Finite Group Databasebeta

← #27158045

Subgroups — #27158045

Order 12416.

260 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 64 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#20261791C211yesmaximal, characteristic, complemented
2#20261745C211yesmaximal, characteristic, complemented
3#20261647C214yesmaximal, complemented
4C2×D8×D97C211yesmaximal, characteristic
5#15690889—21no—
6#15690885—21no—
7#15690884—21no—
8#15690883C2212yescomplemented
9#15690879C2212yescomplemented
10#15690878—21no—
11C2×D4×D97—21no—
12C2×D776C411yescharacteristic, complemented
13C2×#13095496—21no—
14D8×D97—21no—
15D8×D97—21no—
16D8×D97C412yescomplemented
17D8×D97C2212yes—
18C2×C8×D97C2211yescharacteristic
19C2×D8×C97C411yescharacteristic, complemented
20C4⋊#87604—22no—
21C22⋊#87604—41no—
22C4×#87604—41no—
23#13095514—22no—
24C23×D97—41no—
25C2×D388—21no—
26C2×C22⋊D97—41no—
27D4×D97—22no—
28D4×D97—22no—
29D4×D97—41no—
30D4×D97—41no—
31C2×C4×D97D411yescharacteristic
32D776—21no—
33D776C2×C412yes—
34#13095496—22no—
35#13095496—22no—
36C2×#11715895D411yescharacteristic
37C8×D97C2×C411yescharacteristic
38C8×D97D411yescharacteristic
39C8×D97D411yescharacteristic
40C8×D97C2311yescharacteristic
41C2×D4×C97—21no—
42D8×C97—21no—
43D8×C97C2×C412yes—
44C2×C8×C97C2×C411yescharacteristic
45C23×C97—41no—
46C22×D97—41no—
47C22×D97—42no—
48C22×D97—42no—
49C22×D97—84no—
50C22×D97D811yescharacteristic
51C2×#87604—42no—
52D4×C97—22no—
53D4×C97—41no—
54C2×C4×C97C22⋊C411yescharacteristic
55C22⋊D97—42no—
56C22⋊D97—42no—
57C2×Dic97SD1611yescharacteristic
58D388—22no—
59D388—41no—
60C4×D97C22⋊C411yescharacteristic
61C4×D97D811yescharacteristic
62C4×D97SD1611yescharacteristic
63C4×D97C2×D411yescharacteristic
64C8×C97C22⋊C411yescharacteristic
65C8×C97C22×C411yescharacteristic
66#11715895C22⋊C411yescharacteristic
67#11715895C2×D411yescharacteristic
68C22×C97—42no—
69C22×C97—82no—
70C22×C97#10211yescharacteristic
71C2×D97—22no—
72C2×D97—42no—
73C2×D97—84no—
74C2×D97—82no—
75C2×D97—84no—
76C2×D97#10211yescharacteristic
77C2×D97C2×D811yescharacteristic
78#87604—82no—
79C4×C97#10211yescharacteristic
80C4×C97C2×C22⋊C411yescharacteristic
81Dic97#10211yescharacteristic
82Dic97C2×SD1611yescharacteristic
83C2×C97—21no—
84C2×C97—82no—
85C2×C97C2×#10211yescharacteristic
86D97—21no—
87D97—82no—
88D97#50912yes—
89#2140—971nomaximal, Sylow
90C97#214011yescharacteristic, complemented, Sylow
91C22×D8—971no—
92C2×M32—971no—
93#427—971no—
94#357—974no—
95C22×D4—1941no—
96C2×D8—972no—
97C2×D8—972no—
98C2×D8—971no—
99C2×D8—1941no—
100C2×D8—1941no—
101C2×D8—1941no—
102C2×D8#8760411yescharacteristic, complemented
103C22×C8—971no—
104#117—1941no—
105M32—1941no—
106M32—1941no—
107C2×C16—972no—
108#107—972no—
109#106—1941no—
110C24—3881no—
111C2×D4—21no—
112C2×D4—3881no—
113C2×D4—3881no—
114C2×D4—3881no—
115C2×D4—1942no—
116C2×D4—1942no—
117C2×D4—1941no—
118C22×C4—971no—
119D8—21no—
120D8—972no—
121D8—1942no—
122D8—1942no—
123D8—1941no—
124D8C2×#8760412yes—
125C2×C8—971no—
126C2×C8—971no—
127C2×C8—971no—
128C2×C8—971no—
129C2×C8—971no—
130C2×C8C2×#8760411yescharacteristic
131C4⋊C4—1942no—
132C22⋊C4—3881no—
133C42—3881no—
134C16—1942no—
135C23—41no—
136C23—7764no—
137C23—971no—
138C23—3882no—
139C23—3882no—
140C23—3881no—
141D4—22no—
142D4—41no—
143D4—3882no—
144D4—3882no—
145D4—3881no—
146D4—1942no—
147C2×C4—971no—
148C2×C4—971no—
149C2×C4—971no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—3882no—
153C2×C4C22⋊#8760411yescharacteristic
154C8—971no—
155C8—971no—
156C8C22⋊#8760411yescharacteristic
157C8C22×#8760411yescharacteristic
158C22—42no—
159C22—82no—
160C22—7762no—
161C22—7764no—
162C22—7764no—
163C22—971no—
164C22—971no—
165C22—3882no—
166C22—1942no—
167C22#1569087211yescharacteristic
168C4—7762no—
169C4—971no—
170C4—971no—
171C4#1569087211yescharacteristic
172C4C2×C22⋊#8760411yescharacteristic
173C2—21no—
174C2—82no—
175C2—7762no—
176C2—972no—
177C2—1941no—
178C2C2×#1569087211yescharacteristic
subgroupG/H
centerC2C2×#15690872
commutatorC8×C97C22×C4
FrattiniC8C22×#87604
FittingC2×D8×C97C4
socleC2×C97C2×#102
prime radicalC2×D8×D97C2
cosocleC8×D97C23
hypercenterC2×D8#87604
hypocenterC97#2140
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. #20261791C2semidirect
2. #20261745C2semidirect
3. #20261647C2semidirect
8. #15690883C22semidirect
9. #15690879C22semidirect
12. C2×D776C4semidirect
16. D8×D97C4semidirect
19. C2×D8×C97C4semidirect
90. C97#2140semidirect
102. C2×D8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617912
#202617452
#202616472
C2×D8×D972
#214097