Finite Group Databasebeta

← #27157018

Subgroups — #27157018

Order 12416.

214 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 58 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
2#20261751C211yesmaximal, characteristic, complemented
3#20261581C212yesmaximal, complemented
4#20261578C212yesmaximal
5C2×#15690803C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897—21no—
8C22⋊#11715897C2212yes—
9C22⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897—21no—
12#15690803—21no—
13#15690803—21no—
14#15690803—21no—
15#15690803—21no—
16C2×C4⋊Dic97C411yescharacteristic
17C2×#13095490C411yescharacteristic
18C2×#13095490C2211yescharacteristic
19C2×C22⋊Dic97C411yescharacteristic
20C2×C22⋊Dic97C2211yescharacteristic
21C2×C4×Dic97C2211yescharacteristic
22C2×C22⋊C4×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C22×Dic97C2×C411yescharacteristic
26C22×Dic97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2311yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31#13095490—21no—
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35C22⋊Dic97—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39C4×Dic97D411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C24×C97C2×C411yescharacteristic
44C22×C4×C97C2×C411yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C22⋊C4×C97—21no—
47C22⋊C4×C97—21no—
48C23×C97—21no—
49C23×C97—22no—
50C23×C97—41no—
51C23×C97—41no—
52C23×C97C22×C411yescharacteristic
53C2×C4×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—22no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63C2×Dic97—41no—
64C2×Dic97C22⋊C411yescharacteristic
65C2×Dic97C22⋊C411yescharacteristic
66C2×Dic97C2×D411yescharacteristic
67C2×Dic97C2×D411yescharacteristic
68#11715897—42no—
69#11715897—42no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—82no—
76C22×C97C2×C22⋊C411yescharacteristic
77C22×C97#13112yes—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—41no—
81Dic97—21no—
82Dic97—21no—
83Dic97—21no—
84Dic97—21no—
85Dic97—41no—
86Dic97—41no—
87C2×C97—22no—
88C2×C97—81no—
89C2×C97#40811yescharacteristic
90C2×C97#41011yescharacteristic
91C2×C97#41311yescharacteristic
92#1473—971nomaximal, Sylow
93C97#147311yescharacteristic, complemented, Sylow
94C2×#126—971no—
95#432—971no—
96#431—971no—
97#324—972no—
98#323—972no—
99#126—1941no—
100#126—1941no—
101#126—1941no—
102#126—1941no—
103C2×C4⋊C4—971no—
104C2×C4⋊C4—971no—
105C2×C4⋊C4—971no—
106C2×C22⋊C4—971no—
107C2×C22⋊C4—971no—
108C2×C22⋊C4#8760411yescharacteristic
109C2×C42—971no—
110C4⋊C8—1941no—
111C4⋊C8—1941no—
112C22⋊C8—972no—
113C22⋊C8—972no—
114#97—1941no—
115C4×C8—1941no—
116C24C2×#8760411yescharacteristic
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—971no—
120C22×C4—971no—
121C22×C4C2×#8760411yescharacteristic
122C22×C4C2×#8760411yescharacteristic
123C2×C8—1942no—
124C2×C8—1942no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—1941no—
134C22⋊C4—1941no—
135C22⋊C4—1941no—
136C22⋊C4—1941no—
137C42—971no—
138C42—971no—
139C42—971no—
140C42—971no—
141C23—21no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23C22×#8760411yescharacteristic
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—971no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—971no—
153C2×C4—3881no—
154C2×C4—3881no—
155C2×C4—3881no—
156C2×C4—1941no—
157C2×C4—1941no—
158C2×C4—1942no—
159C2×C4C22⋊#8760411yescharacteristic
160C2×C4C22⋊#8760411yescharacteristic
161C8—3882no—
162C8—3882no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—82no—
169C22C2×C22⋊#8760411yescharacteristic
170C22#1569089712yes—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—3881no—
175C4—3881no—
176C4—1941no—
177C4—1941no—
178C4—1941no—
179C4—1941no—
180C2—22no—
181C2—81no—
182C2#2026170811yescharacteristic
183C2#2026171511yescharacteristic
184C2#2026172111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×Dic97C23
hypercenterC2×C22⋊C4#87604
hypocenterC97#1473
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #20261755C2semidirect
2. #20261751C2semidirect
3. #20261581C2semidirect
93. C97#1473semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202617512
#202615812
#202615782
C2×#156908032
#147397