Finite Group Databasebeta

← #27151661

Subgroups — #27151661

Order 12416.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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#20261246C211yesmaximal, characteristic
2C22×C4⋊Dic97C211yesmaximal, characteristic, complemented
3#20261071C212yesmaximal, complemented
4#20261066C212yesmaximal, complemented
5#20261054C212yesmaximal, complemented
6#20261049C212yesmaximal, complemented
7#20260777C212yesmaximal, complemented
8#20260737C212yesmaximal, complemented
9#407×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C23×Dic97—21no—
12#15690793—21no—
13#15690789C2211yescharacteristic
14#15690789C2211yescharacteristic
15#15690784C2211yescharacteristic
16#15690784C2211yescharacteristic
17#15690779—21no—
18C4×C22⋊D97—21no—
19C4×C22⋊D97—21no—
20#15690769C2211yescharacteristic
21C2×C4⋊Dic97—21no—
22C2×C4⋊Dic97—21no—
23C2×C4⋊Dic97—21no—
24C2×C4⋊Dic97C2212yes—
25C2×C4⋊Dic97C2212yes—
26C2×C4⋊Dic97C2212yes—
27#15690744C2214yescomplemented
28#15690742C2214yescomplemented
29#15690735C2214yescomplemented
30#15690732C2214yescomplemented
31C23×C4×C97C2211yescharacteristic
32C2×M16×C97C2212yes—
33C22⋊C8×C97C2212yes—
34C22⋊C8×C97C2212yes—
35#13095511—24no—
36C2×Dic194C2311yescharacteristic
37C2×D388C2311yescharacteristic
38C2×C22⋊D97—21no—
39C2×C22⋊D97—21no—
40C2×C22⋊D97—21no—
41C22×Dic97—22no—
42C22×Dic97—22no—
43C22×Dic97—22no—
44C22×Dic97—44no—
45C2×C4×D97—21no—
46C4⋊Dic97—22no—
47C4⋊Dic97—22no—
48C4⋊Dic97—22no—
49C4⋊Dic97C2314yes—
50C4⋊Dic97C2311yescharacteristic
51C4⋊Dic97C2311yescharacteristic
52#13095490—21no—
53#13095490—21no—
54C22⋊Dic97—21no—
55C22⋊Dic97—21no—
56C22⋊Dic97—21no—
57#13095488—21no—
58#13095488—21no—
59C4×Dic97—21no—
60C24×C97D411yescharacteristic
61C22×C4×C97—24no—
62C22×C4×C97D411yescharacteristic
63C22×C4×C97D411yescharacteristic
64C22×C4×C97D411yescharacteristic
65C22×C4×C97C2311yescharacteristic
66C22×C4×C97C2311yescharacteristic
67C22×C4×C97C2311yescharacteristic
68M16×C97—24no—
69C2×C8×C97C2314yes—
70C23×C97—22no—
71C23×C97—22no—
72C23×C97—22no—
73C23×C97C2×D411yescharacteristic
74C23×C97C2×D411yescharacteristic
75C23×C97C2×D411yescharacteristic
76C22×D97—21no—
77C2×C4×C97—22no—
78C2×C4×C97—22no—
79C2×C4×C97—22no—
80C2×C4×C97—22no—
81C2×C4×C97—22no—
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97#4114yes—
86C2×C4×C97C2411yescharacteristic
87C22⋊D97—42no—
88C22⋊D97—42no—
89C22⋊D97—42no—
90C2×Dic97—21no—
91C2×Dic97—21no—
92C2×Dic97—21no—
93C2×Dic97—24no—
94C2×Dic97—44no—
95C2×Dic97—44no—
96C2×Dic97—44no—
97D388—22no—
98C4×D97—42no—
99Dic194—22no—
100C8×C97—24no—
101C22×C97—28no—
102C22×C97—21no—
103C22×C97—21no—
104C22×C97—21no—
105C22×C97—21no—
106C22×C97—21no—
107C22×C97—21no—
108C22×C97#12312yes—
109C22×C97AGL(1,ℤ/8)12yes—
110C22×C97#13712yes—
111C22×C97C22×D411yescharacteristic
112C2×D97—41no—
113C2×D97—41no—
114C2×D97—41no—
115C4×C97—21no—
116C4×C97—21no—
117C4×C97#12312yes—
118C4×C97C2×#4112yes—
119Dic97—41no—
120Dic97—44no—
121Dic97—41no—
122Dic97—41no—
123C2×C97—22no—
124C2×C97—22no—
125C2×C97—22no—
126C2×C97C2×#12311yescharacteristic
127C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
128C2×C97C2×#13711yescharacteristic
129D97—81no—
