Finite Group Databasebeta

← #27140055

Subgroups — #27140055

Order 12416.

312 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 81 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×#15690770C211yesmaximal, characteristic
2#20260785C212yesmaximal, complemented
3#20260459C211yesmaximal, characteristic, complemented
4#20260458C212yesmaximal, complemented
5#323×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C411yescharacteristic
7C23×Dic97C411yescharacteristic
8C22×C4×D97C411yescharacteristic
9#15690770—22no—
10#15690770—22no—
11C2×#13095490C411yescharacteristic
12C2×#13095488C2211yescharacteristic
13C2×C4×Dic97C2211yescharacteristic
14#15690739—22no—
15C22⋊#11715895C2212yes—
16C8×Dic97—22no—
17C2×C22⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C23×D97C2×C411yescharacteristic
20C2×C22⋊D97—22no—
21C2×C22⋊D97—22no—
22C2×C22⋊D97C814yescomplemented
23C22×Dic97—21no—
24C22×Dic97—21no—
25C22×Dic97—42no—
26C22×Dic97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2×C411yescharacteristic
29C2×C4×D97—21no—
30C2×C4×D97—21no—
31C2×C4×D97—42no—
32C2×#11715895—22no—
33#13095490—22no—
34#13095488—22no—
35C4×Dic97D412yes—
36C4×Dic97D412yes—
37C24×C97C2×C411yescharacteristic
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97C2311yescharacteristic
40C2×C8×C97—22no—
41C22⋊C4×C97—22no—
42C23×C97—21no—
43C23×C97—21no—
44C23×C97—41no—
45C23×C97—41no—
46C23×C97C2×C812yes—
47C23×C97C22×C411yescharacteristic
48C22×D97—21no—
49C22×D97—21no—
50C22×D97—41no—
51C22×D97—41no—
52C22×D97C2×C812yes—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97C2×D412yes—
57C22⋊D97—28no—
58C2×Dic97—22no—
59C2×Dic97—21no—
60C2×Dic97—42no—
61C2×Dic97—42no—
62C2×Dic97—41no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C22⋊C412yes—
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C2×C812yes—
67C2×Dic97M1612yes—
68C4×D97—42no—
69C4×D97—42no—
70C8×C97—42no—
71#11715895—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—22no—
75C22×C97—42no—
76C22×C97—42no—
77C22×C97—41no—
78C22×C97—41no—
79C22×C97C2×C22⋊C411yescharacteristic
80C22×C97C22×C811yescharacteristic
81C22×C97C2×M1611yescharacteristic
82C2×D97—22no—
83C2×D97—42no—
84C2×D97—41no—
85C2×D97—41no—
86C2×D97—42no—
87C4×C97—22no—
88C4×C97—41no—
89Dic97—21no—
90Dic97—41no—
91Dic97C22⋊C814yes—
92Dic97#10012yes—
93C2×C97—21no—
94C2×C97—21no—
95C2×C97—42no—
96C2×C97C2×C22⋊C811yescharacteristic
97C2×C97#41011yescharacteristic
98C2×C97C2×#10011yescharacteristic
99D97—42no—
100#1460—971nomaximal, Sylow
101C97#146011yescharacteristic, complemented, Sylow
102C2×C4×D4—971no—
103#431—972no—
104#323—971no—
105#323—972no—
106#323D9711yescharacteristic, complemented
107C22×D4—971no—
108C23×C4—971no—
109C23×C4—971no—
110C4×D4—1942no—
111C4×D4—1942no—
112C2×C4⋊C4—971no—
113C2×C22⋊C4—971no—
114C2×C22⋊C4C2×D9711yescharacteristic
115C2×C42—971no—
116C4⋊C8—1942no—
117C22⋊C8—972no—
118C22⋊C8C2×D9712yes—
119C4×C8—1942no—
120C24—971no—
121C24C4×D9711yescharacteristic
122C2×D4—974no—
123C2×D4—1942no—
124C2×D4—1942no—
125C22×C4—971no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—3882no—
129C22×C4—3882no—
130C22×C4—1941no—
131C22×C4—1941no—
132C22×C4—1941no—
133C22×C4—1941no—
134C22×C4C4×D9711yescharacteristic
135C22×C4C22×D9711yescharacteristic
136C2×C8—22no—
137C2×C8—1942no—
138C4⋊C4—1942no—
139C22⋊C4—22no—
140C22⋊C4—1942no—
141C42—972no—
142C42—972no—
143C23—21no—
144C23—21no—
145C23—41no—
146C23—41no—
147C23—972no—
148C23—3881no—
149C23—3881no—
150C23—1941no—
151C23—1941no—
152C23C8×D9712yes—
153C23C2×C4×D9711yescharacteristic
154D4—1948no—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—972no—
159C2×C4—971no—
160C2×C4—972no—
161C2×C4—972no—
162C2×C4—971no—
163C2×C4—3882no—
164C2×C4—3882no—
165C2×C4—3882no—
166C2×C4—3882no—
167C2×C4—3881no—
168C2×C4—1942no—
169C2×C4—1941no—
170C2×C4D4×D9712yes—
171C8—42no—
172C8—3882no—
173C22—21no—
174C22—21no—
175C22—22no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—3882no—
181C22—3882no—
182C22—3881no—
183C22—3881no—
184C22—1942no—
185C22C22⋊C4×D9711yescharacteristic
186C22C2×C8×D9711yescharacteristic
187C22M16×D9711yescharacteristic
188C4—22no—
189C4—41no—
190C4—974no—
191C4—972no—
192C4—3881no—
193C4—1941no—
194C2—21no—
195C2—21no—
196C2—42no—
197C2—3882no—
198C2C22⋊C8×D9711yescharacteristic
199C2#2026070711yescharacteristic
200C2#100×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC22×C97C22×C8
FrattiniC22×C4C22×D97
Fitting#323×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×C4×C97C23
hypercenter#323D97
hypocenterC97#1460
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20260785C2semidirect
3. #20260459C2semidirect
4. #20260458C2semidirect
5. #323×C97C2semidirect
22. C2×C22⋊D97C8semidirect
101. C97#1460semidirect
106. #323D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907702
#202607852
#202604592
#202604582
#323×C972
#146097