Finite Group Databasebeta

← #27140073

Subgroups — #27140073

Order 12416.

316 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 69 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×#15690772C211yesmaximal, characteristic
2#20260697C212yesmaximal, complemented
3#20260461C211yesmaximal, characteristic, complemented
4#20260458C212yesmaximal, complemented
5#324×C97C211yesmaximal, characteristic, complemented
6C22×D388C411yescharacteristic
7C22×C4×D97C411yescharacteristic
8C22×C4×D97C2211yescharacteristic
9C2×#13095494—22no—
10#15690772—22no—
11#15690772—22no—
12C2×#13095488C411yescharacteristic
13C2×#13095488C2211yescharacteristic
14C2×C4×Dic97C411yescharacteristic
15C22⋊#11715895C2212yes—
16#15690722—22no—
17C2×C4⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C23×D97C2×C411yescharacteristic
20C23×D97C2×C411yescharacteristic
21C2×D388—22no—
22C2×D388—22no—
23C2×D388—22no—
24C22×Dic97C2×C411yescharacteristic
25C22×Dic97C2×C411yescharacteristic
26C2×C4×D97—21no—
27C2×C4×D97—21no—
28C2×C4×D97—42no—
29C2×C4×D97—42no—
30C2×C4×D97D412yes—
31C2×C4×D97D412yes—
32#13095494—44no—
33C2×#11715895—22no—
34#13095488—22no—
35#13095488—22no—
36C4×Dic97—21no—
37C4×Dic97—21no—
38C22×C4×C97C2×C411yescharacteristic
39C22×C4×C97C2×C411yescharacteristic
40C22×C4×C97C2311yescharacteristic
41C2×C8×C97—22no—
42C4⋊C4×C97—22no—
43C23×C97C22×C411yescharacteristic
44C22×D97—21no—
45C22×D97—21no—
46C22×D97—21no—
47C22×D97—21no—
48C22×D97—41no—
49C22×D97—41no—
50C22×D97—41no—
51C22×D97—41no—
52C22×D97C22⋊C412yes—
53C22×D97M1612yescomplemented
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97M1612yescomplemented
60C2×C4×C97C2×D412yes—
61C2×Dic97—22no—
62C2×Dic97—21no—
63C2×Dic97—41no—
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C22⋊C411yescharacteristic
66D388—28no—
67C4×D97—24no—
68C4×D97—42no—
69C4×D97—42no—
70C4×D97—42no—
71C8×C97—42no—
72#11715895—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97C2×C22⋊C411yescharacteristic
76C22×C97C2×M1611yescharacteristic
77C22×C97C2×M1611yescharacteristic
78C2×D97—22no—
79C2×D97—22no—
80C2×D97—22no—
81C2×D97—22no—
82C2×D97—42no—
83C2×D97—42no—
84C2×D97—42no—
85C2×D97—41no—
86C2×D97—41no—
87C2×D97—42no—
88C4×C97—22no—
89C4×C97—22no—
90C4×C97—41no—
91Dic97—21no—
92Dic97—41no—
93Dic97#9912yes—
94C2×C97—21no—
95C2×C97—21no—
96C2×C97#40711yescharacteristic
97C2×C97C2×#9911yescharacteristic
98C2×C97#41311yescharacteristic
99D97—42no—
100D97—42no—
101#1465—971nomaximal, Sylow
102C97#146511yescharacteristic, complemented, Sylow
103C2×C4×D4—971no—
104#407—972no—
105#324—971no—
106#324D9711yescharacteristic, complemented
107#323—972no—
108C22×D4—971no—
109C23×C4—971no—
110C23×C4—971no—
111C2×M16—1942no—
112C4×D4—1942no—
113C4×D4—1942no—
114C2×C4⋊C4C2×D9711yescharacteristic
115C2×C22⋊C4—971no—
116C2×C22⋊C4—971no—
117C2×C42—971no—
118C22⋊C8—972no—
119C22⋊C8—1942no—
120C22⋊C8C2×D9712yes—
121C24—971no—
122C24—971no—
123C2×D4—1942no—
124C2×D4—1942no—
125C2×D4—1942no—
126C22×C4—971no—
127C22×C4—972no—
128C22×C4—972no—
129C22×C4—971no—
130C22×C4—3882no—
131C22×C4—3882no—
132C22×C4—1941no—
133C22×C4—1941no—
134C22×C4C4×D9711yescharacteristic
135C22×C4C4×D9711yescharacteristic
136C22×C4C22×D9711yescharacteristic
137M16—3884no—
138C2×C8—22no—
139C2×C8—1942no—
140C4⋊C4—22no—
141C22⋊C4—1942no—
142C22⋊C4—1942no—
143C42—1941no—
144C42—1941no—
145C23—972no—
146C23—972no—
147C23—3881no—
148C23—3881no—
149C23—3881no—
150C23—3881no—
151C23—1941no—
152C23—1941no—
153C23—1941no—
154C23—1941no—
155C23C2×C4×D9711yescharacteristic
156D4—1948no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—971no—
163C2×C4—971no—
164C2×C4—3882no—
165C2×C4—3882no—
166C2×C4—3882no—
167C2×C4—3881no—
168C2×C4—1944no—
169C2×C4—1942no—
170C2×C4—1941no—
171C2×C4#1309549412yescomplemented
172C2×C4D4×D9712yes—
173C8—42no—
174C8—3882no—
175C22—21no—
176C22—21no—
177C22—3882no—
178C22—3882no—
179C22—3882no—
180C22—3882no—
181C22—3881no—
182C22—3881no—
183C22—1942no—
184C22—1942no—
185C22—1942no—
186C22—1942no—
187C22C22⋊C4×D9711yescharacteristic
188C22M16×D9711yescharacteristic
189C22C2×#1309549411yescharacteristic
190C4—22no—
191C4—22no—
192C4—41no—
193C4—972no—
194C4—3881no—
195C4—1941no—
196C2—21no—
197C2—21no—
198C2—3882no—
199C2—3882no—
200C2#2026069711yescharacteristic
201C2#99×D9711yescharacteristic
202C2#2026071811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC22×C4×C97C23
hypercenter#324D97
hypocenterC97#1465
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20260697C2semidirect
3. #20260461C2semidirect
4. #20260458C2semidirect
5. #324×C97C2semidirect
53. C22×D97M16semidirect
59. C2×C4×C97M16semidirect
102. C97#1465semidirect
106. #324D97semidirect
171. C2×C4#13095494semidirect

Maximal subgroups

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

subgroupindex
C2×#156907722
#202606972
#202604612
#202604582
#324×C972
#146597