Finite Group Databasebeta

← #27150198

Subgroups — #27150198

Order 12416.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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×#15690857C211yesmaximal, characteristic
2#20261212C212yesmaximal, complemented
3#20260709C211yesmaximal, characteristic, complemented
4#20260708C212yesmaximal, complemented
5#20260707C218yesmaximal, complemented
6#410×C97C211yesmaximal, characteristic, complemented
7#15690857—24no—
8#15690857—24no—
9C2×#13095511C412yescomplemented
10C2×#13095511C2211yescharacteristic
11C2×#13095510C412yescomplemented
12C2×#13095510C2214yes—
13C22×Dic194C412yescomplemented
14C22×Dic194C2211yescharacteristic
15C2×Q8×D97C412yescomplemented
16C4×Dic194—24no—
17#15690775—24no—
18#15690769C2212yes—
19#15690766C2218yes—
20#15690725C2214yes—
21#15690724C2218yes—
22C2×#41×C97C2211yescharacteristic
23#117×C97C2212yes—
24#99×C97C2214yes—
25#13095511—22no—
26#13095511—22no—
27#13095511—24no—
28#13095511—24no—
29#13095510—216no—
30#13095510—28no—
31C2×Dic194—24no—
32C2×Dic194—24no—
33C2×Dic194C2×C414yes—
34C2×Dic194C2×C411yescharacteristic
35C2×Dic194C2×C412yes—
36C2×Dic194C2×C412yes—
37C2×Dic194D414yes—
38C2×Dic194D414yes—
39Q8×D97—28no—
40C2×D388C2×C411yescharacteristic
41C2×C22⋊D97C2×C414yes—
42C2×C22⋊D97C2312yes—
43C22×Dic97C2×C414yes—
44C22×Dic97C2312yes—
45C2×C4×D97C2×C412yes—
46C2×C4×D97C2×C414yes—
47C4⋊Dic97—22no—
48#13095490—28no—
49C22⋊Dic97C2314yes—
50C4×Dic97—22no—
51C4×Dic97—28no—
52#41×C97—22no—
53#41×C97—22no—
54C2×Q8×C97C2×C411yescharacteristic
55C2×D4×C97C2×C411yescharacteristic
56C2×D4×C97C2312yes—
57C22×C4×C97C2×C412yes—
58C22×C4×C97C2311yescharacteristic
59C4⋊C4×C97—22no—
60C22⋊C4×C97C2314yes—
61C42×C97—22no—
62C23×C97C22×C412yes—
63C23×C97C2411yescharacteristic
64C22×D97C22×C412yes—
65Q8×C97—22no—
66D4×C97—24no—
67D4×C97—22no—
68C2×C4×C97—22no—
69C2×C4×C97—22no—
70C2×C4×C97—24no—
71C2×C4×C97C22×C412yes—
72C2×C4×C97C22×C411yescharacteristic
73C2×C4×C97C22×C411yescharacteristic
74C2×C4×C97C2×D412yes—
75C2×C4×C97C2×D412yes—
76C22⋊D97—24no—
77C22⋊D97—28no—
78C2×Dic97—28no—
79C2×Dic97—24no—
80C2×Dic97C22×C412yes—
81C2×Dic97C22×C414yes—
82C2×Dic97C2×D418yes—
83D388—21no—
84D388—21no—
85C4×D97—24no—
86C4×D97—28no—
87Dic194—21no—
88Dic194—21no—
89Dic194—28no—
90Dic194C22⋊C418yes—
91Dic194C22⋊C414yes—
92Dic194C22⋊C414yes—
93C22×C97—22no—
94C22×C97—41no—
95C22×C97—42no—
96C22×C97C23×C411yescharacteristic
97C22×C97C22×D412yes—
98C2×D97—22no—
99C2×D97—42no—
100C4×C97—22no—
101C4×C97—44no—
102C4×C97C2×C22⋊C411yescharacteristic
103C4×C97C2×C22⋊C411yescharacteristic
104C4×C97C2×C22⋊C412yes—
105Dic97—22no—
106Dic97—44no—
107Dic97C2×C22⋊C418yes—
108C2×C97—22no—
109C2×C97—21no—
110C2×C97—42no—
111C2×C97C22×C22⋊C411yescharacteristic
112D97—42no—
