Finite Group Databasebeta

← #27157158

Subgroups — #27157158

Order 12416.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#20261757C211yesmaximal, characteristic
2#20261680C211yesmaximal, characteristic
3#20261589C212yesmaximal
4#20261588C212yesmaximal
5#20261289C211yesmaximal, characteristic
6C4⋊#11715897—21no—
7#15690867C2212yes—
8#15690865—21no—
9(C2×#87604)⋊C4—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—
25C2×Dic194—21no—
26C2×Dic194C2×C411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97C2311yescharacteristic
30C4⋊Dic97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095488—21no—
38#13095488—21no—
39C4×Dic97—21no—
40C4×Dic97C2×C411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C2×Q8×C97C2×C411yescharacteristic
44C4⋊C4×C97—21no—
45C4⋊C4×C97C2×C411yescharacteristic
46C42×C97—21no—
47C42×C97C2×C411yescharacteristic
48C22×D97C2×D411yescharacteristic
49C2×#87604—42no—
50C2×#87604—42no—
51C2×#87604—42no—
52Q8×C97—41no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97C22⋊C411yescharacteristic
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63C4×D97—22no—
64C4×D97—41no—
65C4×D97—41no—
66Dic194—42no—
67Dic194—41no—
68#11715897—42no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—21no—
71C2×D97—21no—
72C2×D97#13712yes—
73#87604—82no—
74C4×C97—21no—
75C4×C97—21no—
76C4×C97—21no—
77C4×C97—41no—
78C4×C97—41no—
79Dic97—21no—
80Dic97—21no—
81Dic97—21no—
82Dic97—41no—
83Dic97—41no—
84C2×C97#41011yescharacteristic
85C2×C97#41911yescharacteristic
86C2×C97#42111yescharacteristic
87D97—22no—
88#1515—971nomaximal, Sylow
89C97#151511yescharacteristic, complemented, Sylow
90#533—971no—
91#432—971no—
92#383—971no—
93#328—972no—
94#328—972no—
95C4⋊Q8—971no—
96C4⋊Q8—971no—
97#125—1941no—
98C22⋊Q8—971no—
99C22⋊Q8—971no—
100C22⋊Q8—1941no—
101C4×Q8—971no—
102C4×Q8—1941no—
103C4×Q8#8760411yescharacteristic, complemented
104#117—1941no—
105C2×C42—971no—
106C4⋊C8—1941no—
107C22⋊C8—972no—
108#97—1941no—
109#95—972no—
110#95—1941no—
111#95—1941no—
112C2×Q8—971no—
113C2×Q8—1941no—
114C2×Q8C2×#8760411yescharacteristic
115C22×C4—971no—
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—1942no—
119C2×C8—1942no—
120C4⋊C4—21no—
121C4⋊C4—971no—
122C4⋊C4—1941no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4C2×#8760411yescharacteristic
129C22⋊C4—1941no—
130C22⋊C4—1941no—
131C42—21no—
132C42—971no—
133C42—971no—
134C42—971no—
135C42—1941no—
136C42C2×#8760411yescharacteristic
137C23—971no—
138Q8—41no—
139Q8—3881no—
140Q8—3882no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—971no—
144C2×C4—971no—
145C2×C4—971no—
146C2×C4—3881no—
147C2×C4—3881no—
148C2×C4—3882no—
149C2×C4—3882no—
150C2×C4—3882no—
151C2×C4—1942no—
152C2×C4—1941no—
153C2×C4—1941no—
154C2×C4C22⋊#8760411yescharacteristic
155C2×C4C22⋊#8760411yescharacteristic
156C2×C4C22×#8760411yescharacteristic
157C8—3882no—
158C22—972no—
159C22—1941no—
160C22—1941no—
161C22C2×C22⋊#8760411yescharacteristic
162C4—21no—
163C4—21no—
164C4—21no—
165C4—41no—
166C4—41no—
167C4—7762no—
168C4—3881no—
169C4—3881no—
170C4—1941no—
171C4—1941no—
172C4—1941no—
173C2—1942no—
174C2#2026171411yescharacteristic
175C2#2026172811yescharacteristic
176C2#2026173211yescharacteristic
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#1515
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
89. C97#1515semidirect
103. C4×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616802
#202615892
#202615882
#202612892
#151597