Finite Group Databasebeta

← #27157205

Subgroups — #27157205

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
2C2×#15690865C211yesmaximal, characteristic, complemented
3#20261591C212yesmaximal, complemented
4#20261590C212yesmaximal, complemented
5#20261281C211yesmaximal, characteristic
6C22×#11715897—21no—
7#15690877—21no—
8C22⋊#11715897—21no—
9#15690865—21no—
10#15690865C2212yes—
11#15690865C2212yes—
12C2×#13095511C411yescharacteristic
13#15690808C411yescharacteristic
14#15690806C411yescharacteristic
15#15690799C2211yescharacteristic
16#15690797C2211yescharacteristic
17#15690782—21no—
18#15690782—21no—
19C4×C22⋊D97—21no—
20C4×C22⋊D97—21no—
21C2×C4×Dic97C2211yescharacteristic
22C2×#41×C97C411yescharacteristic
23C2×#11715897—24no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26#13095511—42no—
27C2×Dic194C2×C411yescharacteristic
28C2×D388C2×C411yescharacteristic
29C2×C22⋊D97—21no—
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C22×Dic97D411yescharacteristic
33C22×Dic97D411yescharacteristic
34C2×C4×D97—21no—
35#13095490—21no—
36#13095490—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45#41×C97—42no—
46C2×Q8×C97C2×C411yescharacteristic
47C2×D4×C97—21no—
48C2×D4×C97C2×C411yescharacteristic
49C22×C4×C97—21no—
50C22×C4×C97C2×C411yescharacteristic
51C23×C97—21no—
52C23×C97C22⋊C411yescharacteristic
53C22×D97—21no—
54Q8×C97—41no—
55D4×C97—41no—
56D4×C97—42no—
57C2×C4×C97—22no—
58C2×C4×C97—21no—
59C2×C4×C97—42no—
60C2×C4×C97C22⋊C411yescharacteristic
61C2×C4×C97C22×C411yescharacteristic
62C22⋊D97—42no—
63C22⋊D97—42no—
64C22⋊D97—42no—
65C2×Dic97—22no—
66C2×Dic97—21no—
67C2×Dic97—41no—
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C2×D411yescharacteristic
71C2×Dic97C2×D411yescharacteristic
72D388—41no—
73C4×D97—42no—
74Dic194—41no—
75#11715897—24no—
76#11715897—42no—
77C22×C97—21no—
78C22×C97—21no—
79C22×C97—41no—
80C22×C97—41no—
81C22×C97—41no—
82C22×C97#10012yes—
83C22×C97C2×C22⋊C411yescharacteristic
84C2×D97—41no—
85C2×D97—41no—
86C2×D97—41no—
87C4×C97—21no—
88C4×C97—21no—
89C4×C97—41no—
90Dic97—21no—
91Dic97—21no—
92Dic97—21no—
93Dic97—41no—
94Dic97#10412yes—
95C2×C97—22no—
96C2×C97—81no—
97C2×C97C2×#10011yescharacteristic
98C2×C97C2×#10411yescharacteristic
99C2×C97#42111yescharacteristic
100D97—81no—
101#1530—971nomaximal, Sylow
102C97#153011yescharacteristic, complemented, Sylow
103#532—971no—
104#432—971no—
105C2×#97—971no—
106#329—972no—
107#329—972no—
108C2×#41—971no—
109C2×#41#8760411yescharacteristic
110C22×C8—1941no—
111C4⋊Q8—971no—
112#127—971no—
113#124—971no—
114#124—971no—
115#121—1941no—
116#121—1941no—
117C4×D4—1941no—
118C4×D4—1941no—
119C2×C42—971no—
120C4⋊C8—1941no—
121C22⋊C8—1941no—
122#97—972no—
123#97—972no—
124#97—1941no—
125#41—42no—
126#41—3882no—
127C2×Q8—971no—
128C2×Q8C2×#8760411yescharacteristic
129C2×D4—21no—
130C2×D4—971no—
131C2×D4—1941no—
132C2×D4—1941no—
133C2×D4—1941no—
134C2×D4C2×#8760411yescharacteristic
135C22×C4—21no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—1941no—
139C22×C4C2×#8760411yescharacteristic
140C2×C8—1942no—
141C2×C8—1942no—
142C2×C8—1944no—
143C4⋊C4—1941no—
144C4⋊C4—1941no—
145C22⋊C4—1941no—
146C22⋊C4—1941no—
147C22⋊C4—1941no—
148C22⋊C4—1941no—
149C42—971no—
150C42—971no—
151C42—971no—
152C42—971no—
153C23—21no—
154C23—1941no—
155C23C22⋊#8760411yescharacteristic
156Q8—41no—
157Q8—3881no—
158D4—41no—
159D4—42no—
160D4—3881no—
161D4—3882no—
162D4—3882no—
163D4—3882no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—42no—
167C2×C4—971no—
168C2×C4—971no—
169C2×C4—971no—
170C2×C4—971no—
171C2×C4—3882no—
172C2×C4—3881no—
173C2×C4—1942no—
174C2×C4—1941no—
175C2×C4C22⋊#8760411yescharacteristic
176C2×C4C22×#8760411yescharacteristic
177C8—3882no—
178C8—1944no—
179C22—21no—
180C22—21no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—3881no—
185C22—3881no—
186C22—3881no—
187C22#1569087012yes—
188C22C2×C22⋊#8760411yescharacteristic
189C4—21no—
190C4—21no—
191C4—41no—
192C4—972no—
193C4—3881no—
194C4—1941no—
195C4—1941no—
196C4—1941no—
197C2—22no—
198C2—81no—
199C2—7761no—
200C2C2×#1569087011yescharacteristic
201C2D97⋊#10411yescharacteristic
202C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×#41×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261281C2
cosocleC4×Dic97C23
hypercenterC2×#41#87604
hypocenterC97#1530
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #20261755C2semidirect
2. C2×#15690865C2semidirect
3. #20261591C2semidirect
4. #20261590C2semidirect
102. C97#1530semidirect

Maximal subgroups

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

subgroupindex
#202617552
C2×#156908652
#202615912
#202615902
#202612812
#153097