Finite Group Databasebeta

← #27152131

Subgroups — #27152131

Order 12416.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#20261286C211yesmaximal, characteristic
2C4⋊Q8×D97C211yesmaximal, characteristic, complemented
3#20261096C214yesmaximal, complemented
4#20261085C211yesmaximal, characteristic, complemented
5#20261083C211yesmaximal, characteristic, complemented
6#20261082C211yesmaximal, characteristic, complemented
7#20260735C214yesmaximal, complemented
8C4×#13095494C211yesmaximal, characteristic, complemented
9#487×C97C211yesmaximal, characteristic, complemented
10C2×#13095512C2212yes—
11C2×Q8×D97—21no—
12C2×#13095494C2212yes—
13#15690810C2211yescharacteristic
14#15690809—21no—
15#15690809C2212yes—
16#15690808—21no—
17#15690807C2211yescharacteristic
18#15690799C2212yes—
19#15690785—21no—
20#15690785—21no—
21#15690772—21no—
22#15690772—21no—
23C4⋊C4×D97—21no—
24C4⋊C4×D97C2212yes—
25C42×D97C2211yescharacteristic
26#15690734C2214yes—
27#15690732C2214yes—
28#15690730C2214yes—
29#15690720C2212yes—
30C4×#11715895C2211yescharacteristic
31C4⋊Q8×C97C2211yescharacteristic
32C4⋊Q8×C97C2211yescharacteristic
33#103×C97C2214yes—
34C4×C8×C97C2211yescharacteristic
35#13095512—28no—
36C2×Dic194—21no—
37C2×Dic194—22no—
38Q8×D97—44no—
39C2×D388—22no—
40C2×D388—21no—
41C2×D388—21no—
42C2×D388C2312yes—
43C2×C4×D97—22no—
44C2×C4×D97—22no—
45C2×C4×D97D411yescharacteristic
46C2×C4×D97D411yescharacteristic
47C2×C4×D97C2311yescharacteristic
48#13095494—24no—
49C2×#11715895C2312yes—
50C4⋊Dic97—21no—
51C4⋊Dic97C2312yes—
52#13095490—22no—
53#13095490—22no—
54#13095488—22no—
55#13095488—22no—
56C4×Dic97D411yescharacteristic
57C4×Dic97D411yescharacteristic
58C4×Dic97C2311yescharacteristic
59C2×Q8×C97—21no—
60C2×Q8×C97C2312yes—
61C2×C8×C97C2312yes—
62C4⋊C4×C97—21no—
63C4⋊C4×C97—21no—
64C4⋊C4×C97—21no—
65C4⋊C4×C97C2312yes—
66C42×C97C2311yescharacteristic
67C22×D97—22no—
68C22×D97C2×D411yescharacteristic
69Q8×C97—24no—
70Q8×C97—42no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97C2×D411yescharacteristic
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2411yescharacteristic
76C2×Dic97—22no—
77C2×Dic97C2×D411yescharacteristic
78C2×Dic97C2×D411yescharacteristic
79C2×Dic97C2×D411yescharacteristic
80D388—24no—
81D388—44no—
82D388—42no—
83D388—42no—
84C4×D97—22no—
85C4×D97—22no—
86C4×D97—44no—
87C4×D97—44no—
88C4×D97#4114yes—
89Dic194—44no—
90Dic194—42no—
91C8×C97—22no—
92#11715895—22no—
93C22×C97C22×D411yescharacteristic
94C2×D97—21no—
95C2×D97—21no—
96C2×D97—42no—
97C2×D97—42no—
98C2×D97—42no—
99C2×D97#12412yes—
100C4×C97—42no—
101C4×C97—42no—
102C4×C97AGL(1,ℤ/8)12yes—
103C4×C97#13712yes—
104C4×C97C2×#4112yes—
105Dic97—21no—
106Dic97—21no—
107Dic97—42no—
108Dic97#12412yes—
109C2×C97C2×#12411yescharacteristic
110C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C97C2×#13711yescharacteristic
