Finite Group Databasebeta

← #27157998

Subgroups — #27157998

Order 12416.

234 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 54 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#20261721C211yesmaximal, characteristic
2#20261714C211yesmaximal, characteristic
3#20261637C214yesmaximal, complemented
4#20261309C211yesmaximal, characteristic
5D97⋊M16—21no—
6#15690889—21no—
7#15690871—21no—
8#15690870C2212yes—
9#15690869—21no—
10#15690868C2212yes—
11C2×#13095511C2211yescharacteristic
12#15690808—21no—
13#15690799—21no—
14#15690793C412yescomplemented
15#15690793C2212yes—
16#15690786—21no—
17#15690786—21no—
18#15690769C411yescharacteristic, complemented
19C22×Q8×C97C411yescharacteristic, complemented
20#13095521—41no—
21#13095520—22no—
22#13095519—41no—
23C4⋊#87604—41no—
24C22⋊#87604—22no—
25C4×#87604—41no—
26#13095511—41no—
27#13095511—41no—
28C2×Dic194D411yescharacteristic
29C2×D388D411yescharacteristic
30C2×C22⋊D97C2×C411yescharacteristic
31C2×C22⋊D97C2311yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C4⋊Dic97—21no—
35#13095490—22no—
36#13095490—22no—
37C22⋊Dic97C2×C412yes—
38#13095488—22no—
39#13095488—22no—
40C4×Dic97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—41no—
44C2×Q8×C97—41no—
45C22×C4×C97C2×C412yes—
46C22×C4×C97C2×C411yescharacteristic
47C23×C97C22×C411yescharacteristic
48C22×D97C22⋊C411yescharacteristic
49C22×D97C2×D411yescharacteristic
50C2×#87604—42no—
51Q8×C97—42no—
52Q8×C97—41no—
53Q8×C97—41no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—22no—
57C2×C4×C97—42no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C22⋊D97—41no—
61C22⋊D97—41no—
62C2×Dic97—22no—
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65D388—41no—
66C4×D97—21no—
67C4×D97—21no—
68C4×D97—21no—
69C4×D97—21no—
70Dic194—41no—
71#11715897—42no—
72C22×C97—21no—
73C22×C97—41no—
74C22×C97C2×C22⋊C411yescharacteristic
75C2×D97—21no—
76C2×D97—21no—
77C2×D97—41no—
78C2×D97—41no—
79#87604—82no—
80C4×C97—21no—
81C4×C97—42no—
82C4×C97#9911yescharacteristic
83C4×C97#9911yescharacteristic
84Dic97—21no—
85Dic97—21no—
86Dic97—42no—
87C2×C97—21no—
88C2×C97—41no—
89C2×C97C2×#9911yescharacteristic
90D97—41no—
91D97—41no—
92#2123—971nomaximal, Sylow
93C97#212311yescharacteristic, complemented, Sylow
94#536—971no—
95#413—971no—
96#410—971no—
97#352—974no—
98C2×#41—971no—
99C22×Q8#8760411yescharacteristic, complemented
100C2×M16—1941no—
101C4⋊Q8—1941no—
102#124—1941no—
103#123—972no—
104#123—972no—
105C22⋊Q8—1941no—
106C22⋊Q8—1941no—
107#117—971no—
108#117—1941no—
109#101—1941no—
110#100—972no—
111#99—972no—
112#99—1941no—
113#41—3881no—
114#41—3881no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—971no—
120C2×D4—971no—
121C2×D4—971no—
122C2×D4—971no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4C2×#8760412yes—
126C22×C4C2×#8760411yescharacteristic
127M16—3881no—
128M16—1942no—
129C2×C8—3881no—
130C4⋊C4—3881no—
131C4⋊C4—1942no—
132C4⋊C4—1942no—
133C4⋊C4—1941no—
134C22⋊C4—972no—
135C22⋊C4—1942no—
136C22⋊C4—1942no—
137C22⋊C4—1942no—
138C42—3881no—
139C42—1941no—
140C23—971no—
141C23—971no—
142C23C22×#8760411yescharacteristic
143Q8—42no—
144Q8—41no—
145Q8—41no—
146Q8—3881no—
147D4—3881no—
148D4—3881no—
149D4—3881no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—971no—
155C2×C4—971no—
156C2×C4—3882no—
157C2×C4—1942no—
158C2×C4—1941no—
159C2×C4—1941no—
160C2×C4—1941no—
161C2×C4—1941no—
162C2×C4C22⋊#8760411yescharacteristic
163C2×C4C22⋊#8760411yescharacteristic
164C8—3882no—
165C22—21no—
166C22—41no—
167C22—3881no—
168C22—3881no—
169C22—1941no—
170C22—1941no—
171C22C2×C22⋊#8760411yescharacteristic
172C4—21no—
173C4—42no—
174C4—7762no—
175C4—3882no—
176C4—1941no—
177C4—1941no—
178C4#1569086811yescharacteristic
179C4#1569086811yescharacteristic
180C2—21no—
181C2—41no—
182C2—3881no—
183C2—3881no—
184C2C2×#1569086811yescharacteristic
subgroupG/H
centerC2C2×#15690868
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC22×Q8×C97C4
socleC2×C97C2×#99
prime radicalC2×#13095511C22
cosocleC2×C22⋊D97C23
hypercenterC22×Q8#87604
hypocenterC97#2123
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
3. #20261637C2semidirect
14. #15690793C4semidirect
18. #15690769C4semidirect
19. C22×Q8×C97C4semidirect
93. C97#2123semidirect
99. C22×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202617142
#202616372
#202613092
#212397