Finite Group Databasebeta

← #27143407

Subgroups — #27143407

Order 12416.

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

normal subgroups only

This group has 270 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#20261299C211yesmaximal, characteristic
2#20260709C211yesmaximal, characteristic, complemented
3#20260527C214yesmaximal, complemented
4#410×C97C211yesmaximal, characteristic, complemented
5C22×D388C411yescharacteristic, complemented
6#15690806C2212yes—
7#15690799C412yescomplemented
8#15690784—21no—
9#15690784—21no—
10#15690778—21no—
11#15690778—21no—
12#15690769—21no—
13#15690769C411yescharacteristic, complemented
14#15690725—21no—
15#15690725C2212yescomplemented
16C2×#41×C97C2211yescharacteristic
17#117×C97—21no—
18#99×C97—21no—
19#99×C97C2212yescomplemented
20C23×D97—21no—
21C2×D388—21no—
22C2×D388—41no—
23C2×D388—41no—
24C2×D388C2×C412yes—
25C2×C22⋊D97—22no—
26C2×C22⋊D97—22no—
27C4⋊Dic97—21no—
28C4⋊Dic97—41no—
29C22⋊Dic97—21no—
30C22⋊Dic97—22no—
31#13095488—22no—
32#13095488—22no—
33C4×Dic97—41no—
34C4×Dic97C2×C412yes—
35#41×C97—41no—
36#41×C97—41no—
37C2×Q8×C97C2×C411yescharacteristic
38C2×D4×C97D411yescharacteristic
39C2×D4×C97D411yescharacteristic
40C2×D4×C97C2311yescharacteristic
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C22×C4×C97D411yescharacteristic
44C4⋊C4×C97—41no—
45C22⋊C4×C97—22no—
46C42×C97—41no—
47C23×C97C22⋊C411yescharacteristic
48C23×C97C2×D411yescharacteristic
49C23×C97C2×D411yescharacteristic
50C22×D97—22no—
51C22×D97—42no—
52C22×D97—82no—
53Q8×C97—41no—
54D4×C97—41no—
55D4×C97—41no—
56D4×C97—41no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—42no—
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22×C411yescharacteristic
68C22⋊D97—82no—
69C22⋊D97—82no—
70C2×Dic97—22no—
71C2×Dic97—42no—
72D388—42no—
73D388—41no—
74D388—41no—
75C22×C97—21no—
76C22×C97—21no—
77C22×C97—21no—
78C22×C97—41no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97C2×C22⋊C411yescharacteristic
82C2×D97—42no—
83C2×D97—42no—
84C2×D97—42no—
85C2×D97—84no—
86C4×C97—21no—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—82no—
90C4×C97#9911yescharacteristic
91C4×C97#9911yescharacteristic
92Dic97—42no—
93Dic97—82no—
94C2×C97—21no—
95C2×C97—41no—
96C2×C97—41no—
97C2×C97—41no—
98C2×C97C2×#9911yescharacteristic
99D97—82no—
100#2128—971nomaximal, Sylow
101C97#212811yescharacteristic, complemented, Sylow
102#535—971no—
103#410—971no—
104#410D9711yescharacteristic, complemented
105#353—974no—
106C2×#41C2×D9711yescharacteristic
107C22×D4—971no—
108#127—972no—
109#124—972no—
110#121—1941no—
111#121—1941no—
112#120—1941no—
113#120—1941no—
114#117—21no—
115#117—971no—
116#117—1941no—
117#99—21no—
118#99—972no—
119#99—1941no—
120#99C2×D9712yescomplemented
121C24—1941no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D9711yescharacteristic
125C2×D4—972no—
126C2×D4—3881no—
127C2×D4—3881no—
128C2×D4—1942no—
129C2×D4—1942no—
130C2×D4—1941no—
131C2×D4D38811yescharacteristic
132C2×D4C22⋊D9711yescharacteristic
133C2×D4C22×D9711yescharacteristic
134C22×C4C4×D9711yescharacteristic
135C22×C4D38811yescharacteristic
136C22×C4C22⋊D9711yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—3881no—
139C4⋊C4—1941no—
140C22⋊C4—22no—
141C22⋊C4—1942no—
142C22⋊C4—1941no—
143C22⋊C4—1942no—
144C22⋊C4—1942no—
145C42—41no—
146C42—972no—
147C42—3881no—
148C23—7762no—
149C23—3882no—
150C23—1942no—
151C23#1309548811yescharacteristic
152C23C2×C22⋊D9711yescharacteristic
153C23C2×D38811yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—7762no—
159D4—7762no—
160D4—3882no—
161D4—3881no—
162D4—3881no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—3882no—
171C2×C4—1942no—
172C2×C4#1309548811yescharacteristic
173C2×C4#1309548811yescharacteristic
174C2×C4#1309548811yescharacteristic
175C2×C4C2×C4×D9711yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—7764no—
183C22—3882no—
184C22—3882no—
185C22—3882no—
186C22C2×#1309548811yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—7762no—
192C4—3882no—
193C4#1569072411yescharacteristic
194C4#1569072411yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—7762no—
200C2C2×#1569072411yescharacteristic
subgroupG/H
centerC2C2×#15690724
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#410×C97C2
socleC2×C97C2×#99
prime radical#20261299C2
cosocleC2×D4×C97C23
hypercenter#410D97
hypocenterC97#2128
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20260709C2semidirect
3. #20260527C2semidirect
4. #410×C97C2semidirect
5. C22×D388C4semidirect
7. #15690799C4semidirect
13. #15690769C4semidirect
15. #15690725C22semidirect
19. #99×C97C22semidirect
101. C97#2128semidirect
104. #410D97semidirect
120. #99C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612992
#202607092
#202605272
#410×C972
#212897