Finite Group Databasebeta

← #44548692

Subgroups — #44548692

Order 32128.

440 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 93 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#30631651C212yesmaximal, complemented
2#30631411C214yesmaximal, complemented
3C22×C22⋊C4×C251C211yesmaximal, characteristic
4C2×#16860794—22no—
5C22⋊(C251⋊C8)—22no—
6C22⋊(C251⋊C8)C2214yescomplemented
7C25×C251C411yescharacteristic
8C23×C4×C251C411yescharacteristic
9C23×C4×C251C2211yescharacteristic
10C2×C22⋊C4×C251—22no—
11C2×C22⋊C4×C251—22no—
12C2×C22⋊C4×C251C412yes—
13C2×C22⋊C4×C251C2212yes—
14#16860794—44no—
15C2×C251⋊C8—24no—
16C24×C251—21no—
17C24×C251—21no—
18C24×C251—24no—
19C24×C251—42no—
20C24×C251—42no—
21C24×C251C2×C411yescharacteristic
22C24×C251C2×C412yes—
23C22×C4×C251—21no—
24C22×C4×C251—21no—
25C22×C4×C251—42no—
26C22×C4×C251—42no—
27C22×C4×C251C2×C412yes—
28C22×C4×C251C2×C411yescharacteristic
29C22×C4×C251D412yes—
30C22×C4×C251D412yes—
31C22×C4×C251C2311yescharacteristic
32C22⋊C4×C251—24no—
33C22⋊C4×C251—24no—
34C251⋊C8—44no—
35C23×C251—22no—
36C23×C251—22no—
37C23×C251—22no—
38C23×C251—24no—
39C23×C251—24no—
40C23×C251—22no—
41C23×C251—22no—
42C23×C251—42no—
43C23×C251—42no—
44C23×C251—416no—
45C23×C251—42no—
46C23×C251—42no—
47C23×C251—42no—
48C23×C251—42no—
49C23×C251C22⋊C411yescharacteristic
50C23×C251C22⋊C411yescharacteristic
51C23×C251M1614yescomplemented
52C23×C251C22×C411yescharacteristic
53C2×C4×C251—24no—
54C2×C4×C251—22no—
55C2×C4×C251—41no—
56C2×C4×C251—42no—
57C2×C4×C251—41no—
58C2×C4×C251—42no—
59C2×C4×C251—42no—
60C2×C4×C251—42no—
61C2×C4×C251C22⋊C412yes—
62C2×C4×C251C2×D412yes—
63C22×C251—28no—
64C22×C251—21no—
65C22×C251—21no—
66C22×C251—21no—
67C22×C251—21no—
68C22×C251—21no—
69C22×C251—21no—
70C22×C251—24no—
71C22×C251—44no—
72C22×C251—48no—
73C22×C251—48no—
74C22×C251—44no—
75C22×C251—42no—
76C22×C251—42no—
77C22×C251#9912yes—
78C22×C251#10012yes—
79C22×C251C2×C22⋊C411yescharacteristic
80C22×C251C2×M1612yes—
81C4×C251—22no—
82C4×C251—22no—
83C4×C251—42no—
84C2×C251—22no—
85C2×C251—22no—
86C2×C251—22no—
87C2×C251—44no—
88C2×C251#40711yescharacteristic
89C2×C251C2×#9911yescharacteristic
90C2×C251C2×#10011yescharacteristic
91C251#146211yescharacteristic, complemented, Sylow
92#1462—2511nomaximal, Sylow
93C22×C22⋊C4D25111yescharacteristic
94#407—2512no—
95#323—2514no—
96C25Dic25111yescharacteristic
97C23×C4Dic25111yescharacteristic
98C23×C4C2×D25111yescharacteristic
99C2×M16—5022no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4Dic25112yes—
103C2×C22⋊C4C2×D25112yes—
104C22⋊C8—5022no—
105C22⋊C8—2514no—
106C24—21no—
107C24—21no—
108C24—24no—
109C24—42no—
110C24—42no—
111C24C2×Dic25111yescharacteristic
112C24C2×Dic25112yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4C2×Dic25112yes—
118C22×C4C2×Dic25111yescharacteristic
119C22×C4C22⋊D25112yes—
120C22×C4C22⋊D25112yes—
121C22×C4C22×D25111yescharacteristic
122M16—10044no—
123C2×C8—5024no—
124C22⋊C4—24no—
125C22⋊C4—24no—
126C23—22no—
127C23—22no—
128C23—22no—
129C23—24no—
130C23—24no—
131C23—22no—
132C23—22no—
133C23—42no—
134C23—42no—
135C23—416no—
136C23—42no—
137C23—42no—
138C23—42no—
139C23—42no—
140C23C22⋊Dic25111yescharacteristic
141C23C22⋊Dic25111yescharacteristic
142C23#1686079414yescomplemented
143C23C22×Dic25111yescharacteristic
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4C22⋊Dic25112yes—
153C2×C4C2×C22⋊D25112yes—
154C8—10044no—
155C22—28no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—24no—
163C22—44no—
164C22—48no—
165C22—48no—
166C22—44no—
167C22—42no—
168C22—42no—
169C22#2254863012yes—
170C22#2254863212yes—
171C22C2×C22⋊Dic25111yescharacteristic
172C22C2×#1686079412yes—
173C4—22no—
174C4—22no—
175C4—42no—
176C2—22no—
177C2—22no—
178C2—22no—
179C2—44no—
180C2#3063165111yescharacteristic
181C2C2×#2254863011yescharacteristic
182C2C2×#2254863211yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic251
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
FittingC22×C22⋊C4×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC25×C251C4
cosocleC22×C4×C251C23
hypercenterC22×C22⋊C4D251
hypocenterC251#1462
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #30631651C2semidirect
2. #30631411C2semidirect
6. C22⋊(C251⋊C8)C22semidirect
51. C23×C251M16semidirect
91. C251#1462semidirect
142. C23#16860794semidirect

Maximal subgroups

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

subgroupindex
#306316512
#306314112
C22×C22⋊C4×C2512
#1462251