Finite Group Databasebeta

← #27140061

Subgroups — #27140061

Order 12416.

440 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 71 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×C22⋊C4×D97C211yesmaximal, characteristic
2#20260697C212yesmaximal, complemented
3#20260459C211yesmaximal, characteristic, complemented
4#20260458C212yesmaximal, complemented
5#323×C97C211yesmaximal, characteristic, complemented
6C24×D97C411yescharacteristic
7C22×C4×D97C411yescharacteristic
8C22×C4×D97C2211yescharacteristic
9C2×#13095494—22no—
10C2×C22⋊Dic97C411yescharacteristic
11C2×#13095488C411yescharacteristic
12C2×#13095488C2211yescharacteristic
13C22⋊C4×D97—22no—
14C22⋊C4×D97—22no—
15C22⋊#11715895C2212yescomplemented
16#15690722—22no—
17C2×C22⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yescomplemented
19C23×D97—22no—
20C23×D97—21no—
21C23×D97—21no—
22C23×D97—22no—
23C23×D97—42no—
24C23×D97—42no—
25C23×D97C2×C411yescharacteristic
26C23×D97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2×C411yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97—42no—
32C2×C4×D97—42no—
33C2×C4×D97D412yes—
34C2×C4×D97D412yes—
35#13095494—44no—
36C2×#11715895—22no—
37C22⋊Dic97—22no—
38#13095488—22no—
39#13095488—22no—
40C24×C97C2×C411yescharacteristic
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97C2311yescharacteristic
43C2×C8×C97—22no—
44C22⋊C4×C97—22no—
45C23×C97—21no—
46C23×C97—21no—
47C23×C97—41no—
48C23×C97—41no—
49C23×C97M1612yescomplemented
50C23×C97C22×C411yescharacteristic
51C22×D97—21no—
52C22×D97—21no—
53C22×D97—22no—
54C22×D97—22no—
55C22×D97—24no—
56C22×D97—24no—
57C22×D97—21no—
58C22×D97—21no—
59C22×D97—48no—
60C22×D97—48no—
61C22×D97—42no—
62C22×D97—42no—
63C22×D97—42no—
64C22×D97—42no—
65C22×D97—41no—
66C22×D97—41no—
67C22×D97C22⋊C411yescharacteristic
68C22×D97C22⋊C411yescharacteristic
69C22×D97M1612yescomplemented
70C2×C4×C97—21no—
71C2×C4×C97—41no—
72C2×C4×C97—41no—
73C2×C4×C97C2×D412yes—
74C2×Dic97—21no—
75C2×Dic97—41no—
76C2×Dic97—41no—
77C2×Dic97C22⋊C412yes—
78C4×D97—24no—
79C4×D97—42no—
80C4×D97—42no—
81C4×D97—42no—
82C8×C97—42no—
83#11715895—42no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—22no—
87C22×C97—42no—
88C22×C97—42no—
89C22×C97—41no—
90C22×C97—41no—
91C22×C97C2×C22⋊C411yescharacteristic
92C22×C97C2×M1611yescharacteristic
93C22×C97C2×M1611yescharacteristic
94C2×D97—24no—
95C2×D97—24no—
96C2×D97—21no—
97C2×D97—21no—
98C2×D97—21no—
99C2×D97—21no—
100C2×D97—22no—
101C2×D97—48no—
102C2×D97—48no—
103C2×D97—42no—
104C2×D97—41no—
105C2×D97—41no—
106C2×D97—42no—
107C2×D97#9912yes—
108C2×D97#10012yes—
109C4×C97—22no—
110C4×C97—41no—
111Dic97—22no—
112Dic97—41no—
113C2×C97—21no—
114C2×C97—21no—
115C2×C97—42no—
116C2×C97#40711yescharacteristic
117C2×C97C2×#9911yescharacteristic
118C2×C97C2×#10011yescharacteristic
119D97—22no—
120D97—22no—
121D97—42no—
