Finite Group Databasebeta

← #27157234

Subgroups — #27157234

Order 12416.

256 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261752C211yesmaximal, characteristic, complemented
2C2×#15690877C211yesmaximal, characteristic, complemented
3#20261594C212yesmaximal, complemented
4#20261593C212yesmaximal, complemented
5#20261280C211yesmaximal, characteristic
6C22×#11715897—21no—
7#15690877—21no—
8#15690877C2212yes—
9#15690877C2212yes—
10C22⋊#11715897—21no—
11C4×#11715897—21no—
12C2×#13095510—21no—
13#15690809C2211yescharacteristic
14#15690806C2211yescharacteristic
15#15690796—21no—
16#15690784C411yescharacteristic
17#15690782—21no—
18D4×Dic97C411yescharacteristic
19#15690772C411yescharacteristic
20#15690770—21no—
21C2×C4×Dic97C2211yescharacteristic
22#121×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C2×#11715897—42no—
26#13095510—42no—
27#13095510—42no—
28C2×Dic194—21no—
29C2×D388C2×C411yescharacteristic
30C2×C22⋊D97—21no—
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C22×Dic97—21no—
34C22×Dic97D411yescharacteristic
35C22×Dic97D411yescharacteristic
36C2×C4×D97—21no—
37C4⋊Dic97C2×C411yescharacteristic
38#13095490—21no—
39C22⋊Dic97—21no—
40#13095488—21no—
41#13095488—21no—
42C4×Dic97C2×C411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97C2311yescharacteristic
46C2×D4×C97—21no—
47C2×D4×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C4⋊C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—21no—
51C23×C97—21no—
52C23×C97C22⋊C411yescharacteristic
53C22×D97—21no—
54D4×C97—41no—
55D4×C97—42no—
56C2×C4×C97—21no—
57C2×C4×C97—41no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22×C411yescharacteristic
60C22⋊D97—42no—
61C22⋊D97—41no—
62C22⋊D97—41no—
63C22⋊D97—42no—
64C2×Dic97—22no—
65C2×Dic97—21no—
66C2×Dic97—42no—
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97D812yes—
70C2×Dic97SD1612yes—
71C2×Dic97C2×D411yescharacteristic
72C2×Dic97C2×D411yescharacteristic
73D388—41no—
74C4×D97—42no—
75Dic194—41no—
76Dic194—41no—
77#11715897—42no—
78#11715897—42no—
79C22×C97—21no—
80C22×C97—21no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97#10212yes—
85C22×C97C2×C22⋊C411yescharacteristic
86C2×D97—41no—
87C2×D97—41no—
88C2×D97—41no—
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—41no—
92Dic97—21no—
93Dic97—41no—
94Dic97#10212yes—
95Dic97C2×D811yescharacteristic
96Dic97C2×SD1611yescharacteristic
97Dic97#13512yes—
98C2×C97—22no—
99C2×C97—81no—
100C2×C97#41311yescharacteristic
101C2×C97C2×#10211yescharacteristic
102C2×C97#41611yescharacteristic
103D97—81no—
104#1540—971nomaximal, Sylow
105C97#154011yescharacteristic, complemented, Sylow
106#532—971no—
107#431—971no—
108C2×C4⋊C8—971no—
109#332—972no—
110#331—972no—
111C2×#41—1941no—
112C22×C8—1941no—
113C4⋊Q8—971no—
114#127—971no—
115#124—1941no—
116#121—971no—
117#121—1941no—
118#121#8760411yescharacteristic
119C4×D4—971no—
120C4×D4—971no—
121C4×D4—1941no—
122C2×C42—971no—
123C4⋊C8—972no—
124C4⋊C8—972no—
125C4⋊C8—1941no—
126C22⋊C8—1941no—
127C4×C8—1941no—
128#41—3882no—
129#41—3882no—
130C2×Q8—1941no—
131C2×D4—21no—
132C2×D4—971no—
133C2×D4—1941no—
134C2×D4—1941no—
135C2×D4—1941no—
136C2×D4C2×#8760411yescharacteristic
137C22×C4—971no—
138C22×C4—971no—
139C22×C4—1941no—
140C22×C4—1941no—
141C22×C4C2×#8760411yescharacteristic
142C2×C8—3882no—
143C2×C8—1942no—
144C2×C8—1942no—
145C4⋊C4—971no—
146C4⋊C4—1941no—
147C4⋊C4C2×#8760411yescharacteristic
148C22⋊C4—21no—
149C22⋊C4—1941no—
150C22⋊C4—1941no—
151C22⋊C4—1941no—
152C42—971no—
153C42—971no—
154C42—971no—
155C42—971no—
156C23—21no—
157C23—1941no—
158C23C22⋊#8760411yescharacteristic
159Q8—3881no—
160Q8—3881no—
161D4—41no—
162D4—42no—
163D4—3881no—
164D4—3881no—
165D4—3881no—
166D4—3882no—
167D4—3882no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—972no—
171C2×C4—972no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—971no—
175C2×C4—971no—
176C2×C4—3882no—
177C2×C4—3882no—
178C2×C4—1942no—
179C2×C4—1941no—
180C2×C4C22⋊#8760411yescharacteristic
181C2×C4C22×#8760411yescharacteristic
182C8—3882no—
183C8—3882no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—3881no—
190C22—3881no—
191C22—3881no—
192C22#1569087212yes—
193C22C2×C22⋊#8760411yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—971no—
198C4—972no—
199C4—971no—
200C4—972no—
201C4—3881no—
202C4—1941no—
203C2—22no—
204C2—81no—
205C2—7761no—
206C2#2026172211yescharacteristic
207C2C2×#1569087211yescharacteristic
208C2#2026172511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#121×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261280C2
cosocleC4×Dic97C23
hypercenter#121#87604
hypocenterC97#1540
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20261752C2semidirect
2. C2×#15690877C2semidirect
3. #20261594C2semidirect
4. #20261593C2semidirect
105. C97#1540semidirect

Maximal subgroups

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

subgroupindex
#202617522
C2×#156908772
#202615942
#202615932
#202612802
#154097