Finite Group Databasebeta

← #27156821

Subgroups — #27156821

Order 12416.

234 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 44 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
1C2×#15690870C211yesmaximal, characteristic
2C2×#15690868C211yesmaximal, characteristic
3#20260612C211yesmaximal, characteristic
4C2×#13095520—21no—
5C2×C22⋊#87604—21no—
6#15690870—21no—
7#15690870C412yescomplemented
8#15690868—21no—
9#15690868C412yescomplemented
10C22×C22⋊D97C2211yescharacteristic
11C23×Dic97C411yescharacteristic, complemented
12C2×#13095488—21no—
13#15690717—21no—
14C2×C22⋊C4×C97C411yescharacteristic, complemented
15C22×#87604—41no—
16#13095520—22no—
17#13095520—41no—
18C2×#11715897—41no—
19C22⋊#87604—22no—
20C22⋊#87604—41no—
21C23×D97D411yescharacteristic
22C2×C22⋊D97—41no—
23C2×C22⋊D97—41no—
24C2×C22⋊D97C2×C411yescharacteristic
25C2×C22⋊D97C2×C411yescharacteristic
26C2×C22⋊D97D411yescharacteristic
27C2×C22⋊D97Q811yescharacteristic
28C22×Dic97—21no—
29C22×Dic97—42no—
30C22×Dic97—41no—
31C22×Dic97D411yescharacteristic
32#13095488—42no—
33C24×C97C2×C411yescharacteristic
34C22×C4×C97—21no—
35C22⋊C4×C97—41no—
36C23×C97—21no—
37C23×C97—21no—
38C23×C97—41no—
39C23×C97—41no—
40C23×C97C4211yescharacteristic
41C23×C97C22⋊C411yescharacteristic
42C23×C97C4⋊C411yescharacteristic
43C22×D97—21no—
44C22×D97—21no—
45C22×D97—41no—
46C22×D97—41no—
47C22×D97C22⋊C411yescharacteristic
48C22×D97C4⋊C411yescharacteristic
49C2×#87604—42no—
50C2×#87604—81no—
51C2×#87604—81no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C22⋊D97—41no—
56C22⋊D97—41no—
57C22⋊D97—41no—
58C22⋊D97—41no—
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97—22no—
62C2×Dic97—41no—
63C2×Dic97—42no—
64C2×Dic97—41no—
65C2×Dic97—42no—
66C2×Dic97C22⋊C411yescharacteristic
67C2×Dic97C4⋊C411yescharacteristic
68#11715897—42no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97—21no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—42no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97#9511yescharacteristic
79C22×C97#9911yescharacteristic
80C22×C97#9911yescharacteristic
81C2×D97—21no—
82C2×D97—21no—
83C2×D97—41no—
84C2×D97—41no—
85C2×D97—41no—
86C2×D97—41no—
87C2×D97—81no—
88#87604—82no—
89C4×C97—81no—
90Dic97—21no—
91Dic97—21no—
92Dic97—42no—
93C2×C97—21no—
94C2×C97—21no—
95C2×C97—41no—
96C2×C97—41no—
97C2×C97#34211yescharacteristic
98C2×C97AΣL(1,𝔽16)11yescharacteristic
99C2×C97#35211yescharacteristic
100D97—41no—
101D97—41no—
102#1390—971nomaximal, Sylow
103C97#139011yescharacteristic, complemented, Sylow
104C2×#100—971no—
105C2×#99—971no—
106#386—971no—
107C22×D4—971no—
108C23×C4—971no—
109C2×M16—1941no—
110C2×C22⋊C4—1941no—
111C2×C22⋊C4—1941no—
112C2×C22⋊C4#8760411yescharacteristic, complemented
113#100—972no—
114#100—1941no—
115#99—972no—
116#99—1941no—
117#95—1941no—
118C24—971no—
119C24C2×#8760411yescharacteristic
120C2×D4—971no—
121C2×D4—971no—
122C2×D4—971no—
123C2×D4—971no—
124C2×D4—3881no—
125C2×D4—3881no—
126C22×C4—21no—
127C22×C4—971no—
128C22×C4—3881no—
129C22×C4—3881no—
130C22×C4—3882no—
131C22×C4—1941no—
132M16—3881no—
133M16—1942no—
134C2×C8—3881no—
135C22⋊C4—41no—
136C22⋊C4—3882no—
137C22⋊C4—3881no—
138C22⋊C4—1942no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—971no—
144C23—971no—
145C23—3881no—
146C23—3881no—
147C23—1941no—
148C23—1941no—
149C23C4×#8760411yescharacteristic
150C23C22⋊#8760411yescharacteristic
151C23C4⋊#8760411yescharacteristic
152D4—3881no—
153D4—3881no—
154D4—3881no—
155D4—3881no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—7761no—
160C2×C4—7761no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—3882no—
164C2×C4—3881no—
165C2×C4—3882no—
166C2×C4—3881no—
167C2×C4—3882no—
168C2×C4—1942no—
169C2×C4—1941no—
170C2×C4—1941no—
171C8—3882no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—7761no—
182C22—3881no—
183C22—3881no—
184C22—3881no—
185C22—3881no—
186C22—1941no—
187C22—1941no—
188C22(C2×#87604)⋊C411yescharacteristic
189C22#1569086811yescharacteristic
190C22#1569086911yescharacteristic
191C4—81no—
192C4—7762no—
193C4—3882no—
194C4—1941no—
195C4—1941no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—3881no—
201C2—3881no—
202C2#2026161311yescharacteristic
203C2#2026163611yescharacteristic
204C2#2026163811yescharacteristic
subgroupG/H
centerC22#15690869
commutatorC23×C97C42
FrattiniC24C2×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97#99
prime radical, cosocleC22×C22⋊D97C22
hypercenterC2×C22⋊C4#87604
hypocenterC97#1390
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. #15690870C4semidirect
9. #15690868C4semidirect
11. C23×Dic97C4semidirect
14. C2×C22⋊C4×C97C4semidirect
103. C97#1390semidirect
112. C2×C22⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908702
C2×#156908682
#202606122
#139097