Finite Group Databasebeta

← #27152126

Subgroups — #27152126

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#20261283C211yesmaximal, characteristic
2C4⋊Q8×D97C211yesmaximal, characteristic, complemented
3#20261093C214yesmaximal, complemented
4#20261087C211yesmaximal, characteristic, complemented
5#20261086C211yesmaximal, characteristic, complemented
6#20261081C211yesmaximal, characteristic, complemented
7#20260733C214yesmaximal, complemented
8C4×#13095494C211yesmaximal, characteristic, complemented
9#486×C97C211yesmaximal, characteristic, complemented
10C2×#13095510C2212yes—
11C2×Q8×D97—21no—
12C2×#13095494C2212yes—
13#15690810C2211yescharacteristic
14#15690810C2211yescharacteristic
15#15690809—21no—
16#15690809C2212yes—
17#15690808—21no—
18#15690797C2212yes—
19#15690781—21no—
20#15690781—21no—
21D4×Dic97—21no—
22D4×Dic97—21no—
23C4⋊C4×D97—21no—
24C4⋊C4×D97C2212yes—
25C42×D97C2211yescharacteristic
26#15690735C2214yes—
27#15690733C2214yes—
28#15690731C2214yes—
29#15690720C2212yes—
30C4×#11715895C2211yescharacteristic
31C4⋊Q8×C97C2211yescharacteristic
32#127×C97C2211yescharacteristic
33#102×C97C2214yes—
34C4×C8×C97C2211yescharacteristic
35#13095510—28no—
36C2×Dic194—21no—
37C2×Dic194—22no—
38C2×Dic194C2312yes—
39Q8×D97—44no—
40C2×C22⋊D97—22no—
41C22×Dic97—22no—
42C2×C4×D97—22no—
43C2×C4×D97D411yescharacteristic
44C2×C4×D97D411yescharacteristic
45C2×C4×D97C2311yescharacteristic
46#13095494—24no—
47C2×#11715895C2312yes—
48C4⋊Dic97—21no—
49C4⋊Dic97—21no—
50C4⋊Dic97—21no—
51C4⋊Dic97C2312yes—
52#13095490—22no—
53#13095490—22no—
54C22⋊Dic97—22no—
55C22⋊Dic97—22no—
56C4×Dic97D411yescharacteristic
57C4×Dic97D411yescharacteristic
58C4×Dic97C2311yescharacteristic
59C2×Q8×C97—21no—
60C2×D4×C97—21no—
61C2×D4×C97—21no—
62C2×D4×C97C2312yes—
63C2×C8×C97C2312yes—
64C4⋊C4×C97—21no—
65C4⋊C4×C97C2312yes—
66C42×C97C2311yescharacteristic
67C23×C97—22no—
68C22×D97C2×D411yescharacteristic
69Q8×C97—42no—
70D4×C97—24no—
71D4×C97—42no—
72D4×C97—42no—
73C2×C4×C97—22no—
74C2×C4×C97C2×D411yescharacteristic
75C2×C4×C97C2×D411yescharacteristic
76C2×C4×C97C2411yescharacteristic
77C22⋊D97—44no—
78C2×Dic97—22no—
79C2×Dic97—22no—
80C2×Dic97—44no—
81C2×Dic97C2×D411yescharacteristic
82C2×Dic97C2×D411yescharacteristic
83C2×Dic97C2×D411yescharacteristic
84C4×D97—22no—
85C4×D97—22no—
86C4×D97—44no—
87C4×D97#4114yes—
88Dic194—24no—
89Dic194—44no—
90Dic194—42no—
91C8×C97—22no—
92#11715895—22no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97—42no—
96C22×C97C22×D411yescharacteristic
97C2×D97—21no—
98C2×D97—21no—
99C2×D97#12412yes—
100C4×C97—42no—
101C4×C97AGL(1,ℤ/8)12yes—
102C4×C97#13712yes—
103C4×C97C2×#4112yes—
104Dic97—21no—
105Dic97—21no—
106Dic97—42no—
107Dic97—42no—
108Dic97#12412yes—
109C2×C97—82no—
110C2×C97C2×#12411yescharacteristic
111C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
112C2×C97C2×#13711yescharacteristic
113D97—22no—
114#3141—971nomaximal, Sylow
115C97#314111yescharacteristic, complemented, Sylow
116#532—971no—
117C2×C4⋊Q8—971no—
118#489—974no—
119#487—971no—
120#487—971no—
121#486—971no—
122#486D9711yescharacteristic, complemented
123#417—974no—
124C4×M16—971no—
125C2×#41—972no—
126C22×Q8—1941no—
127C2×M16—972no—
128C4⋊Q8—972no—
129C4⋊Q8—971no—
130C4⋊Q8—971no—
131C4⋊Q8—1941no—
132C4⋊Q8—1941no—
133C4⋊Q8C2×D9711yescharacteristic
134#127C2×D9711yescharacteristic
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#103—974no—
145#102—974no—
146#102C2×D9714yes—
147#97—972no—
148C4×C8—971no—
149C4×C8C2×D9711yescharacteristic
150#41—1948no—
151C2×Q8—21no—
152C2×Q8—972no—
153C2×Q8—3884no—
154C2×Q8—1942no—
155C2×Q8—1941no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—1942no—
159C2×D4C22×D9712yes—
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—972no—
170C4⋊C4—1942no—
171C4⋊C4—1942no—
172C4⋊C4—1941no—
173C4⋊C4—1941no—
174C4⋊C4—1941no—
175C4⋊C4C22×D9712yes—
176C22⋊C4—1942no—
177C22⋊C4—1942no—
178C42—971no—
179C42—971no—
180C42—971no—
181C42C22×D9711yescharacteristic
182C23—22no—
183C23—971no—
184Q8—42no—
185Q8—3884no—
186Q8—3882no—
187Q8—1944no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—3884no—
192C2×C4—22no—
193C2×C4—971no—
194C2×C4—974no—
195C2×C4—971no—
196C2×C4—971no—
197C2×C4—3884no—
198C2×C4—3884no—
199C2×C4—1942no—
200C2×C4—1942no—
201C2×C4—1942no—
202C2×C4—1942no—
203C2×C4D4×D9711yescharacteristic
204C2×C4D4×D9711yescharacteristic
205C2×C4C23×D9711yescharacteristic
206C8—22no—
207C8—1942no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—972no—
212C22—1941no—
213C22—1941no—
214C22C2×D4×D9711yescharacteristic
215C4—42no—
216C4—972no—
217C4—3882no—
218C4—3882no—
219C4—1941no—
220C4—1941no—
221C4#1569083412yes—
222C4#1569083912yes—
223C4#41×D9712yes—
224C2—82no—
225C2—1942no—
226C2#124×D9711yescharacteristic
227C2C2×#1569083411yescharacteristic
228C2C2×#1569083911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#486×C97C2
socleC22×C97C22×D4
prime radical#20261283C2
hypercenter#486D97
hypocenterC97#3141
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D97C2semidirect
3. #20261093C2semidirect
4. #20261087C2semidirect
5. #20261086C2semidirect
6. #20261081C2semidirect
7. #20260733C2semidirect
8. C4×#13095494C2semidirect
9. #486×C97C2semidirect
115. C97#3141semidirect
122. #486D97semidirect

Maximal subgroups

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

subgroupindex
#202612832
C4⋊Q8×D972
#202610932
#202610872
#202610862
#202610812
#202607332
C4×#130954942
#486×C972
#314197