Finite Group Databasebeta

← #27157197

Subgroups — #27157197

Order 12416.

264 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 62 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#20261756C211yesmaximal, characteristic, complemented
2C4×#13095520C211yesmaximal, characteristic, complemented
3#20261591C214yesmaximal, complemented
4C2×#15690799C211yesmaximal, characteristic
5C2×#13095520—21no—
6C4⋊#11715897—21no—
7C22⋊#11715897—21no—
8#15690865C2212yescomplemented
9#15690865C2212yescomplemented
10C4×#11715897—21no—
11C22×D388C411yescharacteristic
12#15690799—21no—
13#15690799—21no—
14#15690799C412yes—
15#15690799C2212yes—
16C2×#13095488—21no—
17C2×#13095488—21no—
18C2×C4×Dic97C2211yescharacteristic
19C22×Q8×C97C411yescharacteristic
20#13095520—24no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C23×D97—21no—
24C2×D388—21no—
25C2×D388—42no—
26C2×D388C2×C412yes—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29#13095488—22no—
30#13095488—22no—
31#13095488—22no—
32#13095488—22no—
33C4×Dic97C2×C411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97C2311yescharacteristic
37C2×Q8×C97—21no—
38C2×Q8×C97—42no—
39C2×Q8×C97C2×C412yes—
40C22×C4×C97—21no—
41C22×C4×C97C2×C411yescharacteristic
42C23×C97C22⋊C411yescharacteristic
43C22×D97—22no—
44C22×D97—42no—
45C22×D97—42no—
46C22×D97—42no—
47Q8×C97—42no—
48Q8×C97—42no—
49C2×C4×C97—22no—
50C2×C4×C97—22no—
51C2×C4×C97—42no—
52C2×C4×C97C22⋊C411yescharacteristic
53C2×C4×C97C22×C411yescharacteristic
54C2×Dic97—22no—
55C2×Dic97—41no—
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C2×D411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60D388—42no—
61D388—42no—
62#11715897—24no—
63#11715897—42no—
64C22×C97—21no—
65C22×C97—21no—
66C22×C97#10412yes—
67C22×C97C2×C22⋊C411yescharacteristic
68C2×D97—42no—
69C2×D97—42no—
70C2×D97—42no—
71C2×D97—84no—
72C4×C97—21no—
73C4×C97—42no—
74C4×C97#10012yes—
75Dic97—21no—
76Dic97—21no—
77Dic97—21no—
78Dic97—21no—
79C2×C97—22no—
80C2×C97C2×#10011yescharacteristic
81C2×C97C2×#10411yescharacteristic
82C2×C97#42111yescharacteristic
83D97—82no—
84#1527—971nomaximal, Sylow
85C97#152711yescharacteristic, complemented, Sylow
86C2×#124—971no—
87#432—971no—
88C4×M16—971no—
89#329—974no—
90C22×Q8#8760411yescharacteristic
91C22×D4—971no—
92C2×M16—1941no—
93#124—972no—
94#124—972no—
95#124—1941no—
96#124—1941no—
97C2×C22⋊C4—1941no—
98C2×C22⋊C4—1941no—
99C2×C42—971no—
100C4⋊C8—1941no—
101C22⋊C8—1941no—
102#97—972no—
103#97—972no—
104C4×C8—1941no—
105C24—1941no—
106C2×Q8—21no—
107C2×Q8—42no—
108C2×Q8C2×#8760412yes—
109C2×D4—972no—
110C2×D4—3882no—
111C2×D4—1941no—
112C22×C4—21no—
113C22×C4—971no—
114C22×C4—971no—
115C22×C4C2×#8760411yescharacteristic
116M16—1944no—
117C2×C8—1942no—
118C2×C8—1942no—
119C22⋊C4—1942no—
120C22⋊C4—1942no—
121C22⋊C4—1942no—
122C22⋊C4—1942no—
123C42—971no—
124C42—971no—
125C42—971no—
126C42—971no—
127C23—3882no—
128C23—3882no—
129C23—3882no—
130C23—1942no—
131C23C22⋊#8760411yescharacteristic
132Q8—42no—
133Q8—42no—
134D4—3882no—
135D4—3882no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—971no—
140C2×C4—971no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—3881no—
144C2×C4—1942no—
145C2×C4C22⋊#8760411yescharacteristic
146C2×C4C22×#8760411yescharacteristic
147C8—3882no—
148C8—1944no—
149C22—21no—
150C22—21no—
151C22—7764no—
152C22—3882no—
153C22—3882no—
154C22—3882no—
155C22#1569087512yes—
156C22C2×C22⋊#8760411yescharacteristic
157C4—21no—
158C4—42no—
159C4—1941no—
160C4—1941no—
161C4—1941no—
162C4—1941no—
163C4#1569087012yes—
164C2—22no—
165C2—7762no—
166C2C2×#1569087011yescharacteristic
167C2C2×#1569087511yescharacteristic
168C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC22×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC4×Dic97C23
hypercenterC22×Q8#87604
hypocenterC97#1527
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. #20261756C2semidirect
2. C4×#13095520C2semidirect
3. #20261591C2semidirect
8. #15690865C22semidirect
9. #15690865C22semidirect
85. C97#1527semidirect

Maximal subgroups

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

subgroupindex
#202617562
C4×#130955202
#202615912
C2×#156907992
#152797