Finite Group Databasebeta

← #27140051

Subgroups — #27140051

Order 12416.

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

normal subgroups only

This group has 332 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
2C22⋊C8×D97C212yesmaximal, complemented
3#20260460C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#323×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97C411yescharacteristic
7C23×Dic97C411yescharacteristic
8C22×C4×D97C2211yescharacteristic
9C2×C8×D97—22no—
10#15690770—22no—
11#15690770—22no—
12C2×#13095490C2211yescharacteristic
13C2×#13095488C411yescharacteristic
14C2×C4×Dic97C411yescharacteristic
15C22⋊#11715895C2212yes—
16#15690722—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—42no—
30C2×C4×D97D412yes—
31C2×C4×D97D412yes—
32C2×#11715895—22no—
33C8×D97—44no—
34#13095490—22no—
35#13095488—22no—
36C4×Dic97—21no—
37C4×Dic97—21no—
38C24×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2311yescharacteristic
41C2×C8×C97—22no—
42C22⋊C4×C97—22no—
43C23×C97—21no—
44C23×C97—21no—
45C23×C97—41no—
46C23×C97—41no—
47C23×C97C2×C812yescomplemented
48C23×C97C22×C411yescharacteristic
49C22×D97—41no—
50C22×D97—41no—
51C22×D97C22⋊C412yes—
52C22×D97C2×C812yes—
53C22×D97M1612yes—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D412yes—
58C22⋊D97—28no—
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97—22no—
62C2×Dic97—21no—
63C2×Dic97—42no—
64C2×Dic97—42no—
65C2×Dic97—41no—
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C2×C812yescomplemented
69C4×D97—24no—
70C4×D97—42no—
71C8×C97—42no—
72#11715895—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—22no—
76C22×C97—42no—
77C22×C97—42no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97C2×C22⋊C411yescharacteristic
81C22×C97C22×C811yescharacteristic
82C22×C97C2×M1611yescharacteristic
83C2×D97—22no—
84C2×D97—22no—
85C2×D97—42no—
86C2×D97—42no—
87C2×D97C22⋊C814yes—
88C4×C97—22no—
89C4×C97—41no—
90Dic97—22no—
91Dic97—21no—
92Dic97—41no—
93Dic97#9912yes—
94C2×C97—21no—
95C2×C97—21no—
96C2×C97—42no—
97C2×C97C2×C22⋊C811yescharacteristic
98C2×C97C2×#9911yescharacteristic
99C2×C97#41311yescharacteristic
100D97—24no—
101#1459—971nomaximal, Sylow
102C97#145911yescharacteristic, complemented, Sylow
103C2×C4×D4—971no—
104C2×C22⋊C8—972no—
105#324—972no—
106#323—971no—
107#323D9711yescharacteristic, complemented
108C22×D4—971no—
109C23×C4—971no—
110C23×C4—971no—
111C22×C8—1942no—
112C4×D4—1942no—
113C4×D4—1942no—
114C2×C4⋊C4—971no—
115C2×C22⋊C4—971no—
116C2×C22⋊C4C2×D9711yescharacteristic
117C2×C42—971no—
118C22⋊C8—972no—
119C22⋊C8—1942no—
120C22⋊C8C2×D9712yes—
121C24—971no—
122C24C4×D9711yescharacteristic
123C2×D4—974no—
124C2×D4—1942no—
125C2×D4—1942no—
126C22×C4—971no—
127C22×C4—971no—
128C22×C4—972no—
129C22×C4—972no—
130C22×C4—971no—
131C22×C4—3882no—
132C22×C4—3882no—
133C22×C4—1941no—
134C22×C4—1941no—
135C22×C4C4×D9711yescharacteristic
136C22×C4C22×D9711yescharacteristic
137C2×C8—22no—
138C2×C8—3884no—
139C2×C8—1942no—
140C4⋊C4—1942no—
141C22⋊C4—22no—
142C22⋊C4—1942no—
143C42—1941no—
144C42—1941no—
145C23—21no—
146C23—21no—
147C23—41no—
148C23—41no—
149C23—972no—
150C23—972no—
151C23—972no—
152C23—3881no—
153C23—3881no—
154C23C8×D9712yescomplemented
155C23C2×C4×D9711yescharacteristic
156D4—1948no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—971no—
161C2×C4—971no—
162C2×C4—972no—
163C2×C4—3882no—
164C2×C4—3882no—
165C2×C4—3882no—
166C2×C4—3881no—
167C2×C4—1944no—
168C2×C4—1942no—
169C2×C4—1941no—
170C2×C4—1941no—
171C2×C4—1941no—
172C2×C4D4×D9712yes—
173C8—42no—
174C8—3882no—
175C22—21no—
176C22—21no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—974no—
183C22—3882no—
184C22—3882no—
185C22—1942no—
186C22—1942no—
187C22C22⋊C4×D9711yescharacteristic
188C22C2×C8×D9711yescharacteristic
189C22M16×D9711yescharacteristic
190C4—22no—
191C4—41no—
192C4—972no—
193C4—3881no—
194C4—1942no—
195C4—1941no—
196C2—21no—
197C2—21no—
198C2—42no—
199C2—1944no—
200C2C22⋊C8×D9711yescharacteristic
201C2#99×D9711yescharacteristic
202C2#2026071611yescharacteristic
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#1459
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. C22⋊C8×D97C2semidirect
3. #20260460C2semidirect
4. #20260459C2semidirect
5. #323×C97C2semidirect
22. C2×C22⋊D97C8semidirect
47. C23×C97C2×C8semidirect
68. C2×Dic97C2×C8semidirect
102. C97#1459semidirect
107. #323D97semidirect
154. C23C8×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907702
C22⋊C8×D972
#202604602
#202604592
#323×C972
#145997