Finite Group Databasebeta

← #27140924

Subgroups — #27140924

Order 12416.

212 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261470C211yesmaximal, characteristic
2#20261082C211yesmaximal, characteristic
3#20261021C211yesmaximal, characteristic, complemented
4#20260945C211yesmaximal, characteristic, complemented
5#20260478C211yesmaximal, characteristic
6#20260467C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095501—21no—
9#15690810C2211yescharacteristic
10#15690807C2211yescharacteristic
11#15690799—21no—
12#15690798—21no—
13#15690795—22no—
14C4×Dic194C2211yescharacteristic
15C4×D388—21no—
16#15690772—21no—
17#15690742—21no—
18C4⋊#11715895C2211yescharacteristic
19#15690735—21no—
20#15690733—21no—
21#15690732—22no—
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27C2×D388—21no—
28C2×D388—21no—
29C2×D388D412yes—
30C2×C4×D97—22no—
31#13095501—42no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97D411yescharacteristic
36#13095490—21no—
37#13095488—22no—
38#13095488—22no—
39#13095488—22no—
40C4×Dic97—21no—
41C2×Q8×C97D411yescharacteristic
42C2×C8×C97—21no—
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48C22×D97—22no—
49Q8×C97—21no—
50Q8×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—41no—
59D388—42no—
60D388—41no—
61D388—41no—
62D388—41no—
63D388—41no—
64C4×D97—82no—
65Dic194—21no—
66Dic194—21no—
67Dic194—41no—
68Dic194—41no—
69C8×C97—41no—
70C8×C97—41no—
71#11715895—41no—
72C22×C97#12011yescharacteristic
73C2×D97—42no—
74C2×D97—42no—
75C2×D97—42no—
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—41no—
79C4×C97#13511yescharacteristic
80C4×C97#13511yescharacteristic
81C4×C97AGL(1,ℤ/8)11yescharacteristic
82C4×C97#13711yescharacteristic
83Dic97—41no—
84Dic97—41no—
85Dic97—81no—
86C2×C97#44911yescharacteristic
87C2×C97#45211yescharacteristic
88C2×C97#45511yescharacteristic
89D97—82no—
90#1640—971nomaximal, Sylow
91C97#164011yescharacteristic, complemented, Sylow
92#559—971no—
93#486—971no—
94#475—971no—
95#462—971no—
96#331—971no—
97Q8⋊C8—971no—
98Q8⋊C8D9711yescharacteristic, complemented
99C2×Q16—1941no—
100C4⋊Q8—971no—
101#127—971no—
102#124—1941no—
103#124—1941no—
104#123—1942no—
105C4×Q8—971no—
106C4×Q8C2×D9711yescharacteristic
107C4×D4—1941no—
108C4×D4—1941no—
109#106—1941no—
110C4⋊C8—971no—
111C4⋊C8C2×D9711yescharacteristic
112#103—1941no—
113#103—1941no—
114#102—1942no—
115C4×C8C2×D9711yescharacteristic
116C2×Q8—971no—
117C2×Q8—1941no—
118C2×Q8D38811yescharacteristic
119C2×D4—972no—
120C2×D4—1941no—
121C2×D4—1941no—
122C22×C4—1942no—
123Q16—3882no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—1941no—
127C4⋊C4—21no—
128C4⋊C4—971no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C4⋊C4D38811yescharacteristic
133C22⋊C4—1942no—
134C22⋊C4—1942no—
135C22⋊C4—1942no—
136C42—21no—
137C42—1941no—
138C42C22×D9711yescharacteristic
139C23—1942no—
140Q8—21no—
141Q8—21no—
142Q8—3881no—
143Q8—3881no—
144Q8—1941no—
145Q8—1941no—
146D4—3882no—
147D4—3881no—
148D4—3881no—
149D4—3881no—
150D4—3881no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—7762no—
154C2×C4—3881no—
155C2×C4—1941no—
156C2×C4—1941no—
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×D38811yescharacteristic
160C8—41no—
161C8—41no—
162C8—3881no—
163C22—3882no—
164C22—3882no—
165C22—3882no—
166C22#1569078011yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—7761no—
171C4—3881no—
172C4—3881no—
173C4#1569083111yescharacteristic
174C4#1569083211yescharacteristic
175C4#1569083811yescharacteristic
176C4#1569084211yescharacteristic
177C2—7762no—
178C2#2026088211yescharacteristic
179C2#2026090011yescharacteristic
180C2#2026091411yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97#120
prime radical#15690807C22
hypercenterQ8⋊C8D97
hypocenterC97#1640
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #20261021C2semidirect
4. #20260945C2semidirect
6. #20260467C2semidirect
7. Q8⋊C8×C97C2semidirect
91. C97#1640semidirect
98. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202614702
#202610822
#202610212
#202609452
#202604782
#202604672
Q8⋊C8×C972
#164097