Finite Group Databasebeta

← #27140457

Subgroups — #27140457

Order 12416.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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
3#20260757C211yesmaximal, characteristic, complemented
4#20260481C211yesmaximal, characteristic, complemented
5#20260478C211yesmaximal, characteristic, complemented
6#20260478C211yesmaximal, characteristic, complemented
7#332×C97C211yesmaximal, characteristic, complemented
8C2×#13095512—21no—
9C2×#13095494—21no—
10#15690807C2211yescharacteristic
11#15690799—21no—
12#15690785—21no—
13#15690785C412yes—
14#15690772—21no—
15#15690772C412yes—
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—42no—
26#13095512—42no—
27C2×D388—22no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388C2×C412yes—
31C2×C4×D97—22no—
32C2×C4×D97C2×C411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×C4×D97D411yescharacteristic
35#13095494—42no—
36C2×#11715895—21no—
37C2×#11715895—21no—
38#13095488—22no—
39#13095488—22no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C2×Q8×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C412yes—
48C42×C97C2311yescharacteristic
49C22×D97—22no—
50C22×D97C22⋊C411yescharacteristic
51Q8×C97—41no—
52Q8×C97—41no—
53C2×C4×C97—22no—
54C2×C4×C97C22×C411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C22⋊C411yescharacteristic
59C2×Dic97C22⋊C411yescharacteristic
60D388—42no—
61D388—44no—
62D388—41no—
63D388—41no—
64D388—41no—
65D388—41no—
66C4×D97—21no—
67C4×D97—21no—
68C4×D97—21no—
69C4×D97—21no—
70C4×D97—41no—
71C4×D97—44no—
72C8×C97—41no—
73C8×C97—41no—
74#11715895—41no—
75#11715895—41no—
76C22×C97C2×C22⋊C411yescharacteristic
77C2×D97—21no—
78C2×D97—21no—
79C2×D97—21no—
80C2×D97—42no—
81C2×D97—42no—
82C2×D97—42no—
83C4×C97—21no—
84C4×C97—42no—
85C4×C97AGL(1,ℤ/8)11yescharacteristic
86C4×C97AGL(1,ℤ/8)11yescharacteristic
87C4×C97AGL(1,ℤ/8)11yescharacteristic
88C4×C97#13711yescharacteristic
89Dic97—21no—
90Dic97—21no—
91Dic97—21no—
92C2×C97#41311yescharacteristic
93C2×C97#41711yescharacteristic
94C2×C97#41811yescharacteristic
95D97—41no—
96D97—82no—
97#1557—971nomaximal, Sylow
98C97#155711yescharacteristic, complemented, Sylow
99#532—971no—
100#432—971no—
101#423—971no—
102#332—971no—
103#332D9711yescharacteristic, complemented
104#331—971no—
105#331—971no—
106C2×#41—1941no—
107C2×M16—1941no—
108C4⋊Q8C2×D9711yescharacteristic
109#127—971no—
110#124—1941no—
111#121—972no—
112#121—1941no—
113C4×D4—972no—
114C4×D4—1941no—
115C2×C42—971no—
116C4⋊C8—971no—
117C4⋊C8—971no—
118C4⋊C8—1941no—
119C4⋊C8C2×D9711yescharacteristic
120C4⋊C8C2×D9711yescharacteristic
121C22⋊C8—1941no—
122#97—1941no—
123#41—3882no—
124#41—3882no—
125C2×Q8—21no—
126C2×D4—972no—
127C2×D4—1942no—
128C2×D4—1941no—
129C2×D4—1941no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—1942no—
134M16—3882no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—1941no—
138C2×C8—1941no—
139C4⋊C4—21no—
140C4⋊C4C4×D9712yes—
141C22⋊C4—1942no—
142C22⋊C4—1942no—
143C42—971no—
144C42—971no—
145C42—971no—
146C42C22×D9711yescharacteristic
147C23—971no—
148C23—1942no—
149Q8—41no—
150Q8—41no—
151D4—3882no—
152D4—3884no—
153D4—3881no—
154D4—3881no—
155D4—3881no—
156D4—3881no—
157C2×C4—22no—
158C2×C4—971no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—3881no—
162C2×C4—3884no—
163C2×C4—1941no—
164C2×C4—1941no—
165C2×C4—1941no—
166C2×C4—1941no—
167C2×C4C2×C4×D9711yescharacteristic
168C2×C4D4×D9711yescharacteristic
169C2×C4D4×D9711yescharacteristic
170C8—41no—
171C8—41no—
172C8—3881no—
173C8—3881no—
174C22—3882no—
175C22—3882no—
176C22—3882no—
177C22—1941no—
178C22—1941no—
179C22—1941no—
180C22C22⋊C4×D9711yescharacteristic
181C4—21no—
182C4—42no—
183C4—1941no—
184C4—1941no—
185C4—1941no—
186C4#1569083411yescharacteristic
187C4#1569083711yescharacteristic
188C4#1569083711yescharacteristic
189C4#1569084211yescharacteristic
190C2—7762no—
191C2—3881no—
192C2#2026071811yescharacteristic
193C2#2026073511yescharacteristic
194C2#2026073811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#332×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261286C2
cosocleC42×C97C23
hypercenter#332D97
hypocenterC97#1557
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20260791C2semidirect
3. #20260757C2semidirect
4. #20260481C2semidirect
5. #20260478C2semidirect
6. #20260478C2semidirect
7. #332×C97C2semidirect
98. C97#1557semidirect
103. #332D97semidirect

Maximal subgroups

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

subgroupindex
#202612862
#202607912
#202607572
#202604812
#202604782
#332×C972
#155797