Finite Group Databasebeta

← #27157258

Subgroups — #27157258

Order 12416.

208 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261736C211yesmaximal, characteristic
3#20261594C212yesmaximal, complemented
4#20261594C212yesmaximal, complemented
5#20261291C211yesmaximal, characteristic
6C2×#13095519—21no—
7#15690877C2212yes—
8#15690877C2212yes—
9C4⋊#11715897—21no—
10#15690867—21no—
11#15690865—21no—
12#15690809C2211yescharacteristic
13#15690809C2211yescharacteristic
14#15690802C411yescharacteristic
15#15690788—21no—
16#15690788—21no—
17#15690775—21no—
18#15690775—21no—
19#15690768C411yescharacteristic
20#15690768C411yescharacteristic
21C42×D97C2211yescharacteristic
22#125×C97C411yescharacteristic
23#13095519—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C4⋊Dic97C2×C411yescharacteristic
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095488—21no—
38#13095488—21no—
39C4×Dic97—21no—
40C4×Dic97—21no—
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97C2311yescharacteristic
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C4⋊C4×C97C2×C411yescharacteristic
48C42×C97C2×C411yescharacteristic
49C22×D97C22⋊C411yescharacteristic
50C2×C4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97C22⋊C411yescharacteristic
53C2×C4×C97C22⋊C411yescharacteristic
54C2×C4×C97C22×C411yescharacteristic
55C2×Dic97—21no—
56C2×Dic97—21no—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C2×D411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60C4×D97—22no—
61C4×D97—22no—
62C4×D97—21no—
63C4×D97—21no—
64Dic194—42no—
65Dic194—41no—
66Dic194—41no—
67#11715897—42no—
68#11715897—42no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—21no—
71C2×D97—21no—
72C2×D97#10112yes—
73C4×C97—21no—
74C4×C97—21no—
75C4×C97—21no—
76C4×C97—41no—
77C4×C97—41no—
78Dic97—21no—
79Dic97—41no—
80Dic97—41no—
81Dic97#13512yes—
82Dic97AGL(1,ℤ/8)11yescharacteristic
83Dic97#13711yescharacteristic
84C2×C97C2×#10111yescharacteristic
85C2×C97#41611yescharacteristic
86C2×C97#41711yescharacteristic
87D97—22no—
88#1552—971nomaximal, Sylow
89C97#155211yescharacteristic, complemented, Sylow
90#533—971no—
91#432—971no—
92C2×C4⋊C8—971no—
93#332—972no—
94#332—972no—
95C22×C8—1941no—
96C4⋊Q8—971no—
97C4⋊Q8—971no—
98#125—971no—
99#125#8760411yescharacteristic
100C22⋊Q8—1941no—
101C22⋊Q8—1941no—
102C4×Q8—1941no—
103C4×Q8—1941no—
104#117—971no—
105#117—971no—
106C2×C42—971no—
107C4⋊C8—972no—
108C4⋊C8—972no—
109C4⋊C8—1941no—
110C22⋊C8—1941no—
111#97—1941no—
112C2×Q8—1941no—
113C2×Q8—1941no—
114C22×C4—971no—
115C22×C4—971no—
116C22×C4—971no—
117C2×C8—3882no—
118C2×C8—1942no—
119C2×C8—1942no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—971no—
123C4⋊C4—971no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4C2×#8760411yescharacteristic
129C4⋊C4C2×#8760411yescharacteristic
130C22⋊C4—1941no—
131C22⋊C4—1941no—
132C42—971no—
133C42—971no—
134C42—971no—
135C42—1941no—
136C42—1941no—
137C42C2×#8760411yescharacteristic
138C23—971no—
139Q8—3882no—
140Q8—3881no—
141Q8—3881no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—971no—
145C2×C4—971no—
146C2×C4—971no—
147C2×C4—1941no—
148C2×C4—1942no—
149C2×C4—1941no—
150C2×C4—1942no—
151C2×C4—1941no—
152C2×C4—1941no—
153C2×C4C22⋊#8760411yescharacteristic
154C2×C4C22⋊#8760411yescharacteristic
155C2×C4C22×#8760411yescharacteristic
156C8—3882no—
157C8—3882no—
158C22—972no—
159C22—1941no—
160C22—1941no—
161C22C2×C22⋊#8760411yescharacteristic
162C4—21no—
163C4—21no—
164C4—21no—
165C4—41no—
166C4—41no—
167C4—972no—
168C4—971no—
169C4—971no—
170C4—3881no—
171C4—3881no—
172C4—1941no—
173C2—1942no—
174C2D97⋊#10111yescharacteristic
175C2#2026172511yescharacteristic
176C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenter#125#87604
hypocenterC97#1552
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
3. #20261594C2semidirect
4. #20261594C2semidirect
89. C97#1552semidirect

Maximal subgroups

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

subgroupindex
#202617582
#202617362
#202615942
#202612912
#155297