Finite Group Databasebeta

← #27140397

Subgroups — #27140397

Order 12416.

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

normal subgroups only

This group has 246 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#20261286C211yesmaximal, characteristic
2#20260787C212yesmaximal, complemented
3#20260484C211yesmaximal, characteristic, complemented
4#20260478C212yesmaximal, complemented
5#333×C97C211yesmaximal, characteristic, complemented
6C2×#13095512C412yes—
7#15690807C2211yescharacteristic
8#15690799C412yes—
9#15690785—22no—
10#15690772—22no—
11C42×D97C2211yescharacteristic
12C4⋊#11715895C2212yes—
13#15690722—22no—
14C8×Dic97—22no—
15C4⋊Q8×C97C2211yescharacteristic
16C4⋊C8×C97C2212yes—
17#13095512—44no—
18C2×D388—22no—
19C2×D388—22no—
20C2×D388C2×C412yes—
21C2×C4×D97—22no—
22C2×C4×D97C2×C411yescharacteristic
23C2×C4×D97D412yes—
24C2×#11715895—22no—
25#13095488—24no—
26C4×Dic97C2×C411yescharacteristic
27C4×Dic97D412yes—
28C2×Q8×C97C2×C412yes—
29C2×C8×C97—22no—
30C4⋊C4×C97—22no—
31C42×C97C2311yescharacteristic
32C22×D97—22no—
33C22×D97C22⋊C411yescharacteristic
34Q8×C97—42no—
35C2×C4×C97—22no—
36C2×C4×C97C22×C411yescharacteristic
37C2×C4×C97C2×D412yes—
38C2×Dic97C22⋊C412yes—
39C2×Dic97C22⋊C411yescharacteristic
40D388—42no—
41D388—44no—
42D388—42no—
43D388—42no—
44C4×D97—22no—
45C4×D97—22no—
46C4×D97—22no—
47C4×D97—44no—
48C8×C97—42no—
49#11715895—42no—
50C22×C97C2×C22⋊C411yescharacteristic
51C2×D97—21no—
52C2×D97—22no—
53C2×D97—42no—
54C2×D97—44no—
55C4×C97—21no—
56C4×C97—42no—
57C4×C97#13512yes—
58C4×C97#13512yes—
59Dic97—22no—
60Dic97#10012yes—
61C2×C97C2×#10011yescharacteristic
62C2×C97#41612yes—
63D97—41no—
64D97—82no—
65#1543—971nomaximal, Sylow
66C97#154311yescharacteristic, complemented, Sylow
67#532—971no—
68#431—972no—
69#333—971no—
70#333D9711yescharacteristic, complemented
71#331—972no—
72C2×#41—972no—
73C4⋊Q8C2×D9711yescharacteristic
74#127—971no—
75#124—972no—
76#121—1942no—
77C4×D4—1942no—
78C2×C42—971no—
79C4⋊C8—972no—
80C4⋊C8C2×D9712yes—
81C22⋊C8—1942no—
82C4×C8—1942no—
83#41—3884no—
84C2×Q8C4×D9712yes—
85C2×D4—972no—
86C2×D4—1942no—
87C2×D4—1942no—
88C22×C4—971no—
89C22×C4—972no—
90C22×C4—1942no—
91C2×C8—22no—
92C2×C8—1942no—
93C4⋊C4—22no—
94C22⋊C4—1944no—
95C42—971no—
96C42—972no—
97C42C22×D9711yescharacteristic
98C23—971no—
99C23—1942no—
100Q8—42no—
101D4—3882no—
102D4—3884no—
103D4—3882no—
104D4—3882no—
105C2×C4—22no—
106C2×C4—971no—
107C2×C4—972no—
108C2×C4—3884no—
109C2×C4—1942no—
110C2×C4—1942no—
111C2×C4—1942no—
112C2×C4C2×C4×D9711yescharacteristic
113C2×C4D4×D9712yes—
114C8—42no—
115C8—3882no—
116C22—3882no—
117C22—3884no—
118C22—1941no—
119C22—1942no—
120C22C22⋊C4×D9711yescharacteristic
121C4—21no—
122C4—42no—
123C4—972no—
124C4—1942no—
125C4#1569083112yes—
126C4#1569083312yes—
127C2—7762no—
128C2—3881no—
129C2#100×D9711yescharacteristic
130C2#2026073012yes—
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#333×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261286C2
cosocleC42×C97C23
hypercenter#333D97
hypocenterC97#1543
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #20260787C2semidirect
3. #20260484C2semidirect
4. #20260478C2semidirect
5. #333×C97C2semidirect
66. C97#1543semidirect
70. #333D97semidirect

Maximal subgroups

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

subgroupindex
#202612862
#202607872
#202604842
#202604782
#333×C972
#154397