Finite Group Databasebeta

← #27140248

Subgroups — #27140248

Order 12416.

228 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 81 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#20260792C211yesmaximal, characteristic
2#20260581C211yesmaximal, characteristic
3#20260472C214yesmaximal
4#533×C97C211yesmaximal, characteristic
5C2×C4×Dic97—21no—
6C4⋊#11715895—21no—
7C22⋊#11715895C2212yes—
8#15690721—21no—
9#15690717—21no—
10#15690717C2212yes—
11C4⋊Q8×C97C411yescharacteristic, complemented
12C4⋊Q8×C97C411yescharacteristic, complemented
13#125×C97—21no—
14C22⋊Q8×C97—21no—
15C22⋊Q8×C97C2212yes—
16C4×Q8×C97—21no—
17C4×Q8×C97C412yescomplemented
18#117×C97—21no—
19C2×C42×C97C2211yescharacteristic
20C22×Dic97—22no—
21C2×#11715895—22no—
22C4×Dic97—24no—
23C2×Q8×C97—21no—
24C2×Q8×C97C2×C412yes—
25C22×C4×C97D411yescharacteristic
26C22×C4×C97D411yescharacteristic
27C22×C4×C97C2311yescharacteristic
28C4⋊C4×C97—21no—
29C4⋊C4×C97—21no—
30C4⋊C4×C97—21no—
31C4⋊C4×C97—22no—
32C4⋊C4×C97—22no—
33C4⋊C4×C97C2×C412yes—
34C22⋊C4×C97—22no—
35C42×C97—22no—
36C42×C97C2×C411yescharacteristic
37C42×C97C2×C411yescharacteristic
38C42×C97D411yescharacteristic
39C42×C97D411yescharacteristic
40C23×C97C2×D411yescharacteristic
41Q8×C97—42no—
42Q8×C97—42no—
43C2×C4×C97—22no—
44C2×C4×C97—22no—
45C2×C4×C97—22no—
46C2×C4×C97—22no—
47C2×C4×C97—41no—
48C2×C4×C97C22⋊C411yescharacteristic
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22⋊C411yescharacteristic
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×Dic97—24no—
55C2×Dic97—42no—
56C2×Dic97—42no—
57#11715895—42no—
58C22×C97—21no—
59C22×C97—21no—
60C22×C97C2×C22⋊C411yescharacteristic
61C22×C97#13712yes—
62C4×C97—21no—
63C4×C97—21no—
64C4×C97—21no—
65C4×C97—21no—
66C4×C97—21no—
67C4×C97—42no—
68C4×C97—42no—
69C4×C97#10412yes—
70Dic97—44no—
71C2×C97—22no—
72C2×C97#41011yescharacteristic
73C2×C97#41911yescharacteristic
74C2×C97C2×#10411yescharacteristic
75#1506—971nomaximal, Sylow
76C97#150611yescharacteristic, complemented, Sylow
77#533D9711yescharacteristic
78#432—971no—
79#376—971no—
80#328—974no—
81C4⋊Q8Dic9711yescharacteristic, complemented
82C4⋊Q8Dic9711yescharacteristic, complemented
83#125—21no—
84C22⋊Q8—21no—
85C22⋊Q8C2×D9712yes—
86C4×Q8—21no—
87C4×Q8Dic9712yescomplemented
88#117—21no—
89C2×C42—1941no—
90C2×C42C2×D9711yescharacteristic
91C4⋊C8—1941no—
92C22⋊C8—972no—
93#97—1941no—
94#95—972no—
95#95—1941no—
96C2×Q8—21no—
97C2×Q8C2×Dic9712yes—
98C22×C4—1942no—
99C22×C4C22⋊D9711yescharacteristic
100C22×C4C22⋊D9711yescharacteristic
101C22×C4C22×D9711yescharacteristic
102C2×C8—1942no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4C2×Dic9712yes—
109C22⋊C4—22no—
110C42—22no—
111C42—1944no—
112C42C2×Dic9711yescharacteristic
113C42C2×Dic9711yescharacteristic
114C42C22⋊D9711yescharacteristic
115C42C22⋊D9711yescharacteristic
116C23C2×C22⋊D9711yescharacteristic
117Q8—42no—
118Q8—42no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4—3882no—
125C2×C4—3882no—
126C2×C4—1944no—
127C2×C4C22⋊Dic9711yescharacteristic
128C2×C4C22⋊Dic9711yescharacteristic
129C2×C4C22⋊Dic9711yescharacteristic
130C2×C4C22⋊Dic9711yescharacteristic
131C2×C4C22×Dic9711yescharacteristic
132C2×C4C2×C22⋊D9711yescharacteristic
133C8—3882no—
134C22—21no—
135C22—21no—
136C22C2×C22⋊Dic9711yescharacteristic
137C22#1569084112yes—
138C4—21no—
139C4—21no—
140C4—21no—
141C4—21no—
142C4—21no—
143C4—42no—
144C4—42no—
145C4—3884no—
146C4#1569073712yes—
147C2—22no—
148C2#2026070911yescharacteristic
149C2#2026074411yescharacteristic
150C2C2×#1569073711yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#533×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#533D97
hypocenterC97#1506
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
11. C4⋊Q8×C97C4semidirect
12. C4⋊Q8×C97C4semidirect
17. C4×Q8×C97C4semidirect
76. C97#1506semidirect
81. C4⋊Q8Dic97semidirect
82. C4⋊Q8Dic97semidirect
87. C4×Q8Dic97semidirect

Maximal subgroups

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

subgroupindex
#202607922
#202605812
#202604722
#533×C972
#150697