Finite Group Databasebeta

← #27157005

Subgroups — #27157005

Order 12416.

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

normal subgroups only

This group has 256 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#20261755C211yesmaximal, characteristic, complemented
2#20261751C211yesmaximal, characteristic, complemented
3#20261579C212yesmaximal, complemented
4#20261578C212yesmaximal, complemented
5C2×#15690796C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897—21no—
8C22⋊#11715897C2212yes—
9C22⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897—21no—
12C22×Dic194C411yescharacteristic
13C22×C22⋊D97C411yescharacteristic
14#15690796—21no—
15#15690796—21no—
16#15690796—21no—
17#15690796—21no—
18C2×C22⋊Dic97C2211yescharacteristic
19C2×#13095488C411yescharacteristic
20C2×#13095488C2211yescharacteristic
21C2×C4×Dic97C2211yescharacteristic
22C2×C22⋊C4×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C23×D97C2×C411yescharacteristic
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—42no—
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C2×C22⋊D97—42no—
32C22×Dic97C2×C411yescharacteristic
33C22×Dic97C2×C411yescharacteristic
34C22×Dic97C2311yescharacteristic
35C22⋊Dic97—21no—
36C22⋊Dic97—21no—
37#13095488—21no—
38#13095488—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C24×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C22×C4×C97C2×C411yescharacteristic
48C22⋊C4×C97—21no—
49C22⋊C4×C97—21no—
50C23×C97—21no—
51C23×C97—22no—
52C23×C97—41no—
53C23×C97—41no—
54C23×C97C22×C411yescharacteristic
55C22×D97—21no—
56C22×D97—22no—
57C22×D97—41no—
58C22×D97—41no—
59C2×C4×C97—22no—
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22⋊C411yescharacteristic
64C22⋊D97—42no—
65C22⋊D97—42no—
66C2×Dic97—22no—
67C2×Dic97—21no—
68C2×Dic97—41no—
69C2×Dic97—41no—
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73C2×Dic97C2×D411yescharacteristic
74Dic194—42no—
75Dic194—42no—
76#11715897—42no—
77#11715897—42no—
78C22×C97—21no—
79C22×C97—21no—
80C22×C97—41no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—82no—
84C22×C97C2×C22⋊C411yescharacteristic
85C22×C97#13112yes—
86C2×D97—41no—
87C2×D97—41no—
88C2×D97—41no—
89C2×D97—82no—
90C4×C97—21no—
91C4×C97—21no—
92C4×C97—41no—
93Dic97—21no—
94Dic97—21no—
95Dic97—21no—
96Dic97—41no—
97Dic97#10012yes—
98C2×C97—22no—
99C2×C97—81no—
100C2×C97#40811yescharacteristic
101C2×C97#41011yescharacteristic
102C2×C97C2×#10011yescharacteristic
103D97—81no—
104#1469—971nomaximal, Sylow
105C97#146911yescharacteristic, complemented, Sylow
106C2×#124—971no—
107#432—971no—
108#431—971no—
109#323—972no—
110#323—972no—
111C22×Q8—971no—
112C22×D4—971no—
113#124—1941no—
114#124—1941no—
115#124—1941no—
116#124—1941no—
117C2×C22⋊C4—971no—
118C2×C22⋊C4—971no—
119C2×C22⋊C4—971no—
120C2×C22⋊C4#8760411yescharacteristic
121C2×C42—971no—
122C4⋊C8—1941no—
123C4⋊C8—1941no—
124C22⋊C8—972no—
125C22⋊C8—972no—
126#97—1941no—
127C4×C8—1941no—
128C24—971no—
129C24C2×#8760411yescharacteristic
130C2×Q8—3882no—
131C2×Q8—1941no—
132C2×Q8—1941no—
133C2×D4—3882no—
134C2×D4—1941no—
135C2×D4—1941no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4C2×#8760411yescharacteristic
140C22×C4C2×#8760411yescharacteristic
141C2×C8—1942no—
142C2×C8—1942no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—1941no—
146C22⋊C4—1941no—
147C22⋊C4—1941no—
148C22⋊C4—1941no—
149C22⋊C4—1941no—
150C22⋊C4—1941no—
151C42—971no—
152C42—971no—
153C42—971no—
154C42—971no—
155C23—21no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—3881no—
160C23—3881no—
161C23—1942no—
162C23—1941no—
163C23C22×#8760411yescharacteristic
164Q8—3882no—
165Q8—3882no—
166D4—3882no—
167D4—3882no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—3881no—
176C2×C4—3881no—
177C2×C4—1942no—
178C2×C4—1941no—
179C2×C4C22⋊#8760411yescharacteristic
180C2×C4C22⋊#8760411yescharacteristic
181C8—3882no—
182C8—3882no—
183C22—21no—
184C22—21no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—82no—
189C22—7762no—
190C22—3881no—
191C22—3881no—
192C22—3881no—
193C22C2×C22⋊#8760411yescharacteristic
194C22#1569089712yes—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—972no—
199C4—3881no—
200C4—1941no—
201C4—1941no—
202C4—1941no—
203C2—22no—
204C2—81no—
205C2—7761no—
206C2#2026170811yescharacteristic
207C2#2026171411yescharacteristic
208C2D97⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×C22⋊D97C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1469
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261755C2semidirect
2. #20261751C2semidirect
3. #20261579C2semidirect
4. #20261578C2semidirect
105. C97#1469semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202617512
#202615792
#202615782
C2×#156907962
#146997