Finite Group Databasebeta

← #27157037

Subgroups — #27157037

Order 12416.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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#20261751C211yesmaximal, characteristic, complemented
2#20261751C211yesmaximal, characteristic, complemented
3#20261583C214yesmaximal, complemented
4C2×#15690772C211yesmaximal, characteristic
5C4⋊#11715897—21no—
6C4⋊#11715897C2212yescomplemented
7C22⋊#11715897—21no—
8C22⋊#11715897—21no—
9C4×#11715897—21no—
10C4×#11715897C2212yescomplemented
11C22×D388C411yescharacteristic
12C22×C4×D97—21no—
13#15690772—21no—
14#15690772—21no—
15#15690772C412yes—
16#15690772C2212yes—
17C2×#13095488—21no—
18C2×C4×Dic97C2211yescharacteristic
19C2×C4⋊C4×C97C411yescharacteristic
20C2×#11715897—22no—
21C2×#11715897—22no—
22C23×D97—21no—
23C2×D388—21no—
24C2×D388—22no—
25C2×D388C814yescomplemented
26C2×D388C2×C412yes—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C2×C4×D97—22no—
30C2×C4×D97—22no—
31C2×C4×D97—44no—
32#13095488—22no—
33#13095488—22no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C4×Dic97C2311yescharacteristic
38C22×C4×C97—21no—
39C22×C4×C97C2×C411yescharacteristic
40C4⋊C4×C97—21no—
41C4⋊C4×C97C2×C412yes—
42C23×C97C22⋊C411yescharacteristic
43C22×D97—24no—
44C22×D97—22no—
45C22×D97—42no—
46C22×D97—42no—
47C2×C4×C97—22no—
48C2×C4×C97—42no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C2×C812yes—
51C2×C4×C97M1612yes—
52C2×C4×C97C22×C411yescharacteristic
53C2×Dic97—22no—
54C2×Dic97—41no—
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C2×D411yescharacteristic
58C2×Dic97C2×D411yescharacteristic
59D388—24no—
60D388—22no—
61D388C2×C814yes—
62C4×D97—44no—
63C4×D97—44no—
64#11715897—42no—
65#11715897—42no—
66C22×C97—21no—
67C22×C97—21no—
68C22×C97C22⋊C812yes—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—24no—
71C2×D97—48no—
72C2×D97—42no—
73C2×D97—42no—
74C4×C97—42no—
75C4×C97C22⋊C812yes—
76C4×C97C22×C811yescharacteristic
77C4×C97C2×M1611yescharacteristic
78Dic97—21no—
79Dic97—21no—
80Dic97—21no—
81Dic97AGL(1,ℤ/8)12yes—
82C2×C97—22no—
83C2×C97C2×C22⋊C811yescharacteristic
84C2×C97#41811yescharacteristic
85C2×C97#42111yescharacteristic
86D97—44no—
87#1478—971nomaximal, Sylow
88C97#147811yescharacteristic, complemented, Sylow
89C2×C4×D4—971no—
90#431—971no—
91#431—971no—
92D4⋊C8—974no—
93C22×D4—971no—
94C23×C4—1941no—
95C4×D4—972no—
96C4×D4—972no—
97C4×D4—1941no—
98C4×D4—1941no—
99C2×C4⋊C4#8760411yescharacteristic
100C2×C22⋊C4—1941no—
101C2×C42—971no—
102C4⋊C8—972no—
103C4⋊C8—1941no—
104C22⋊C8—1941no—
105C22⋊C8—1941no—
106C4×C8—972no—
107C4×C8—1941no—
108C24—1941no—
109C2×D4—974no—
110C2×D4—972no—
111C2×D4—1941no—
112C2×D4—1942no—
113C22×C4—21no—
114C22×C4—971no—
115C22×C4—971no—
116C22×C4—3884no—
117C22×C4—1942no—
118C22×C4—1942no—
119C22×C4C2×#8760411yescharacteristic
120C2×C8—1942no—
121C2×C8—1942no—
122C4⋊C4—21no—
123C4⋊C4C2×#8760412yes—
124C22⋊C4—1942no—
125C22⋊C4—1942no—
126C42—971no—
127C42—971no—
128C42—971no—
129C42—971no—
130C23—3882no—
131C23—3882no—
132C23—1944no—
133C23—1942no—
134C23C22⋊#8760411yescharacteristic
135D4—974no—
136D4—1944no—
137D4—1942no—
138C2×C4—22no—
139C2×C4—42no—
140C2×C4—971no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—3884no—
145C2×C4—3884no—
146C2×C4—3881no—
147C2×C4—1942no—
148C2×C4C22⋊#8760411yescharacteristic
149C2×C4#1309551912yes—
150C2×C4#1309552112yes—
151C2×C4C22×#8760411yescharacteristic
152C8—3882no—
153C8—3882no—
154C22—21no—
155C22—21no—
156C22—3888no—
157C22—3882no—
158C22—3882no—
159C22—1944no—
160C22#1569086712yes—
161C22C2×C22⋊#8760411yescharacteristic
162C4—42no—
163C4—972no—
164C4—1941no—
165C4—1941no—
166C4—1941no—
167C4#1569086712yes—
168C4C2×#1309551911yescharacteristic
169C4C2×#1309552111yescharacteristic
170C2—22no—
171C2—3884no—
172C2C2×#1569086711yescharacteristic
173C2#2026172711yescharacteristic
174C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC4×C97C22×C8
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D388C4
cosocleC4×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1478
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261751C2semidirect
2. #20261751C2semidirect
3. #20261583C2semidirect
6. C4⋊#11715897C22semidirect
10. C4×#11715897C22semidirect
25. C2×D388C8semidirect
88. C97#1478semidirect

Maximal subgroups

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

subgroupindex
#202617512
#202615832
C2×#156907722
#147897