130#3092—971nomaximal, Sylow
131C97#309211yescharacteristic, complemented, Sylow
132#525—971no—
133C22×C4⋊C4—971no—
134#484—972no—
135#483—972no—
136#481—972no—
137#480—972no—
138#428—972no—
139#417—972no—
140#407D9711yescharacteristic, complemented
141C2×#41—971no—
142C23×C4—1941no—
143C23×C4C2×D9711yescharacteristic
144C2×M16C2×D9712yes—
145#123—1941no—
146C22⋊Q8—971no—
147C22⋊Q8—971no—
148#121—971no—
149#121—971no—
150#120—1941no—
151C4×D4—1941no—
152C4×D4—1941no—
153#117—971no—
154C2×C4⋊C4—972no—
155C2×C4⋊C4—972no—
156C2×C4⋊C4—972no—
157C2×C4⋊C4—1941no—
158C2×C4⋊C4—1941no—
159C2×C4⋊C4—1941no—
160#107—974no—
161#106—974no—
162#103—974no—
163#102—974no—
164C22⋊C8C2×D9712yes—
165C22⋊C8C2×D9712yes—
166C24D38811yescharacteristic
167#41—1944no—
168C2×Q8—971no—
169C2×D4—971no—
170C2×D4—1941no—
171C2×D4—1941no—
172C2×D4—1941no—
173C22×C4—24no—
174C22×C4—3884no—
175C22×C4—1942no—
176C22×C4—1942no—
177C22×C4—1942no—
178C22×C4—1941no—
179C22×C4D38811yescharacteristic
180C22×C4D38811yescharacteristic
181C22×C4D38811yescharacteristic
182C22×C4C22×D9711yescharacteristic
183C22×C4C22×D9711yescharacteristic
184C22×C4C22×D9711yescharacteristic
185M16—24no—
186C2×C8C22×D9714yes—
187C4⋊C4—974no—
188C4⋊C4—971no—
189C4⋊C4—971no—
190C4⋊C4—1941no—
191C4⋊C4—1941no—
192C4⋊C4—1942no—
193C4⋊C4—1942no—
194C4⋊C4—1942no—
195C22⋊C4—1941no—
196C22⋊C4—1941no—
197C22⋊C4—1941no—
198C22⋊C4—1941no—
199C22⋊C4—1941no—
200C42—1941no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—1941no—
205C23C2×D38811yescharacteristic
206C23C2×D38811yescharacteristic
207C23C2×D38811yescharacteristic
208Q8—1942no—
209D4—3882no—
210D4—3882no—
211D4—3882no—
212D4—1942no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—3884no—
219C2×C4—3884no—
220C2×C4—3884no—
221C2×C4—3882no—
222C2×C4—1941no—
223C2×C4—1941no—
224C2×C4—1941no—
225C2×C4—1944no—
226C2×C4C2×D38811yescharacteristic
227C2×C4C2×D38811yescharacteristic
228C2×C4C2×D38811yescharacteristic
229C2×C4#1309551014yes—
230C2×C4C23×D9711yescharacteristic
231C8—24no—
232C22—28no—
233C22—21no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—3881no—
240C22—3881no—
241C22—3881no—
242C22#1569079412yes—
243C22#1569083812yes—
244C22#1569084312yes—
245C22C22×D38811yescharacteristic
246C4—21no—
247C4—21no—
248C4—3881no—
249C4—3884no—
250C4—3881no—
251C4—3881no—
252C4#1569079412yes—
253C4C2×#1309551012yes—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—7761no—
258C2C2×#1569079411yescharacteristic
259C2C2×#1569083811yescharacteristic
260C2C2×#1569084311yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#407×C97C2
socleC22×C97C22×D4
prime radical#20261246C2
hypercenter#407D97
hypocenterC97#3092
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic97C2semidirect
3. #20261071C2semidirect
4. #20261066C2semidirect
5. #20261054C2semidirect
6. #20261049C2semidirect
7. #20260777C2semidirect
8. #20260737C2semidirect
9. #407×C97C2semidirect
27. #15690744C22semidirect
28. #15690742C22semidirect
29. #15690735C22semidirect
30. #15690732C22semidirect
131. C97#3092semidirect
140. #407D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
C22×C4⋊Dic972
#202610712
#202610662
#202610542
#202610492
#202607772
#202607372
#407×C972
#309297