Finite Group Databasebeta

← #27157129

Subgroups — #27157129

Order 12416.

228 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 58 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#20261758C211yesmaximal, characteristic
2#20261671C211yesmaximal, characteristic
3#20261589C212yesmaximal
4#20261588C212yesmaximal
5#20261289C211yesmaximal, characteristic
6C2×C4×#87604—21no—
7#15690877—21no—
8#15690867C2212yes—
9#15690865—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12#15690809C411yescharacteristic, complemented
13#15690808C411yescharacteristic, complemented
14#15690800—21no—
15#15690790C2211yescharacteristic
16#15690788—21no—
17#15690786C2211yescharacteristic
18C4×Dic194—21no—
19C4×Dic194C411yescharacteristic, complemented
20#15690767—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C4×#87604—24no—
26C2×Dic194—21no—
27C2×Dic194C2×C411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97C2311yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C2×Q8×C97C2×C411yescharacteristic
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C42×C97—21no—
48C42×C97C2×C411yescharacteristic
49C22×D97C2×D411yescharacteristic
50C2×#87604—24no—
51C2×#87604—42no—
52C2×#87604—42no—
53Q8×C97—41no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C4×D97—22no—
65C4×D97—22no—
66C4×D97—41no—
67Dic194—42no—
68Dic194—41no—
69#11715897—42no—
70C22×C97C2×C22⋊C411yescharacteristic
71C2×D97—21no—
72C2×D97—21no—
73C2×D97#13712yes—
74#87604—44no—
75C4×C97—21no—
76C4×C97—21no—
77C4×C97—21no—
78C4×C97—41no—
79C4×C97—41no—
80Dic97—21no—
81Dic97—21no—
82Dic97—41no—
83Dic97—41no—
84Dic97#10412yes—
85C2×C97#41011yescharacteristic
86C2×C97#41911yescharacteristic
87C2×C97C2×#10411yescharacteristic
88D97—22no—
89#1506—971nomaximal, Sylow
90C97#150611yescharacteristic, complemented, Sylow
91#533—971no—
92#432—971no—
93#376—971no—
94#328—972no—
95#328—972no—
96C4⋊Q8—971no—
97C4⋊Q8—971no—
98#125—1941no—
99C22⋊Q8—971no—
100C22⋊Q8—971no—
101C22⋊Q8—1941no—
102C4×Q8—971no—
103C4×Q8—1941no—
104C4×Q8#8760411yescharacteristic, complemented
105#117—1941no—
106C2×C42—971no—
107C2×C42—1941no—
108C4⋊C8—1941no—
109C22⋊C8—972no—
110#97—1941no—
111#95—972no—
112#95—1941no—
113C2×Q8—971no—
114C2×Q8—1941no—
115C2×Q8C2×#8760411yescharacteristic
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—1942no—
120C2×C8—1942no—
121C4⋊C4—21no—
122C4⋊C4—971no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4C2×#8760411yescharacteristic
130C22⋊C4—1941no—
131C22⋊C4—1941no—
132C42—21no—
133C42—971no—
134C42—971no—
135C42—971no—
136C42—1944no—
137C42—1941no—
138C42C2×#8760411yescharacteristic
139C23—971no—
140Q8—41no—
141Q8—3881no—
142Q8—3882no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—971no—
146C2×C4—971no—
147C2×C4—971no—
148C2×C4—3881no—
149C2×C4—3882no—
150C2×C4—3882no—
151C2×C4—1942no—
152C2×C4—1942no—
153C2×C4—1944no—
154C2×C4—1941no—
155C2×C4—1941no—
156C2×C4C22⋊#8760411yescharacteristic
157C2×C4C22⋊#8760411yescharacteristic
158C2×C4C22×#8760411yescharacteristic
159C8—3882no—
160C22—972no—
161C22—1941no—
162C22—1941no—
163C22C2×C22⋊#8760411yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—972no—
170C4—3884no—
171C4—3881no—
172C4—3881no—
173C4—1941no—
174C4—1941no—
175C2—1942no—
176C2#2026171411yescharacteristic
177C2#2026172811yescharacteristic
178C2D97⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4×Q8#87604
hypocenterC97#1506
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
12. #15690809C4semidirect
13. #15690808C4semidirect
19. C4×Dic194C4semidirect
22. C4×Q8×C97C4semidirect
90. C97#1506semidirect
104. C4×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202616712
#202615892
#202615882
#202612892
#150697