122#1462—971nomaximal, Sylow
123C97#146211yescharacteristic, complemented, Sylow
124C22×C22⋊C4—971no—
125#407—972no—
126#323—971no—
127#323—972no—
128#323D9711yescharacteristic, complemented
129C25—971no—
130C23×C4—971no—
131C23×C4—971no—
132C2×M16—1942no—
133C2×C22⋊C4—971no—
134C2×C22⋊C4—971no—
135C2×C22⋊C4—971no—
136C2×C22⋊C4—1942no—
137C2×C22⋊C4—1942no—
138C2×C22⋊C4C2×D9711yescharacteristic
139C22⋊C8—972no—
140C22⋊C8—1942no—
141C22⋊C8C2×D9712yescomplemented
142C24—971no—
143C24—971no—
144C24—3882no—
145C24—3882no—
146C24—1941no—
147C24—1941no—
148C24—1942no—
149C24—1942no—
150C24C4×D9711yescharacteristic
151C22×C4—971no—
152C22×C4—972no—
153C22×C4—972no—
154C22×C4—971no—
155C22×C4—3882no—
156C22×C4—3882no—
157C22×C4—1941no—
158C22×C4—1941no—
159C22×C4C4×D9711yescharacteristic
160C22×C4C22×D9711yescharacteristic
161M16—3884no—
162C2×C8—22no—
163C2×C8—1942no—
164C22⋊C4—22no—
165C22⋊C4—1942no—
166C22⋊C4—1942no—
167C22⋊C4—1942no—
168C23—21no—
169C23—21no—
170C23—41no—
171C23—41no—
172C23—971no—
173C23—971no—
174C23—972no—
175C23—3881no—
176C23—3881no—
177C23—3882no—
178C23—3882no—
179C23—3888no—
180C23—3888no—
181C23—3882no—
182C23—3882no—
183C23—1942no—
184C23—1942no—
185C23—1944no—
186C23—1944no—
187C23—1941no—
188C23—1941no—
189C23—1941no—
190C23—1941no—
191C23#1309549412yescomplemented
192C23C2×C4×D9711yescharacteristic
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—972no—
197C2×C4—3882no—
198C2×C4—3882no—
199C2×C4—3882no—
200C2×C4—3881no—
201C2×C4—3881no—
202C2×C4—1944no—
203C2×C4—1941no—
204C2×C4D4×D9712yes—
205C8—42no—
206C8—3882no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—42no—
211C22—42no—
212C22—41no—
213C22—41no—
214C22—972no—
215C22—972no—
216C22—3882no—
217C22—3882no—
218C22—3888no—
219C22—3888no—
220C22—3881no—
221C22—3881no—
222C22—1944no—
223C22—1944no—
224C22—1942no—
225C22—1941no—
226C22—1941no—
227C22—1941no—
228C22—1941no—
229C22C22⋊C4×D9711yescharacteristic
230C22M16×D9711yescharacteristic
231C22C2×#1309549411yescharacteristic
232C4—22no—
233C4—41no—
234C4—3881no—
235C4—1942no—
236C2—21no—
237C2—21no—
238C2—42no—
239C2—3882no—
240C2—1942no—
241C2—1942no—
242C2#2026069711yescharacteristic
243C2#99×D9711yescharacteristic
244C2#100×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#323×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×D97C4
cosocleC22×C4×C97C23
hypercenter#323D97
hypocenterC97#1462
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20260697C2semidirect
3. #20260459C2semidirect
4. #20260458C2semidirect
5. #323×C97C2semidirect
15. C22⋊#11715895C22semidirect
18. C22⋊C8×C97C22semidirect
49. C23×C97M16semidirect
69. C22×D97M16semidirect
123. C97#1462semidirect
128. #323D97semidirect
141. C22⋊C8C2×D97semidirect
191. C23#13095494semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D972
#202606972
#202604592
#202604582
#323×C972
#146297