112D97—22no—
113D97—82no—
114#3141—971nomaximal, Sylow
115C97#314111yescharacteristic, complemented, Sylow
116#532—971no—
117C2×C4⋊Q8—971no—
118#489—974no—
119#487—971no—
120#487D9711yescharacteristic, complemented
121#486—971no—
122#486—971no—
123#417—974no—
124C4×M16—971no—
125C2×#41—972no—
126C22×Q8—1941no—
127C2×M16—972no—
128C4⋊Q8—972no—
129C4⋊Q8—971no—
130C4⋊Q8—1941no—
131C4⋊Q8—1941no—
132C4⋊Q8C2×D9711yescharacteristic
133C4⋊Q8C2×D9711yescharacteristic
134#127—971no—
135#124—972no—
136#121—1941no—
137#121—1941no—
138C4×D4—1941no—
139C4×D4—1941no—
140C2×C4⋊C4—972no—
141C2×C4⋊C4—1941no—
142C2×C42—971no—
143#103—974no—
144#103C2×D9714yes—
145#102—974no—
146#102—974no—
147#97—972no—
148C4×C8—971no—
149C4×C8C2×D9711yescharacteristic
150#41—1948no—
151C2×Q8—21no—
152C2×Q8—3884no—
153C2×Q8—1942no—
154C2×Q8—1941no—
155C2×Q8C22×D9712yes—
156C2×D4—972no—
157C2×D4—1942no—
158C2×D4—1941no—
159C2×D4—1941no—
160C22×C4—971no—
161C22×C4—971no—
162C22×C4—971no—
163C22×C4—1942no—
164C22×C4—1942no—
165M16—1944no—
166C2×C8—972no—
167C2×C8C22×D9712yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—972no—
172C4⋊C4—1942no—
173C4⋊C4—1942no—
174C4⋊C4—1941no—
175C4⋊C4C22×D9712yes—
176C22⋊C4—1942no—
177C22⋊C4—1942no—
178C42—971no—
179C42—971no—
180C42—971no—
181C42C22×D9711yescharacteristic
182C23—971no—
183C23—1942no—
184Q8—24no—
185Q8—42no—
186Q8—3884no—
187Q8—3882no—
188D4—3884no—
189D4—3882no—
190D4—3882no—
191D4—1944no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—971no—
195C2×C4—974no—
196C2×C4—971no—
197C2×C4—971no—
198C2×C4—3884no—
199C2×C4—3884no—
200C2×C4—1942no—
201C2×C4—1942no—
202C2×C4—1942no—
203C2×C4D4×D9711yescharacteristic
204C2×C4D4×D9711yescharacteristic
205C2×C4C23×D9711yescharacteristic
206C8—22no—
207C8—1942no—
208C22—972no—
209C22—3882no—
210C22—3882no—
211C22—3882no—
212C22—1941no—
213C22—1941no—
214C22C2×D4×D9711yescharacteristic
215C4—42no—
216C4—42no—
217C4—972no—
218C4—3882no—
219C4—1941no—
220C4—1941no—
221C4#1569083712yes—
222C4#1569084212yes—
223C4#41×D9712yes—
224C2—7762no—
225C2—1942no—
226C2#124×D9711yescharacteristic
227C2C2×#1569083711yescharacteristic
228C2C2×#1569084211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#487×C97C2
socleC22×C97C22×D4
prime radical#20261286C2
hypercenter#487D97
hypocenterC97#3141
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D97C2semidirect
3. #20261096C2semidirect
4. #20261085C2semidirect
5. #20261083C2semidirect
6. #20261082C2semidirect
7. #20260735C2semidirect
8. C4×#13095494C2semidirect
9. #487×C97C2semidirect
115. C97#3141semidirect
120. #487D97semidirect

Maximal subgroups

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

subgroupindex
#202612862
C4⋊Q8×D972
#202610962
#202610852
#202610832
#202610822
#202607352
C4×#130954942
#487×C972
#314197