Finite Group Databasebeta

← #27140430

Subgroups — #27140430

Order 12416.

252 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 67 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#20261286C211yesmaximal, characteristic
2#20260791C211yesmaximal, characteristic, complemented
3C4⋊C8×D97C211yesmaximal, characteristic, complemented
4#20260484C211yesmaximal, characteristic, complemented
5#20260478C211yesmaximal, characteristic, complemented
6#20260478C211yesmaximal, characteristic, complemented
7#333×C97C211yesmaximal, characteristic, complemented
8C2×#13095512C412yes—
9C2×C8×D97—21no—
10#15690807C2211yescharacteristic
11#15690799C412yes—
12#15690785—21no—
13#15690785—21no—
14#15690772—21no—
15#15690772—21no—
16C42×D97C2211yescharacteristic
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690720—21no—
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25#13095512—44no—
26C2×D388—22no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388C2×C412yes—
30C2×C4×D97—22no—
31C2×C4×D97C2×C411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×#11715895—21no—
35C2×#11715895—21no—
36C8×D97—42no—
37#13095488—22no—
38#13095488—22no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C2×Q8×C97C2×C412yes—
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C22×D97—22no—
49C22×D97C22⋊C411yescharacteristic
50Q8×C97—42no—
51C2×C4×C97—22no—
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58D388—42no—
59D388—44no—
60D388—41no—
61D388—41no—
62D388—41no—
63D388—41no—
64C4×D97—22no—
65C4×D97—21no—
66C4×D97—21no—
67C4×D97—21no—
68C4×D97—21no—
69C4×D97—44no—
70C8×C97—41no—
71C8×C97—41no—
72#11715895—41no—
73#11715895—41no—
74C22×C97C2×C22⋊C411yescharacteristic
75C2×D97—21no—
76C2×D97—21no—
77C2×D97—42no—
78C2×D97—42no—
79C2×D97—42no—
80C2×D97#10012yes—
81C4×C97—21no—
82C4×C97—42no—
83C4×C97#13511yescharacteristic
84C4×C97#13511yescharacteristic
85C4×C97AGL(1,ℤ/8)11yescharacteristic
86C4×C97#13711yescharacteristic
87Dic97—21no—
88Dic97—21no—
89Dic97—21no—
90C2×C97C2×#10011yescharacteristic
91C2×C97#41611yescharacteristic
92C2×C97#41711yescharacteristic
93D97—22no—
94D97—82no—
95#1551—971nomaximal, Sylow
96C97#155111yescharacteristic, complemented, Sylow
97#532—971no—
98#432—971no—
99C2×C4⋊C8—971no—
100#333—971no—
101#333D9711yescharacteristic, complemented
102#331—971no—
103#331—971no—
104C2×#41—972no—
105C22×C8—1941no—
106C4⋊Q8C2×D9711yescharacteristic
107#127—971no—
108#124—972no—
109#121—1941no—
110#121—1941no—
111C4×D4—1941no—
112C4×D4—1941no—
113C2×C42—971no—
114C4⋊C8—971no—
115C4⋊C8—971no—
116C4⋊C8—1941no—
117C4⋊C8C2×D9711yescharacteristic
118C4⋊C8C2×D9711yescharacteristic
119C22⋊C8—1941no—
120#97—1941no—
121#41—3884no—
122C2×Q8C4×D9712yes—
123C2×D4—972no—
124C2×D4—1942no—
125C2×D4—1941no—
126C2×D4—1941no—
127C22×C4—971no—
128C22×C4—971no—
129C22×C4—971no—
130C22×C4—1942no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—3882no—
134C2×C8—1941no—
135C2×C8—1941no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C22⋊C4—1942no—
139C22⋊C4—1942no—
140C42—971no—
141C42—971no—
142C42—971no—
143C42C22×D9711yescharacteristic
144C23—971no—
145C23—1942no—
146Q8—42no—
147D4—3882no—
148D4—3884no—
149D4—3881no—
150D4—3881no—
151D4—3881no—
152D4—3881no—
153C2×C4—22no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—971no—
157C2×C4—3884no—
158C2×C4—1942no—
159C2×C4—1941no—
160C2×C4—1941no—
161C2×C4—1941no—
162C2×C4—1941no—
163C2×C4C2×C4×D9711yescharacteristic
164C2×C4D4×D9711yescharacteristic
165C2×C4D4×D9711yescharacteristic
166C8—41no—
167C8—41no—
168C8—3881no—
169C8—3881no—
170C22—972no—
171C22—3882no—
172C22—3882no—
173C22—3882no—
174C22—1941no—
175C22—1941no—
176C22C22⋊C4×D9711yescharacteristic
177C4—21no—
178C4—42no—
179C4—1941no—
180C4—1941no—
181C4—1941no—
182C4#1569083111yescharacteristic
183C4#1569083311yescharacteristic
184C4#1569083711yescharacteristic
185C4#1569084211yescharacteristic
186C2—7762no—
187C2—1942no—
188C2#100×D9711yescharacteristic
189C2#2026073011yescharacteristic
190C2#2026073511yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#333×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261286C2
cosocleC42×C97C23
hypercenter#333D97
hypocenterC97#1551
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #20260791C2semidirect
3. C4⋊C8×D97C2semidirect
4. #20260484C2semidirect
5. #20260478C2semidirect
6. #20260478C2semidirect
7. #333×C97C2semidirect
96. C97#1551semidirect
101. #333D97semidirect

Maximal subgroups

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

subgroupindex
#202612862
#202607912
C4⋊C8×D972
#202604842
#202604782
#333×C972
#155197