Finite Group Databasebeta

← #27140448

Subgroups — #27140448

Order 12416.

332 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 67 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#127×D97C211yesmaximal, characteristic
2#20260791C212yesmaximal, complemented
3#20260479C211yesmaximal, characteristic, complemented
4#20260478C212yesmaximal, complemented
5#331×C97C211yesmaximal, characteristic, complemented
6C2×D4×D97—22no—
7C2×D4×D97C412yes—
8#15690807C2211yescharacteristic
9#15690806—22no—
10#15690806C412yes—
11C42×D97C2211yescharacteristic
12C4⋊#11715895C2212yescomplemented
13#15690722—22no—
14#15690720—22no—
15#127×C97C2211yescharacteristic
16C4⋊C8×C97C2212yescomplemented
17C23×D97—22no—
18C2×D388—22no—
19C2×D388C2×C412yes—
20C2×C22⋊D97—22no—
21C2×C22⋊D97—24no—
22D4×D97—44no—
23D4×D97—44no—
24D4×D97—44no—
25C2×C4×D97C2×C411yescharacteristic
26C2×C4×D97D412yes—
27C2×#11715895—22no—
28C4×Dic97C2×C411yescharacteristic
29C4×Dic97D412yes—
30C2×D4×C97—22no—
31C2×D4×C97C2×C412yes—
32C2×C8×C97—22no—
33C42×C97C2311yescharacteristic
34C23×C97—22no—
35C22×D97—22no—
36C22×D97—48no—
37C22×D97—44no—
38C22×D97C22⋊C411yescharacteristic
39D4×C97—42no—
40D4×C97—42no—
41D4×C97—42no—
42C2×C4×C97C22×C411yescharacteristic
43C2×C4×C97C2×D412yes—
44C22⋊D97—44no—
45C22⋊D97—48no—
46C2×Dic97C22⋊C412yes—
47C2×Dic97C22⋊C411yescharacteristic
48D388—42no—
49D388—42no—
50D388—42no—
51C4×D97—22no—
52C4×D97—22no—
53C4×D97—22no—
54C8×C97—42no—
55#11715895—42no—
56C22×C97—42no—
57C22×C97—44no—
58C22×C97C2×C22⋊C411yescharacteristic
59C2×D97—22no—
60C2×D97—42no—
61C2×D97—44no—
62C2×D97—88no—
63C2×D97#10012yes—
64C4×C97—21no—
65C4×C97AGL(1,ℤ/8)12yes—
66C4×C97AGL(1,ℤ/8)12yes—
67Dic97—21no—
68Dic97—22no—
69C2×C97—82no—
70C2×C97C2×#10011yescharacteristic
71C2×C97#41812yes—
72D97—22no—
73D97—82no—
74#1555—971nomaximal, Sylow
75C97#155511yescharacteristic, complemented, Sylow
76C2×#127—971no—
77#432—972no—
78#331—971no—
79#331—972no—
80#331D9711yescharacteristic, complemented
81C22×D4—972no—
82C22×D4—1942no—
83#127—972no—
84#127—971no—
85#127—1942no—
86#127C2×D9711yescharacteristic
87C2×C42—971no—
88C4⋊C8—972no—
89C4⋊C8C2×D9712yescomplemented
90C22⋊C8—1942no—
91#97—1942no—
92C24—1942no—
93C2×D4—22no—
94C2×D4—972no—
95C2×D4—3884no—
96C2×D4—3884no—
97C2×D4—3884no—
98C2×D4—1944no—
99C2×D4—1942no—
100C2×D4—1942no—
101C2×D4C4×D9712yes—
102C22×C4—971no—
103C22×C4—972no—
104C2×C8—22no—
105C2×C8—1942no—
106C42—971no—
107C42—972no—
108C42C22×D9711yescharacteristic
109C23—22no—
110C23—971no—
111C23—3884no—
112C23—3888no—
113C23—1942no—
114D4—42no—
115D4—42no—
116D4—42no—
117D4—3882no—
118D4—3884no—
119D4—3888no—
120D4—3882no—
121D4—3882no—
122C2×C4—971no—
123C2×C4—972no—
124C2×C4—1942no—
125C2×C4—1942no—
126C2×C4—1942no—
127C2×C4C2×C4×D9711yescharacteristic
128C2×C4D4×D9712yes—
129C8—42no—
130C8—3882no—
131C22—42no—
132C22—44no—
133C22—7768no—
134C22—972no—
135C22—3882no—
136C22—3884no—
137C22—1942no—
138C22C22⋊C4×D9711yescharacteristic
139C4—21no—
140C4—1941no—
141C4—1942no—
142C4#1569083412yes—
143C4#1569083712yes—
144C2—82no—
145C2—7762no—
146C2—1942no—
147C2#100×D9711yescharacteristic
148C2#2026073812yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#331×C97C2
socleC22×C97C2×C22⋊C4
prime radical#127×D97C2
cosocleC42×C97C23
hypercenter#331D97
hypocenterC97#1555
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20260791C2semidirect
3. #20260479C2semidirect
4. #20260478C2semidirect
5. #331×C97C2semidirect
12. C4⋊#11715895C22semidirect
16. C4⋊C8×C97C22semidirect
75. C97#1555semidirect
80. #331D97semidirect
89. C4⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#127×D972
#202607912
#202604792
#202604782
#331×C972
#155597