113#2884—971nomaximal, Sylow
114C97#288411yescharacteristic, complemented, Sylow
115C2×2−1+4—971no—
116#519—972no—
117#410—971no—
118#410—972no—
119#410—978no—
120#410D9711yescharacteristic, complemented
1212−1+4—1944no—
1222−1+4—1944no—
123C2×#41—972no—
124C2×#41—974no—
125C2×#41—972no—
126C2×#41—971no—
127C2×#41C2×D9711yescharacteristic
128C22×Q8—972no—
129C22×Q8—972no—
130C22×Q8—971no—
131C4×Q8—1944no—
132C4×Q8—1944no—
133#117—972no—
134#117—978no—
135#117C2×D9712yes—
136#99—974no—
137#99—978no—
138#99C2×D9714yes—
139#41—22no—
140#41—22no—
141#41—1944no—
142#41—1944no—
143#41—19416no—
144#41—1948no—
145#41—1942no—
146#41—1942no—
147C2×Q8—972no—
148C2×Q8—972no—
149C2×Q8—974no—
150C2×Q8—971no—
151C2×Q8—974no—
152C2×Q8—974no—
153C2×Q8—1944no—
154C2×Q8—1944no—
155C2×Q8—1948no—
156C2×Q8C4×D9711yescharacteristic
157C2×D4—974no—
158C2×D4—972no—
159C2×D4—971no—
160C2×D4C4×D9711yescharacteristic
161C2×D4C22×D9712yes—
162C22×C4—972no—
163C22×C4—974no—
164C22×C4—974no—
165C22×C4—972no—
166C22×C4C4×D9712yes—
167C22×C4C22×D9711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—1948no—
170C4⋊C4—1942no—
171C22⋊C4—974no—
172C22⋊C4C22×D9714yes—
173C42—22no—
174C42—1948no—
175C42—1942no—
176C23—972no—
177C23C2×C4×D9712yes—
178C23C23×D9711yescharacteristic
179Q8—22no—
180Q8—974no—
181Q8—974no—
182Q8—978no—
183Q8—1948no—
184Q8—1941no—
185Q8—1941no—
186D4—24no—
187D4—22no—
188D4—1948no—
189D4—1944no—
190D4—1941no—
191D4—1941no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—974no—
196C2×C4—978no—
197C2×C4—972no—
198C2×C4—1948no—
199C2×C4—1948no—
200C2×C4—1944no—
201C2×C4—1944no—
202C2×C4C2×C4×D9712yes—
203C2×C4C2×C4×D9711yescharacteristic
204C2×C4C2×C4×D9711yescharacteristic
205C2×C4D4×D9712yes—
206C2×C4D4×D9712yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—3882no—
211C22—1942no—
212C22C22×C4×D9711yescharacteristic
213C22C2×D4×D9712yes—
214C4—22no—
215C4—44no—
216C4—978no—
217C4—3884no—
218C4—1942no—
219C4C22⋊C4×D9711yescharacteristic
220C4C22⋊C4×D9711yescharacteristic
221C4C22⋊C4×D9712yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—3882no—
226C2C2×C22⋊C4×D9711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D97
commutatorC22×C97C23×C4
FrattiniC23C23×D97
Fitting#410×C97C2
socleC2×C97C22×C22⋊C4
prime radicalC2×#15690857C2
cosocleC23×C97C24
hypercenter#410D97
hypocenterC97#2884
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20261212C2semidirect
3. #20260709C2semidirect
4. #20260708C2semidirect
5. #20260707C2semidirect
6. #410×C97C2semidirect
9. C2×#13095511C4semidirect
11. C2×#13095510C4semidirect
13. C22×Dic194C4semidirect
15. C2×Q8×D97C4semidirect
114. C97#2884semidirect
120. #410D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908572
#202612122
#202607092
#202607082
#202607072
#410×C972
#288497