Finite Group Databasebeta

← #27143411

Subgroups — #27143411

Order 12416.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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#20260709C212yesmaximal
2#20260530C214yesmaximal, complemented
3#536×C97C211yesmaximal, characteristic
4#15690769—22no—
5#15690725—22no—
6#15690725C2214yes—
7C2×#41×C97C2211yescharacteristic
8C22×Q8×C97C411yescharacteristic, complemented
9C4⋊Q8×C97C412yescomplemented
10#124×C97C2212yes—
11#123×C97—22no—
12C22⋊Q8×C97—22no—
13#117×C97C411yescharacteristic, complemented
14C4⋊Dic97—42no—
15C22⋊Dic97—24no—
16C4×Dic97—42no—
17#41×C97—42no—
18C2×Q8×C97—21no—
19C2×Q8×C97—42no—
20C2×Q8×C97C2×C412yes—
21C2×Q8×C97C2×C411yescharacteristic
22C2×D4×C97D412yes—
23C2×D4×C97C2311yescharacteristic
24C22×C4×C97—21no—
25C22×C4×C97C2×C411yescharacteristic
26C22×C4×C97D412yes—
27C4⋊C4×C97—21no—
28C4⋊C4×C97—24no—
29C22⋊C4×C97—21no—
30C22⋊C4×C97—24no—
31C42×C97C2×C412yes—
32C23×C97C22⋊C411yescharacteristic
33C23×C97C2×D412yes—
34Q8×C97—41no—
35Q8×C97—42no—
36Q8×C97—42no—
37D4×C97—42no—
38D4×C97—41no—
39C2×C4×C97—22no—
40C2×C4×C97—22no—
41C2×C4×C97—22no—
42C2×C4×C97—22no—
43C2×C4×C97—22no—
44C2×C4×C97—42no—
45C2×C4×C97C22⋊C412yes—
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22×C411yescharacteristic
48C2×Dic97—44no—
49C22×C97—22no—
50C22×C97—21no—
51C22×C97—41no—
52C22×C97—42no—
53C22×C97C2×C22⋊C411yescharacteristic
54C4×C97—22no—
55C4×C97—21no—
56C4×C97—42no—
57C4×C97—42no—
58C4×C97#9912yes—
59Dic97—84no—
60C2×C97—21no—
61C2×C97—42no—
62C2×C97—41no—
63C2×C97C2×#9911yescharacteristic
64#2129—971nomaximal, Sylow
65C97#212911yescharacteristic, complemented, Sylow
66#536D9711yescharacteristic
67#410—972no—
68#354—974no—
69C2×#41C2×D9711yescharacteristic
70C22×Q8Dic9711yescharacteristic, complemented
71C4⋊Q8Dic9712yescomplemented
72#124C2×D9712yes—
73#123—22no—
74C22⋊Q8—22no—
75#117—1942no—
76#117Dic9711yescharacteristic, complemented
77#99—974no—
78#99—1942no—
79#41—42no—
80C2×Q8—21no—
81C2×Q8—42no—
82C2×Q8C2×Dic9712yes—
83C2×Q8C2×Dic9711yescharacteristic
84C2×D4C22⋊D9712yes—
85C2×D4C22×D9711yescharacteristic
86C22×C4—21no—
87C22×C4C2×Dic9711yescharacteristic
88C22×C4C22⋊D9712yes—
89C4⋊C4—24no—
90C4⋊C4—21no—
91C4⋊C4—3882no—
92C22⋊C4—21no—
93C22⋊C4—24no—
94C22⋊C4—1944no—
95C42—3882no—
96C42C2×Dic9712yes—
97C23C22⋊Dic9711yescharacteristic
98C23C2×C22⋊D9712yes—
99Q8—41no—
100Q8—42no—
101Q8—42no—
102D4—42no—
103D4—41no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—22no—
109C2×C4—42no—
110C2×C4—3884no—
111C2×C4C22⋊Dic9712yes—
112C2×C4C22⋊Dic9711yescharacteristic
113C2×C4C22×Dic9711yescharacteristic
114C22—21no—
115C22—22no—
116C22—41no—
117C22—42no—
118C22C2×C22⋊Dic9711yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C4—7764no—
124C4#1569072512yes—
125C2—21no—
126C2—42no—
127C2—41no—
128C2C2×#1569072511yescharacteristic
subgroupG/H
centerC2C2×#15690725
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#536×C97C2
socleC2×C97C2×#99
prime radicalC2×#41×C97C22
cosocleC2×D4×C97C23
hypercenter#536D97
hypocenterC97#2129
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #20260530C2semidirect
8. C22×Q8×C97C4semidirect
9. C4⋊Q8×C97C4semidirect
13. #117×C97C4semidirect
65. C97#2129semidirect
70. C22×Q8Dic97semidirect
71. C4⋊Q8Dic97semidirect
76. #117Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607092
#202605302
#536×C972
#212997