Finite Group Databasebeta

← #10736

Subgroups — #10736

Order 256.

755 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 755 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C22×#117C211yesmaximal, characteristic
2C2×#381C211yesmaximal, characteristic
3C2×#381C211yesmaximal, characteristic
4C2×#342C214yesmaximal
5C24×C4C2211yescharacteristic
6C2×#117—22no—
7C2×#117—28no—
8C2×#117C414yescomplemented
9C22×C4⋊C4—21no—
10C22×C22⋊C4C2212yes—
11C22×C22⋊C4C2212yes—
12C22×C22⋊C4C2212yes—
13C22×C42—21no—
14#381—22no—
15#381—21no—
16#381—22no—
17#381—21no—
18#381C414yescomplemented
19C2×#95—22no—
20C2×#95C414yescomplemented
21#342—28no—
22C25C2311yescharacteristic
23C23×C4—24no—
24C23×C4—24no—
25C23×C4—22no—
26C23×C4—22no—
27C23×C4C2×C412yes—
28C23×C4C2×C412yes—
29C23×C4C2×C412yes—
30C23×C4C2×C414yes—
31C23×C4D411yescharacteristic
32C23×C4D412yes—
33C23×C4D412yes—
34C23×C4D411yescharacteristic
35C23×C4Q812yes—
36#117—216no—
37#117—216no—
38C2×C4⋊C4—28no—
39C2×C4⋊C4—24no—
40C2×C22⋊C4—24no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—24no—
43C2×C22⋊C4—24no—
44C2×C22⋊C4—24no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—24no—
49C2×C22⋊C4C2×C414yes—
50C2×C22⋊C4C2×C414yes—
51C2×C22⋊C4D418yes—
52C2×C42—28no—
53C2×C42—24no—
54#95—28no—
55#95—24no—
56#95—24no—
57C24—44no—
58C24C22×C411yescharacteristic
59C24C22×C411yescharacteristic
60C24C22×C411yescharacteristic
61C24C2×D411yescharacteristic
62C24C2×D411yescharacteristic
63C24C2×D411yescharacteristic
64C24C2×D414yes—
65C24C2×Q811yescharacteristic
66C24#4114yes—
67C22×C4—24no—
68C22×C4—24no—
69C22×C4—28no—
70C22×C4—28no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—24no—
74C22×C4—28no—
75C22×C4—28no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—24no—
79C22×C4—24no—
80C22×C4—28no—
81C22×C4—24no—
82C22×C4—24no—
83C22×C4—28no—
84C22×C4—48no—
85C22×C4—48no—
86C22×C4—44no—
87C22×C4—48no—
88C22×C4C4214yes—
89C22×C4C22⋊C414yes—
90C22×C4C22⋊C412yes—
91C22×C4C22⋊C412yes—
92C22×C4C22⋊C412yes—
93C22×C4C22⋊C412yes—
94C22×C4C4⋊C414yes—
95C22×C4C4⋊C414yes—
96C22×C4C4⋊C412yes—
97C22×C4C4⋊C412yes—
98C4⋊C4—216no—
99C22⋊C4—24no—
100C22⋊C4—28no—
101C22⋊C4—28no—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—44no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4C22⋊C418yes—
108C42—216no—
109C23—24no—
110C23—24no—
111C23—24no—
112C23—42no—
113C23—42no—
114C23—48no—
115C23—42no—
116C23—42no—
117C23—48no—
118C23C2×C4211yescharacteristic
119C23C2×C22⋊C411yescharacteristic
120C23C2×C22⋊C411yescharacteristic
121C23C2×C22⋊C412yes—
122C23C2×C22⋊C411yescharacteristic
123C23C2×C4⋊C411yescharacteristic
124C23C2×C4⋊C411yescharacteristic
125C23C2×C4⋊C411yescharacteristic
126C23#11712yes—
127C23C4×D414yes—
128C23C4×D414yes—
129C23#12012yes—
130C23#12012yes—
131C23#12114yes—
132C23C22⋊Q814yes—
133C23#12312yes—
134C23#12312yes—
135C2×C4—28no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—28no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—24no—
142C2×C4—24no—
143C2×C4—28no—
144C2×C4—28no—
145C2×C4—24no—
146C2×C4—24no—
147C2×C4—28no—
148C2×C4—416no—
149C2×C4—48no—
150C2×C4—48no—
151C2×C4—44no—
152C2×C4—44no—
153C2×C4—48no—
154C2×C4—88no—
155C2×C4—88no—
156C2×C4#9514yes—
157C2×C4#9514yes—
158C22—24no—
159C22—28no—
160C22—28no—
161C22—24no—
162C22—22no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—44no—
171C22—42no—
172C22—44no—
173C22—42no—
174C22C2×#9511yescharacteristic
175C22C4×C22⋊C412yes—
176C22#37911yescharacteristic
177C22#38012yes—
178C22#38111yescharacteristic
179C22#38512yes—
180C22#38512yes—
181C22#38612yes—
182C22#38612yes—
183C22#41014yes—
184C4—24no—
185C4—24no—
186C4—48no—
187C4—84no—
188C4—84no—
189C2—21no—
190C2—22no—
191C2—22no—
192C2—22no—
193C2—21no—
194C2—22no—
195C2—22no—
196C2C22⋊#9511yescharacteristic
197C2#174012yes—
198C2#189914yes—
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C2×C42
Frattini, prime radical, cosocleC25C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#117C4semidirect
18. #381C4semidirect
20. C2×#95C4semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#3812
C2×#3422