Finite Group Databasebeta

← #27140468

Subgroups — #27140468

Order 12416.

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

normal subgroups only

This group has 254 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#20261283C211yesmaximal, characteristic
2#20260757C212yesmaximal, complemented
3#20260483C212yesmaximal, complemented
4#20260479C211yesmaximal, characteristic, complemented
5#331×C97C211yesmaximal, characteristic, complemented
6C2×#13095510C412yes—
7C2×#13095494—22no—
8#15690810C2211yescharacteristic
9#15690797C412yes—
10#15690781—22no—
11D4×Dic97—22no—
12C42×D97C2211yescharacteristic
13C4⋊#11715895C2212yes—
14#15690739—22no—
15#127×C97C2211yescharacteristic
16C4⋊C8×C97C2212yes—
17#13095510—44no—
18C2×Dic194C2×C412yes—
19C2×C22⋊D97—22no—
20C22×Dic97—22no—
21C2×C4×D97C2×C411yescharacteristic
22C2×C4×D97D412yes—
23#13095494—44no—
24C2×#11715895—22no—
25C4⋊Dic97—22no—
26C22⋊Dic97—24no—
27C4×Dic97C2×C411yescharacteristic
28C4×Dic97D412yes—
29C2×D4×C97—22no—
30C2×D4×C97C2×C412yes—
31C2×C8×C97—22no—
32C42×C97C2311yescharacteristic
33C23×C97—22no—
34C22×D97C22⋊C411yescharacteristic
35D4×C97—42no—
36D4×C97—42no—
37D4×C97—42no—
38C2×C4×C97C22×C411yescharacteristic
39C2×C4×C97C2×D412yes—
40C22⋊D97—44no—
41C2×Dic97—22no—
42C2×Dic97—44no—
43C2×Dic97C22⋊C412yes—
44C2×Dic97C22⋊C411yescharacteristic
45C4×D97—22no—
46C4×D97—22no—
47C4×D97—22no—
48Dic194—42no—
49C8×C97—42no—
50#11715895—42no—
51C22×C97—42no—
52C22×C97—44no—
53C22×C97C2×C22⋊C411yescharacteristic
54C2×D97—21no—
55C2×D97—22no—
56C4×C97—21no—
57C4×C97AGL(1,ℤ/8)12yes—
58C4×C97#13712yes—
59Dic97—22no—
60Dic97—42no—
61Dic97#10012yes—
62C2×C97—82no—
63C2×C97C2×#10011yescharacteristic
64C2×C97#41712yes—
65D97—41no—
66#1560—971nomaximal, Sylow
67C97#156011yescharacteristic, complemented, Sylow
68#532—971no—
69#423—972no—
70#333—972no—
71#331—971no—
72#331D9711yescharacteristic, complemented
73C2×#41—972no—
74C2×M16—1942no—
75C4⋊Q8—971no—
76#127C2×D9711yescharacteristic
77#124—972no—
78#121—1942no—
79C4×D4—1942no—
80C2×C42—971no—
81C4⋊C8—972no—
82C4⋊C8—1942no—
83C4⋊C8C2×D9712yes—
84#41—3884no—
85C2×Q8—972no—
86C2×D4—22no—
87C2×D4—1942no—
88C2×D4C4×D9712yes—
89C22×C4—971no—
90C22×C4—972no—
91C22×C4—1942no—
92M16—3884no—
93C2×C8—22no—
94C2×C8—1942no—
95C4⋊C4—1942no—
96C22⋊C4—1944no—
97C42—971no—
98C42—972no—
99C42C22×D9711yescharacteristic
100C23—22no—
101C23—971no—
102Q8—3882no—
103D4—42no—
104D4—42no—
105D4—42no—
106D4—3884no—
107C2×C4—971no—
108C2×C4—972no—
109C2×C4—3884no—
110C2×C4—1942no—
111C2×C4—1942no—
112C2×C4—1942no—
113C2×C4—1942no—
114C2×C4C2×C4×D9711yescharacteristic
115C2×C4D4×D9712yes—
116C8—42no—
117C8—3882no—
118C22—42no—
119C22—44no—
120C22—1941no—
121C22—1942no—
122C22C22⋊C4×D9711yescharacteristic
123C4—21no—
124C4—972no—
125C4—3882no—
126C4—1942no—
127C4#1569083412yes—
128C4#1569083912yes—
129C2—82no—
130C2—3881no—
131C2#100×D9711yescharacteristic
132C2#2026073312yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#331×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261283C2
cosocleC42×C97C23
hypercenter#331D97
hypocenterC97#1560
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. #20260757C2semidirect
3. #20260483C2semidirect
4. #20260479C2semidirect
5. #331×C97C2semidirect
67. C97#1560semidirect
72. #331D97semidirect

Maximal subgroups

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

subgroupindex
#202612832
#202607572
#202604832
#202604792
#331×C972
#156097