Finite Group Databasebeta

← #27140269

Subgroups — #27140269

Order 12416.

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

normal subgroups only

This group has 274 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
1#20260699C211yesmaximal, characteristic
2#20260598C211yesmaximal, characteristic
3#20260472C214yesmaximal, complemented
4C2×C22⋊Q8×C97C211yesmaximal, characteristic
5C2×#13095495—21no—
6C2×C22⋊Dic97—21no—
7C22⋊#11715895—21no—
8C22⋊#11715895C2212yes—
9#15690717—21no—
10#15690717C2212yes—
11C22×Q8×C97C411yescharacteristic, complemented
12C23×C4×C97C2211yescharacteristic
13C22⋊Q8×C97—21no—
14C22⋊Q8×C97—21no—
15C22⋊Q8×C97C412yescomplemented
16C22⋊Q8×C97C2212yes—
17C2×C4⋊C4×C97—21no—
18C2×C4⋊C4×C97C411yescharacteristic, complemented
19C2×C22⋊C4×C97—21no—
20C22×Dic97—22no—
21#13095495—42no—
22C2×#11715895—22no—
23C22⋊Dic97—42no—
24C24×C97D411yescharacteristic
25C2×Q8×C97—21no—
26C2×Q8×C97—42no—
27C2×Q8×C97C2×C412yes—
28C22×C4×C97—21no—
29C22×C4×C97—21no—
30C22×C4×C97—42no—
31C22×C4×C97C2×C411yescharacteristic
32C22×C4×C97C2×C411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C22×C4×C97D411yescharacteristic
35C22×C4×C97D411yescharacteristic
36C22×C4×C97C2311yescharacteristic
37C4⋊C4×C97—22no—
38C4⋊C4×C97—21no—
39C4⋊C4×C97—22no—
40C4⋊C4×C97C2×C412yes—
41C22⋊C4×C97—22no—
42C22⋊C4×C97—22no—
43C23×C97—22no—
44C23×C97—22no—
45C23×C97—41no—
46C23×C97C22⋊C411yescharacteristic
47C23×C97C22⋊C411yescharacteristic
48C23×C97C2×D411yescharacteristic
49Q8×C97—42no—
50Q8×C97—42no—
51C2×C4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—42no—
60C2×C4×C97—42no—
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22×C411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×Dic97—42no—
66C2×Dic97—42no—
67C2×Dic97—42no—
68#11715895—42no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—21no—
76C22×C97—44no—
77C22×C97#9912yes—
78C22×C97C2×C22⋊C411yescharacteristic
79C22×C97#13712yes—
80C4×C97—21no—
81C4×C97—21no—
82C4×C97—21no—
83C4×C97—21no—
84C4×C97—42no—
85C4×C97—42no—
86Dic97—82no—
87C2×C97—22no—
88C2×C97—22no—
89C2×C97—41no—
90C2×C97C2×#9911yescharacteristic
91C2×C97#41911yescharacteristic
92C2×C97#42111yescharacteristic
93#1511—971nomaximal, Sylow
94C97#151111yescharacteristic, complemented, Sylow
95C2×C22⋊Q8D9711yescharacteristic
96#407—971no—
97#381—971no—
98#328—974no—
99C22×Q8Dic9711yescharacteristic, complemented
100C23×C4C2×D9711yescharacteristic
101C2×M16—1941no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C22⋊Q8Dic9712yescomplemented
105C22⋊Q8C2×D9712yes—
106C2×C4⋊C4—21no—
107C2×C4⋊C4Dic9711yescharacteristic, complemented
108C2×C22⋊C4—21no—
109C2×C22⋊C4—1941no—
110C22⋊C8—972no—
111C22⋊C8—1941no—
112#95—972no—
113#95—1941no—
114C24C22⋊D9711yescharacteristic
115C2×Q8—21no—
116C2×Q8—42no—
117C2×Q8C2×Dic9712yes—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4—42no—
121C22×C4—1942no—
122C22×C4C2×Dic9711yescharacteristic
123C22×C4C2×Dic9711yescharacteristic
124C22×C4C22⋊D9711yescharacteristic
125C22×C4C22⋊D9711yescharacteristic
126C22×C4C22⋊D9711yescharacteristic
127C22×C4C22×D9711yescharacteristic
128M16—3882no—
129C2×C8—1942no—
130C4⋊C4—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4C2×Dic9712yes—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—3882no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23C22⋊Dic9711yescharacteristic
141C23C22⋊Dic9711yescharacteristic
142C23C2×C22⋊D9711yescharacteristic
143Q8—42no—
144Q8—42no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—3882no—
156C2×C4—3882no—
157C2×C4—3882no—
158C2×C4C22⋊Dic9711yescharacteristic
159C2×C4C22⋊Dic9711yescharacteristic
160C2×C4C22×Dic9711yescharacteristic
161C2×C4C2×C22⋊D9711yescharacteristic
162C8—3882no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#1569072512yes—
172C22C2×C22⋊Dic9711yescharacteristic
173C22#1569084112yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—42no—
179C4—42no—
180C4—7762no—
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#1569072511yescharacteristic
185C2#2026074411yescharacteristic
186C2#2026075211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊Q8×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC2×C22⋊Q8D97
hypocenterC97#1511
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20260472C2semidirect
11. C22×Q8×C97C4semidirect
15. C22⋊Q8×C97C4semidirect
18. C2×C4⋊C4×C97C4semidirect
94. C97#1511semidirect
99. C22×Q8Dic97semidirect
104. C22⋊Q8Dic97semidirect
107. C2×C4⋊C4Dic97semidirect

Maximal subgroups

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

subgroupindex
#202606992
#202605982
#202604722
C2×C22⋊Q8×C972
#151197