Finite Group Databasebeta

← #27140243

Subgroups — #27140243

Order 12416.

272 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 73 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#20261285C211yesmaximal, characteristic
2#20260792C211yesmaximal, characteristic, complemented
3#20260468C214yesmaximal, complemented
4#376×C97C211yesmaximal, characteristic, complemented
5C2×#13095511—21no—
6#15690810C411yescharacteristic, complemented
7#15690807C411yescharacteristic, complemented
8#15690798—21no—
9#15690784—21no—
10#15690784C2212yes—
11C4×D388—21no—
12C4×D388C412yescomplemented
13C4⋊#11715895—21no—
14C22⋊#11715895C2212yes—
15#15690721—21no—
16C2×C42×C97—21no—
17C2×C42×C97C2211yescharacteristic
18#95×C97—21no—
19#95×C97C2212yes—
20#13095511—44no—
21C2×Dic194—21no—
22C2×D388—21no—
23C2×D388—21no—
24C2×D388C2×C412yes—
25C2×C22⋊D97—22no—
26C2×C4×D97—22no—
27C2×#11715895—22no—
28C4⋊Dic97—21no—
29C4⋊Dic97C2×C412yes—
30#13095488—22no—
31#13095488—22no—
32C22×C4×C97—22no—
33C22×C4×C97D411yescharacteristic
34C22×C4×C97D411yescharacteristic
35C22×C4×C97C2311yescharacteristic
36C42×C97—24no—
37C42×C97C2×C411yescharacteristic
38C42×C97C2×C411yescharacteristic
39C42×C97D411yescharacteristic
40C42×C97D411yescharacteristic
41C23×C97C2×D411yescharacteristic
42C22×D97—22no—
43C2×C4×C97—22no—
44C2×C4×C97—22no—
45C2×C4×C97—24no—
46C2×C4×C97—41no—
47C2×C4×C97—42no—
48C2×C4×C97—42no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22⋊C411yescharacteristic
52C2×C4×C97C22⋊C411yescharacteristic
53C2×C4×C97C22×C411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C22⋊D97—44no—
56C2×Dic97—22no—
57D388—42no—
58D388—42no—
59D388—42no—
60C4×D97—44no—
61Dic194—42no—
62#11715895—42no—
63C22×C97—21no—
64C22×C97—21no—
65C22×C97C2×C22⋊C411yescharacteristic
66C22×C97AGL(1,ℤ/8)12yes—
67C2×D97—42no—
68C2×D97—42no—
69C2×D97—42no—
70C4×C97—21no—
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—21no—
74C4×C97—21no—
75C4×C97—44no—
76C4×C97#10412yes—
77Dic97—42no—
78C2×C97—22no—
79C2×C97#41011yescharacteristic
80C2×C97#41811yescharacteristic
81C2×C97C2×#10411yescharacteristic
82D97—82no—
83#1505—971nomaximal, Sylow
84C97#150511yescharacteristic, complemented, Sylow
85#532—971no—
86#432—971no—
87#376D9711yescharacteristic, complemented
88#327—974no—
89C2×#41—1941no—
90C4⋊Q8—971no—
91#127—971no—
92#124—1941no—
93#121—972no—
94#121—1941no—
95C4×D4—972no—
96C4×D4—1941no—
97C2×C42—21no—
98C2×C42C2×D9711yescharacteristic
99C4⋊C8—1941no—
100C22⋊C8—972no—
101#97—1941no—
102#95—21no—
103#95C2×D9712yes—
104#41—3884no—
105C2×Q8—1941no—
106C2×D4—972no—
107C2×D4—1942no—
108C2×D4—1941no—
109C2×D4—1941no—
110C22×C4—22no—
111C22×C4—1942no—
112C22×C4C22⋊D9711yescharacteristic
113C22×C4C22⋊D9711yescharacteristic
114C22×C4C22×D9711yescharacteristic
115C2×C8—1942no—
116C4⋊C4—972no—
117C4⋊C4—1941no—
118C22⋊C4—1942no—
119C22⋊C4—1942no—
120C42—24no—
121C42C4×D9711yescharacteristic
122C42C4×D9711yescharacteristic
123C42D38811yescharacteristic
124C42D38811yescharacteristic
125C23—1942no—
126C23C2×C22⋊D9711yescharacteristic
127Q8—3882no—
128D4—3882no—
129D4—3882no—
130D4—3884no—
131D4—3882no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—24no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—3884no—
139C2×C4—1942no—
140C2×C4#1309548811yescharacteristic
141C2×C4#1309548811yescharacteristic
142C2×C4#1309548811yescharacteristic
143C2×C4#1309548811yescharacteristic
144C2×C4C2×C4×D9711yescharacteristic
145C2×C4C2×D38811yescharacteristic
146C8—3882no—
147C22—21no—
148C22—21no—
149C22—3882no—
150C22—3882no—
151C22—3882no—
152C22C2×#1309548811yescharacteristic
153C22#1569083512yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—44no—
160C4—3882no—
161C4#1569073812yes—
162C2—22no—
163C2—7762no—
164C2#2026070811yescharacteristic
165C2#2026073911yescharacteristic
166C2C2×#1569073811yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#376×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261285C2
cosocleC22×C4×C97C23
hypercenter#376D97
hypocenterC97#1505
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. #20260792C2semidirect
3. #20260468C2semidirect
4. #376×C97C2semidirect
6. #15690810C4semidirect
7. #15690807C4semidirect
12. C4×D388C4semidirect
84. C97#1505semidirect
87. #376D97semidirect

Maximal subgroups

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

subgroupindex
#202612852
#202607922
#202604682
#376×C972
#150597