C2×#484
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #20261072C2semidirect
4. #20261071C2semidirect
5. #20261070C2semidirect
6. #20261069C2semidirect
7. #20261068C2semidirect
8. #107×D97C2semidirect
9. #103×D97C2semidirect
11. #484×C97C2semidirect
30. #15690744C22semidirect
31. #15690743C22semidirect
32. #15690743C22semidirect
33. #15690735C22semidirect
34. #15690734C22semidirect
35. #15690733C22semidirect
40. #107×C97C22semidirect
41. #103×C97C22semidirect
141. D97#484direct (N × H)
143. C97C2×#484semidirect
151. #484D97direct (N × H)
178. #107C2×D97semidirect
182. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D972
C2×C4⋊C4×D972
#202610722
#202610712
#202610702
#202610692
#202610682
#107×D972
#103×D972
C22⋊C8×D972
#484×C972
C2×#48497