Finite Group Databasebeta

← #27157013

Subgroups — #27157013

Order 12416.

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

normal subgroups only

This group has 322 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, complemented
2#20261703C211yesmaximal, characteristic, complemented
3#20261579C214yesmaximal, complemented
4C2×#15690785C211yesmaximal, characteristic
5C2×#13095521—21no—
6C2×#13095519—21no—
7#15690867—21no—
8#15690867—21no—
9C22⋊#11715897C2212yes—
10C22⋊#11715897C2212yes—
11C22×D388C412yes—
12C22×D388C411yescharacteristic
13C22×C4×D97C2211yescharacteristic
14#15690785—21no—
15#15690785—21no—
16#15690785—21no—
17#15690785—21no—
18C2×#13095488C2212yes—
19C2×C4⋊C4×C97C411yescharacteristic
20#13095521—42no—
21#13095519—42no—
22C2×#11715897—22no—
23C2×#11715897—22no—
24C23×D97C2×C411yescharacteristic
25C23×D97C2×C412yes—
26C2×D388—22no—
27C2×D388—22no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—42no—
31C2×D388—44no—
32C22×Dic97C2311yescharacteristic
33C2×C4×D97—42no—
34C2×C4×D97D411yescharacteristic
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97D411yescharacteristic
38#13095488—22no—
39#13095488—22no—
40C22×C4×C97C2×C412yes—
41C22×C4×C97C2×C411yescharacteristic
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C23×C97C22×C411yescharacteristic
45C22×D97—22no—
46C22×D97—22no—
47C22×D97—22no—
48C22×D97—24no—
49C22×D97—41no—
50C22×D97—42no—
51C22×D97—42no—
52C22×D97C22⋊C411yescharacteristic
53C22×D97C22⋊C411yescharacteristic
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—42no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×Dic97—41no—
60C2×Dic97C2×D411yescharacteristic
61C2×Dic97C2×D411yescharacteristic
62D388—42no—
63D388—44no—
64D388—44no—
65D388—42no—
66C4×D97—22no—
67C4×D97—22no—
68C4×D97—41no—
69C4×D97—41no—
70#11715897—42no—
71#11715897—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97C2×C22⋊C411yescharacteristic
75C22×C97#13112yes—
76C2×D97—21no—
77C2×D97—21no—
78C2×D97—21no—
79C2×D97—21no—
80C2×D97—21no—
81C2×D97—44no—
82C2×D97—42no—
83C2×D97—42no—
84C2×D97—42no—
85C2×D97—84no—
86C2×D97#10012yes—
87C4×C97—21no—
88C4×C97—42no—
89C4×C97#9912yes—
90Dic97—21no—
91Dic97—21no—
92C2×C97—22no—
93C2×C97#40811yescharacteristic
94C2×C97C2×#9911yescharacteristic
95C2×C97C2×#10011yescharacteristic
96D97—22no—
97D97—41no—
98D97—82no—
99#1471—971nomaximal, Sylow
100C97#147111yescharacteristic, complemented, Sylow
101C2×#121—971no—
102#407—971no—
103C2×C22⋊C8—971no—
104#323—974no—
105C22×D4—972no—
106C22×D4—971no—
107C23×C4—971no—
108C2×M16—1941no—
109C22×C8—1941no—
110#121—1941no—
111#121—1941no—
112#121—1941no—
113#121—1941no—
114C2×C4⋊C4#8760411yescharacteristic
115C2×C22⋊C4—972no—
116C22⋊C8—972no—
117C22⋊C8—972no—
118C22⋊C8—1941no—
119C22⋊C8—1941no—
120C24—971no—
121C24—972no—
122C2×D4—3882no—
123C2×D4—3884no—
124C2×D4—1942no—
125C2×D4—1942no—
126C2×D4—1941no—
127C2×D4—1941no—
128C22×C4—971no—
129C22×C4—971no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—3882no—
134C22×C4C2×#8760412yes—
135C22×C4C2×#8760411yescharacteristic
136M16—3882no—
137C2×C8—3882no—
138C2×C8—1942no—
139C2×C8—1942no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C22⋊C4—1942no—
143C22⋊C4—1942no—
144C23—971no—
145C23—971no—
146C23—3882no—
147C23—3882no—
148C23—3881no—
149C23—1942no—
150C23—1944no—
151C23—1942no—
152C23—1942no—
153C23C22×#8760411yescharacteristic
154D4—3882no—
155D4—3884no—
156D4—3884no—
157D4—3882no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—971no—
162C2×C4—971no—
163C2×C4—3881no—
164C2×C4—3881no—
165C2×C4—3881no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4C22⋊#8760411yescharacteristic
169C2×C4C22⋊#8760411yescharacteristic
170C8—3882no—
171C8—3882no—
172C22—21no—
173C22—21no—
174C22—7764no—
175C22—972no—
176C22—3884no—
177C22—3882no—
178C22—3882no—
179C22—3882no—
180C22—1941no—
181C22—1941no—
182C22—1941no—
183C22—1941no—
184C22—1941no—
185C22C2×C22⋊#8760411yescharacteristic
186C22#1569089812yes—
187C4—21no—
188C4—42no—
189C4—1941no—
190C4—1941no—
191C4#1569086912yes—
192C2—22no—
193C2—7762no—
194C2—3881no—
195C2—1942no—
196C2#2026171011yescharacteristic
197C2C2×#1569086911yescharacteristic
198C2D97⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690785C2
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1471
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261706C2semidirect
2. #20261703C2semidirect
3. #20261579C2semidirect
100. C97#1471semidirect

Maximal subgroups

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

subgroupindex
#202617062
#202617032
#202615792
C2×#156907852
#147197