Finite Group Databasebeta

← #27140049

Subgroups — #27140049

Order 12416.

264 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 79 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×#15690768C211yesmaximal, characteristic
2#20260785C212yesmaximal, complemented
3#20260461C211yesmaximal, characteristic, complemented
4#20260458C212yesmaximal
5#324×C97C211yesmaximal, characteristic, complemented
6C22×C4×D97C411yescharacteristic
7#15690768—22no—
8#15690768—22no—
9C2×C4⋊Dic97C411yescharacteristic
10C2×#13095488C411yescharacteristic
11C2×#13095488C2211yescharacteristic
12C2×C4×Dic97C411yescharacteristic
13C2×C4×Dic97C2211yescharacteristic
14#15690739—22no—
15C22⋊#11715895C2212yes—
16C8×Dic97—22no—
17C2×C4⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C23×D97C2×C411yescharacteristic
20C22×Dic97C2×C411yescharacteristic
21C22×Dic97C2×C411yescharacteristic
22C22×Dic97C2×C411yescharacteristic
23C2×C4×D97—21no—
24C2×C4×D97—21no—
25C2×C4×D97—22no—
26C2×C4×D97C814yescomplemented
27C2×#11715895—22no—
28C4⋊Dic97—22no—
29#13095488—22no—
30#13095488—22no—
31C4×Dic97—21no—
32C4×Dic97—21no—
33C4×Dic97D412yes—
34C4×Dic97D412yes—
35C22×C4×C97C2×C411yescharacteristic
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C2×C8×C97—22no—
39C4⋊C4×C97—22no—
40C23×C97C22×C411yescharacteristic
41C22×D97—21no—
42C22×D97—21no—
43C22×D97—41no—
44C22×D97—41no—
45C22×D97C2×C812yes—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97—41no—
50C2×C4×C97—41no—
51C2×C4×C97C2×C812yes—
52C2×C4×C97C2×D412yes—
53C2×Dic97—21no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56C2×Dic97C22⋊C412yes—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C22⋊C411yescharacteristic
59C2×Dic97C2×C812yes—
60C2×Dic97M1612yes—
61C4×D97—24no—
62C4×D97—24no—
63C8×C97—42no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97C2×C22⋊C411yescharacteristic
68C22×C97C22×C811yescharacteristic
69C22×C97C2×M1611yescharacteristic
70C2×D97—22no—
71C2×D97—42no—
72C2×D97—41no—
73C2×D97—41no—
74C2×D97—42no—
75C4×C97—22no—
76C4×C97—22no—
77C4×C97—41no—
78Dic97—22no—
79Dic97—41no—
80Dic97C22⋊C812yes—
81Dic97C22⋊C812yes—
82C2×C97—21no—
83C2×C97—21no—
84C2×C97C2×C22⋊C811yescharacteristic
85C2×C97#41011yescharacteristic
86C2×C97#41311yescharacteristic
87D97—42no—
88#1458—971nomaximal, Sylow
89C97#145811yescharacteristic, complemented, Sylow
90C2×#117—971no—
91#431—972no—
92#324—971no—
93#324D9711yescharacteristic, complemented
94#323—972no—
95C23×C4—971no—
96#117—1942no—
97#117—1942no—
98C2×C4⋊C4—971no—
99C2×C4⋊C4C2×D9711yescharacteristic
100C2×C22⋊C4—971no—
101C2×C22⋊C4—971no—
102C2×C42—971no—
103C2×C42—971no—
104C4⋊C8—1942no—
105C22⋊C8—972no—
106C22⋊C8C2×D9712yes—
107C4×C8—1942no—
108C24—971no—
109C22×C4—971no—
110C22×C4—971no—
111C22×C4—974no—
112C22×C4—971no—
113C22×C4—1941no—
114C22×C4—1941no—
115C22×C4—1942no—
116C22×C4C4×D9711yescharacteristic
117C22×C4C4×D9711yescharacteristic
118C22×C4C22×D9711yescharacteristic
119C2×C8—22no—
120C2×C8—1942no—
121C4⋊C4—22no—
122C4⋊C4—1942no—
123C22⋊C4—1942no—
124C22⋊C4—1942no—
125C42—972no—
126C42—972no—
127C42—1941no—
128C42—1941no—
129C23—972no—
130C23—3881no—
131C23—3881no—
132C23—1941no—
133C23—1941no—
134C23C2×C4×D9711yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—971no—
141C2×C4—971no—
142C2×C4—972no—
143C2×C4—972no—
144C2×C4—972no—
145C2×C4—3881no—
146C2×C4—3881no—
147C2×C4—1944no—
148C2×C4—1944no—
149C2×C4—1941no—
150C2×C4C8×D9712yes—
151C2×C4D4×D9712yes—
152C8—42no—
153C8—3882no—
154C22—21no—
155C22—21no—
156C22—3882no—
157C22—3882no—
158C22—3881no—
159C22—3881no—
160C22—1942no—
161C22C22⋊C4×D9711yescharacteristic
162C22C2×C8×D9711yescharacteristic
163C22M16×D9711yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—972no—
168C4—972no—
169C4—3881no—
170C4—1942no—
171C2—21no—
172C2—21no—
173C2—3882no—
174C2C22⋊C8×D9711yescharacteristic
175C2#2026070711yescharacteristic
176C2#2026071811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC22×C97C22×C8
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×C4×C97C23
hypercenter#324D97
hypocenterC97#1458
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20260785C2semidirect
3. #20260461C2semidirect
5. #324×C97C2semidirect
26. C2×C4×D97C8semidirect
89. C97#1458semidirect
93. #324D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907682
#202607852
#202604612
#202604582
#324×C972
#145897