Finite Group Databasebeta

← #27157127

Subgroups — #27157127

Order 12416.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261757C211yesmaximal, characteristic, complemented
2#20261671C211yesmaximal, characteristic, complemented
3#20261587C214yesmaximal, complemented
4#20261286C211yesmaximal, characteristic
5C2×C4×#87604—21no—
6C4⋊#11715897—21no—
7#15690867C2212yes—
8#15690865—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4C2212yes—
11C2×#13095512—21no—
12#15690807C411yescharacteristic, complemented
13#15690799—21no—
14#15690785—21no—
15#15690785C2212yes—
16#15690772—21no—
17#15690772C412yescomplemented
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C2×#11715897—22no—
22C4×#87604—24no—
23#13095512—44no—
24C2×D388—22no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388C2×C412yes—
28C2×C4×D97—22no—
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97C2311yescharacteristic
32#13095488—22no—
33#13095488—22no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97D411yescharacteristic
37C2×Q8×C97—21no—
38C4⋊C4×C97—21no—
39C4⋊C4×C97C2×C412yes—
40C42×C97C2×C411yescharacteristic
41C22×D97—22no—
42C22×D97C2×D411yescharacteristic
43C2×#87604—24no—
44C2×#87604—42no—
45C2×#87604—42no—
46Q8×C97—42no—
47C2×C4×C97—22no—
48C2×C4×C97C22⋊C411yescharacteristic
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22×C411yescharacteristic
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C22⋊C411yescharacteristic
53C2×Dic97C2×D411yescharacteristic
54D388—42no—
55D388—42no—
56D388—44no—
57D388—42no—
58C4×D97—22no—
59C4×D97—22no—
60C4×D97—41no—
61C4×D97—44no—
62#11715897—42no—
63C22×C97C2×C22⋊C411yescharacteristic
64C2×D97—21no—
65C2×D97—21no—
66C2×D97—42no—
67C2×D97—42no—
68C2×D97—42no—
69C2×D97AGL(1,ℤ/8)12yes—
70#87604—44no—
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—42no—
74C4×C97#10412yes—
75Dic97—21no—
76Dic97—21no—
77Dic97—21no—
78C2×C97#41011yescharacteristic
79C2×C97#41811yescharacteristic
80C2×C97C2×#10411yescharacteristic
81D97—22no—
82D97—82no—
83#1505—971nomaximal, Sylow
84C97#150511yescharacteristic, complemented, Sylow
85#532—971no—
86#432—971no—
87#376—971no—
88#327—974no—
89C2×#41—1941no—
90C4⋊Q8#8760411yescharacteristic, complemented
91#127—971no—
92#124—1941no—
93#121—972no—
94#121—1941no—
95C4×D4—972no—
96C4×D4—1941no—
97C2×C42—971no—
98C2×C42—1941no—
99C4⋊C8—1941no—
100C22⋊C8—972no—
101#97—1941no—
102#95—972no—
103#95—1941no—
104#41—3884no—
105C2×Q8—21no—
106C2×D4—972no—
107C2×D4—1942no—
108C2×D4—1941no—
109C2×D4—1941no—
110C22×C4—971no—
111C22×C4—971no—
112C22×C4—971no—
113C22×C4—1942no—
114C22×C4—1942no—
115C2×C8—1942no—
116C4⋊C4—21no—
117C4⋊C4C2×#8760412yes—
118C22⋊C4—1942no—
119C22⋊C4—1942no—
120C42—971no—
121C42—971no—
122C42—971no—
123C42—1944no—
124C42C2×#8760411yescharacteristic
125C23—971no—
126C23—1942no—
127Q8—42no—
128D4—3882no—
129D4—3882no—
130D4—3884no—
131D4—3882no—
132C2×C4—22no—
133C2×C4—971no—
134C2×C4—971no—
135C2×C4—971no—
136C2×C4—3881no—
137C2×C4—3884no—
138C2×C4—3882no—
139C2×C4—3882no—
140C2×C4—1942no—
141C2×C4—1944no—
142C2×C4—1942no—
143C2×C4C22⋊#8760411yescharacteristic
144C2×C4C22⋊#8760411yescharacteristic
145C2×C4C22×#8760411yescharacteristic
146C8—3882no—
147C22—972no—
148C22—3882no—
149C22—3882no—
150C22—3882no—
151C22—1941no—
152C22—1941no—
153C22C2×C22⋊#8760411yescharacteristic
154C4—21no—
155C4—21no—
156C4—42no—
157C4—3884no—
158C4—1941no—
159C4—1941no—
160C4—1941no—
161C4#1569087512yes—
162C2—7762no—
163C2—1942no—
164C2#2026171611yescharacteristic
165C2#2026172711yescharacteristic
166C2C2×#1569087511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261286C2
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1505
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261757C2semidirect
2. #20261671C2semidirect
3. #20261587C2semidirect
12. #15690807C4semidirect
17. #15690772C4semidirect
19. C4⋊Q8×C97C4semidirect
84. C97#1505semidirect
90. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616712
#202615872
#202612862
#150597