Finite Group Databasebeta

← #27157135

Subgroups — #27157135

Order 12416.

294 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 64 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#20261706C211yesmaximal, characteristic
2C4×C22⋊#87604C211yesmaximal, characteristic
3#20261588C214yesmaximal, complemented
4C2×#15690790C211yesmaximal, characteristic
5C2×#13095521—21no—
6C2×C22⋊#87604—21no—
7C2×C4×#87604—21no—
8#15690867C2212yes—
9C22⋊#11715897—21no—
10(C2×#87604)⋊C4C2212yes—
11C22×Dic194C411yescharacteristic, complemented
12C22×C4×D97C2211yescharacteristic
13#15690790—21no—
14#15690790—21no—
15#15690790C412yescomplemented
16#15690790C2212yes—
17C2×C4⋊Dic97—21no—
18C2×#13095488—21no—
19C2×C4⋊C4×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21#13095521—42no—
22C2×#11715897—22no—
23C22⋊#87604—24no—
24C4×#87604—42no—
25C23×D97D411yescharacteristic
26C2×Dic194—21no—
27C2×Dic194—42no—
28C2×Dic194C2×C412yes—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C2×C4×D97—42no—
32C2×C4×D97C2×C411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C2×C4×D97C2311yescharacteristic
36C4⋊Dic97—22no—
37C4⋊Dic97—22no—
38#13095488—22no—
39#13095488—22no—
40C22×C4×C97—21no—
41C22×C4×C97C2×C411yescharacteristic
42C4⋊C4×C97—21no—
43C4⋊C4×C97C2×C412yes—
44C23×C97C22⋊C411yescharacteristic
45C22×D97—22no—
46C22×D97—22no—
47C22×D97—41no—
48C22×D97C22⋊C411yescharacteristic
49C22×D97C2×D411yescharacteristic
50C2×#87604—24no—
51C2×#87604—42no—
52C2×#87604—42no—
53C2×C4×C97—22no—
54C2×C4×C97—22no—
55C2×C4×C97—42no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22×C411yescharacteristic
58C2×Dic97—22no—
59C2×Dic97—41no—
60C2×Dic97—42no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C2×D411yescharacteristic
63C4×D97—22no—
64C4×D97—22no—
65C4×D97—41no—
66C4×D97—41no—
67Dic194—42no—
68Dic194—42no—
69#11715897—42no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97#10412yes—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—21no—
75C2×D97—21no—
76C2×D97—21no—
77C2×D97—21no—
78C2×D97—21no—
79C2×D97—44no—
80C2×D97#13712yes—
81#87604—44no—
82C4×C97—21no—
83C4×C97—42no—
84C4×C97#9912yes—
85Dic97—21no—
86Dic97—21no—
87Dic97—42no—
88C2×C97—22no—
89C2×C97C2×#9911yescharacteristic
90C2×C97#41911yescharacteristic
91C2×C97C2×#10411yescharacteristic
92D97—22no—
93D97—41no—
94#1508—971nomaximal, Sylow
95C97#150811yescharacteristic, complemented, Sylow
96C2×C22⋊Q8—971no—
97#407—971no—
98C4×C22⋊C4—971no—
99#328—974no—
100C22×Q8—971no—
101C23×C4—971no—
102C2×M16—1941no—
103C22⋊Q8—972no—
104C22⋊Q8—972no—
105C22⋊Q8—1941no—
106C22⋊Q8—1941no—
107C2×C4⋊C4—1941no—
108C2×C4⋊C4#8760411yescharacteristic, complemented
109C2×C22⋊C4—1941no—
110C2×C22⋊C4—1941no—
111C2×C42—1941no—
112C22⋊C8—972no—
113C22⋊C8—1941no—
114#95—972no—
115C24—971no—
116C2×Q8—972no—
117C2×Q8—3882no—
118C2×Q8—1941no—
119C22×C4—21no—
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—971no—
123C22×C4—971no—
124C22×C4—971no—
125C22×C4—3882no—
126C22×C4—1942no—
127C22×C4—1941no—
128C22×C4C2×#8760411yescharacteristic
129M16—3882no—
130C2×C8—1942no—
131C4⋊C4—21no—
132C4⋊C4—1942no—
133C4⋊C4—1942no—
134C4⋊C4C2×#8760412yes—
135C22⋊C4—1942no—
136C22⋊C4—1942no—
137C22⋊C4—1944no—
138C42—3882no—
139C23—971no—
140C23—971no—
141C23—3881no—
142C23—1942no—
143C23—1942no—
144C23C22⋊#8760411yescharacteristic
145Q8—3882no—
146Q8—3882no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—42no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4—3882no—
155C2×C4—3882no—
156C2×C4—3882no—
157C2×C4—3881no—
158C2×C4—1944no—
159C2×C4—1942no—
160C2×C4—1942no—
161C2×C4—1942no—
162C2×C4C22⋊#8760411yescharacteristic
163C2×C4C22×#8760411yescharacteristic
164C8—3882no—
165C22—21no—
166C22—21no—
167C22—972no—
168C22—3884no—
169C22—1941no—
170C22—1941no—
171C22—1941no—
172C22—1941no—
173C22—1941no—
174C22#1569087412yes—
175C22C2×C22⋊#8760411yescharacteristic
176C4—21no—
177C4—42no—
178C4—3884no—
179C4—3882no—
180C4—1941no—
181C4—1941no—
182C4#1569086912yes—
183C2—22no—
184C2—3881no—
185C2—1942no—
186C2C2×#1569086911yescharacteristic
187C2#2026172811yescharacteristic
188C2C2×#1569087411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97D4
cosocleC2×C4×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1508
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261588C2semidirect
11. C22×Dic194C4semidirect
15. #15690790C4semidirect
19. C2×C4⋊C4×C97C4semidirect
95. C97#1508semidirect
108. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617062
C4×C22⋊#876042
#202615882
C2×#156907902
#150897