Finite Group Databasebeta

← #27157019

Subgroups — #27157019

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#20261756C211yesmaximal, characteristic, complemented
2#20261752C211yesmaximal, characteristic, complemented
3#20261581C212yesmaximal, complemented
4#20261579C212yesmaximal
5C2×#15690805C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897—21no—
8C22⋊#11715897C2212yes—
9C22⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897—21no—
12#15690805—21no—
13#15690805—21no—
14#15690805—21no—
15#15690805—21no—
16C2×C4⋊Dic97C411yescharacteristic
17C2×#13095490C2211yescharacteristic
18C2×#13095488C411yescharacteristic
19C2×#13095488C411yescharacteristic
20C2×#13095488C2211yescharacteristic
21C2×C4×Dic97C2211yescharacteristic
22C2×C4⋊C4×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C23×D97C2×C411yescharacteristic
26C22×Dic97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2311yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31#13095490—21no—
32#13095490—21no—
33#13095488—21no—
34#13095488—21no—
35#13095488—21no—
36#13095488—21no—
37#13095488—21no—
38#13095488—21no—
39C4×Dic97D411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C22×C4×C97C2×C411yescharacteristic
44C22×C4×C97C2×C411yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97—21no—
48C23×C97C22×C411yescharacteristic
49C22×D97—21no—
50C22×D97—22no—
51C22×D97—41no—
52C22×D97—41no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×Dic97—22no—
61C2×Dic97—21no—
62C2×Dic97—22no—
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×C97C2×C22⋊C411yescharacteristic
73C22×C97#13112yes—
74C2×D97—41no—
75C2×D97—41no—
76C2×D97—41no—
77C2×D97—82no—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—41no—
81C4×C97—41no—
82Dic97—21no—
83Dic97—21no—
84Dic97—21no—
85Dic97—21no—
86Dic97—41no—
87C2×C97—22no—
88C2×C97#40811yescharacteristic
89C2×C97#41011yescharacteristic
90C2×C97#41311yescharacteristic
91D97—81no—
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#8760411yescharacteristic
106C2×C22⋊C4—971no—
107C2×C22⋊C4—971no—
108C2×C22⋊C4—971no—
109C2×C42—971no—
110C4⋊C8—1941no—
111C4⋊C8—1941no—
112C22⋊C8—972no—
113C22⋊C8—972no—
114#97—1941no—
115C4×C8—1941no—
116C24—971no—
117C22×C4—971no—
118C22×C4—971no—
119C22×C4—971no—
120C22×C4C2×#8760411yescharacteristic
121C22×C4C2×#8760411yescharacteristic
122C22×C4C2×#8760411yescharacteristic
123C2×C8—1942no—
124C2×C8—1942no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C22⋊C4—1941no—
132C22⋊C4—1941no—
133C22⋊C4—1941no—
134C22⋊C4—1941no—
135C22⋊C4—1941no—
136C22⋊C4—1941no—
137C42—971no—
138C42—971no—
139C42—971no—
140C42—971no—
141C23—3881no—
142C23—3881no—
143C23—1942no—
144C23—1941no—
145C23C22×#8760411yescharacteristic
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—971no—
152C2×C4—971no—
153C2×C4—971no—
154C2×C4—971no—
155C2×C4—3881no—
156C2×C4—1941no—
157C2×C4—1942no—
158C2×C4—1942no—
159C2×C4C22⋊#8760411yescharacteristic
160C2×C4C22⋊#8760411yescharacteristic
161C8—3882no—
162C8—3882no—
163C22—21no—
164C22—21no—
165C22—7762no—
166C22—3881no—
167C22—3881no—
168C22—3881no—
169C22C2×C22⋊#8760411yescharacteristic
170C22#1569089812yes—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—3881no—
176C4—1941no—
177C4—1941no—
178C4—1941no—
179C4—1941no—
180C2—22no—
181C2—7761no—
182C2#2026171011yescharacteristic
183C2#2026171411yescharacteristic
184C2#2026172211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1473
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. #20261756C2semidirect
2. #20261752C2semidirect
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
#202617562
#202617522
#202615812
#202615792
C2×#156908052
#147397