Finite Group Databasebeta

← #27140284

Subgroups — #27140284

Order 12416.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 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#20260603C211yesmaximal, characteristic, complemented
3#20260470C214yesmaximal, complemented
4#432×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
13#15690717—21no—
14#15690717—21no—
15#15690717C2212yes—
16C2×C42×C97C2211yescharacteristic
17C4⋊C8×C97—21no—
18C22⋊C8×C97C2212yes—
19#97×C97—21no—
20#13095511—44no—
21C2×Dic194—21no—
22C2×D388—21no—
23C2×D388—21no—
24C2×D388C2×C412yes—
25C2×C22⋊D97—22no—
26C22×Dic97—22no—
27C2×C4×D97—22no—
28C4⋊Dic97—21no—
29C4⋊Dic97C2×C412yes—
30#13095488—22no—
31#13095488—22no—
32C22×C4×C97D411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C22×C4×C97C2311yescharacteristic
35C2×C8×C97—22no—
36C42×C97C2×C411yescharacteristic
37C42×C97C2×C411yescharacteristic
38C42×C97D411yescharacteristic
39C42×C97D411yescharacteristic
40C23×C97C2×D411yescharacteristic
41C22×D97—22no—
42C2×C4×C97—22no—
43C2×C4×C97—41no—
44C2×C4×C97—41no—
45C2×C4×C97C22⋊C411yescharacteristic
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22⋊C411yescharacteristic
49C2×C4×C97C22×C411yescharacteristic
50C2×C4×C97C2×D411yescharacteristic
51C22⋊D97—44no—
52C2×Dic97—22no—
53C2×Dic97—42no—
54C2×Dic97—42no—
55C2×Dic97—42no—
56D388—42no—
57D388—42no—
58D388—42no—
59C4×D97—44no—
60Dic194—42no—
61C8×C97—42no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97C2×C22⋊C411yescharacteristic
65C22×C97AGL(1,ℤ/8)12yes—
66C2×D97—42no—
67C2×D97—42no—
68C2×D97—42no—
69C4×C97—21no—
70C4×C97—21no—
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—21no—
74C4×C97—21no—
75Dic97—42no—
76Dic97—82no—
77C2×C97—22no—
78C2×C97#41011yescharacteristic
79C2×C97#41811yescharacteristic
80C2×C97#42111yescharacteristic
81D97—82no—
82#1514—971nomaximal, Sylow
83C97#151411yescharacteristic, complemented, Sylow
84#532—971no—
85#432D9711yescharacteristic, complemented
86#383—971no—
87#327—974no—
88C2×#41—1941no—
89C4⋊Q8—971no—
90#127—971no—
91#124—1941no—
92#121—972no—
93#121—1941no—
94C4×D4—972no—
95C4×D4—1941no—
96C2×C42C2×D9711yescharacteristic
97C4⋊C8—21no—
98C22⋊C8C2×D9712yes—
99#97—21no—
100#95—972no—
101#95—1941no—
102#95—1941no—
103#41—3884no—
104C2×Q8—1941no—
105C2×D4—972no—
106C2×D4—1942no—
107C2×D4—1941no—
108C2×D4—1941no—
109C22×C4—1942no—
110C22×C4—1942no—
111C22×C4D38811yescharacteristic
112C22×C4D38811yescharacteristic
113C22×C4C22×D9711yescharacteristic
114C2×C8—22no—
115C4⋊C4—972no—
116C4⋊C4—1941no—
117C22⋊C4—1942no—
118C22⋊C4—1942no—
119C42C4×D9711yescharacteristic
120C42C4×D9711yescharacteristic
121C42C22⋊D9711yescharacteristic
122C42C22⋊D9711yescharacteristic
123C23—1942no—
124C23C2×D38811yescharacteristic
125Q8—3882no—
126D4—3882no—
127D4—3882no—
128D4—3882no—
129D4—3884no—
130C2×C4—22no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—3884no—
134C2×C4—3882no—
135C2×C4—3882no—
136C2×C4—3882no—
137C2×C4—1942no—
138C2×C4#1309548811yescharacteristic
139C2×C4#1309548811yescharacteristic
140C2×C4#1309548811yescharacteristic
141C2×C4#1309548811yescharacteristic
142C2×C4C2×C4×D9711yescharacteristic
143C2×C4C2×C22⋊D9711yescharacteristic
144C8—42no—
145C22—21no—
146C22—21no—
147C22—3882no—
148C22—3882no—
149C22—3882no—
150C22C2×#1309548811yescharacteristic
151C22#1569083812yes—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—7762no—
159C4—3882no—
160C2—22no—
161C2—7762no—
162C2#2026070811yescharacteristic
163C2#2026074111yescharacteristic
164C2#2026075111yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#432×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261285C2
cosocleC22×C4×C97C23
hypercenter#432D97
hypocenterC97#1514
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #20260603C2semidirect
3. #20260470C2semidirect
4. #432×C97C2semidirect
6. #15690810C4semidirect
7. #15690807C4semidirect
12. C4×D388C4semidirect
83. C97#1514semidirect
85. #432D97semidirect

Maximal subgroups

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

subgroupindex
#202612852
#202606032
#202604702
#432×C972
#151497