Finite Group Databasebeta

← #27157011

Subgroups — #27157011

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#20261578C214yesmaximal, complemented
4C2×#15690781C211yesmaximal, characteristic
5C2×#13095521—21no—
6C2×#13095519—21no—
7#15690867—21no—
8#15690867—21no—
9C22⋊#11715897C2212yes—
10C22⋊#11715897C2212yes—
11C22×C22⋊D97C412yes—
12C22×C4×D97C2211yescharacteristic
13#15690781—21no—
14#15690781—21no—
15#15690781—21no—
16#15690781—21no—
17C2×C4⋊Dic97C411yescharacteristic
18C2×C22⋊Dic97C2212yes—
19C22×D4×C97C411yescharacteristic
20#13095521—42no—
21#13095519—42no—
22C2×#11715897—22no—
23C2×#11715897—22no—
24C23×D97C2×C411yescharacteristic
25C2×C22⋊D97—22no—
26C2×C22⋊D97—22no—
27C2×C22⋊D97—44no—
28C22×Dic97C2×C412yes—
29C22×Dic97C2311yescharacteristic
30C2×C4×D97—42no—
31C2×C4×D97D411yescharacteristic
32C2×C4×D97D411yescharacteristic
33C2×C4×D97D411yescharacteristic
34C2×C4×D97D411yescharacteristic
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C22⋊Dic97—22no—
38C22⋊Dic97—22no—
39C24×C97C2×C412yes—
40C2×D4×C97—21no—
41C2×D4×C97—21no—
42C2×D4×C97—42no—
43C22×C4×C97C2×C411yescharacteristic
44C23×C97—22no—
45C23×C97—24no—
46C23×C97—42no—
47C23×C97—42no—
48C23×C97C22×C411yescharacteristic
49C22×D97—22no—
50C22×D97—22no—
51C22×D97—41no—
52C22×D97C22⋊C411yescharacteristic
53C22×D97C22⋊C411yescharacteristic
54D4×C97—42no—
55D4×C97—42no—
56C2×C4×C97—22no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C22⋊D97—44no—
60C22⋊D97—44no—
61C2×Dic97—22no—
62C2×Dic97—41no—
63C2×Dic97—42no—
64C2×Dic97C2×D411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
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×C97—42no—
75C22×C97—42no—
76C22×C97—42no—
77C22×C97—84no—
78C22×C97C2×C22⋊C411yescharacteristic
79C22×C97#13112yes—
80C2×D97—21no—
81C2×D97—21no—
82C2×D97—21no—
83C2×D97—21no—
84C2×D97—21no—
85C2×D97—44no—
86C2×D97#10012yes—
87C4×C97—21no—
88C4×C97#9912yes—
89Dic97—21no—
90Dic97—21no—
91Dic97—42no—
92C2×C97—22no—
93C2×C97—82no—
94C2×C97#40811yescharacteristic
95C2×C97C2×#9911yescharacteristic
96C2×C97C2×#10011yescharacteristic
97D97—22no—
98D97—41no—
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#8760411yescharacteristic
107C23×C4—971no—
108C2×M16—1941no—
109C22×C8—1941no—
110#121—1941no—
111#121—1941no—
112#121—1941no—
113#121—1941no—
114C2×C4⋊C4—971no—
115C2×C22⋊C4—972no—
116C22⋊C8—972no—
117C22⋊C8—972no—
118C22⋊C8—1941no—
119C22⋊C8—1941no—
120C24—971no—
121C24C2×#8760412yes—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—42no—
125C2×D4—3884no—
126C2×D4—1942no—
127C2×D4—1942no—
128C22×C4—971no—
129C22×C4—971no—
130C22×C4—971no—
131C22×C4—971no—
132C22×C4—971no—
133C22×C4—972no—
134C22×C4—3882no—
135C22×C4C2×#8760411yescharacteristic
136M16—3882no—
137C2×C8—3882no—
138C2×C8—1942no—
139C2×C8—1942no—
140C4⋊C4—1941no—
141C4⋊C4—1941no—
142C22⋊C4—1942no—
143C22⋊C4—1942no—
144C23—22no—
145C23—24no—
146C23—42no—
147C23—42no—
148C23—971no—
149C23—971no—
150C23—3881no—
151C23—1942no—
152C23—1942no—
153C23C22×#8760411yescharacteristic
154D4—42no—
155D4—42no—
156D4—3884no—
157D4—3884no—
158C2×C4—22no—
159C2×C4—971no—
160C2×C4—971no—
161C2×C4—3881no—
162C2×C4—3881no—
163C2×C4—3882no—
164C2×C4—3881no—
165C2×C4—1942no—
166C2×C4—1942no—
167C2×C4—1942no—
168C2×C4C22⋊#8760411yescharacteristic
169C2×C4C22⋊#8760411yescharacteristic
170C8—3882no—
171C8—3882no—
172C22—21no—
173C22—21no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—84no—
178C22—972no—
179C22—3884no—
180C22—1941no—
181C22—1941no—
182C22—1941no—
183C22—1941no—
184C22—1941no—
185C22C2×C22⋊#8760411yescharacteristic
186C22#1569089712yes—
187C4—21no—
188C4—3882no—
189C4—1941no—
190C4—1941no—
191C4#1569086812yes—
192C2—22no—
193C2—82no—
194C2—3881no—
195C2—1942no—
196C2#2026170911yescharacteristic
197C2C2×#1569086811yescharacteristic
198C2D97⋊#10011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC22×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#15690781C2
cosocleC22×Dic97C23
hypercenterC22×D4#87604
hypocenterC97#1471
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20261706C2semidirect
2. #20261703C2semidirect
3. #20261578C2semidirect
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
#202615782
C2×#156907812
#147197