Finite Group Databasebeta

← #9653

Subgroups — #9653

Order 256.

460 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#390C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3#1749C211yesmaximal, characteristic, complemented
4C2×#323C214yesmaximal, complemented
5C23×D4C411yescharacteristic
6C22×C4⋊C4C411yescharacteristic
7C22×C22⋊C4C412yes—
8C22×C22⋊C4C2212yes—
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8C2212yes—
12C2×C22⋊C8C2212yes—
13C2×#97—21no—
14#390—21no—
15#390—21no—
16#390—21no—
17#390—21no—
18#336—21no—
19#336—21no—
20#323—28no—
21C25C2×C412yes—
22C22×D4—24no—
23C22×D4—21no—
24C22×D4—21no—
25C22×D4—42no—
26C22×D4—42no—
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C412yes—
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—42no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—24no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—24no—
42C2×C22⋊C4—44no—
43C2×C42—42no—
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C2×C42D411yescharacteristic
48C4⋊C8—42no—
49C22⋊C8—22no—
50C22⋊C8—24no—
51C22⋊C8D414yescomplemented
52#97—42no—
53C24—22no—
54C24—22no—
55C24—24no—
56C24—24no—
57C24—42no—
58C24—42no—
59C24—42no—
60C24—42no—
61C24M1614yes—
62C24C22×C411yescharacteristic
63C2×D4—24no—
64C2×D4—24no—
65C2×D4—416no—
66C2×D4—42no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—42no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—24no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D412yes—
89C22×C4#4112yes—
90C2×C8—24no—
91C2×C8—42no—
92C2×C8—42no—
93C2×C8—42no—
94C2×C8—42no—
95C2×C8—42no—
96C4⋊C4—42no—
97C4⋊C4—42no—
98C22⋊C4—44no—
99C22⋊C4—44no—
100C22⋊C4—44no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C42—22no—
104C42—22no—
105C42—41no—
106C42—41no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—24no—
112C23—28no—
113C23—44no—
114C23—44no—
115C23—44no—
116C23—44no—
117C23—42no—
118C23—42no—
119C23—42no—
120C23—42no—
121C23—42no—
122C23—42no—
123C23—44no—
124C23—84no—
125C23—84no—
126C23—84no—
127C23—84no—
128C23C2×C22⋊C411yescharacteristic
129C23C4×D412yes—
130C23C2×M1612yes—
131C23#13112yes—
132D4—48no—
133D4—48no—
134C2×C4—28no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—44no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—84no—
155C2×C4#9912yes—
156C2×C4#10012yes—
157C2×C4#10012yes—
158C2×C4#12112yes—
159C2×C4#12411yescharacteristic
160C2×C4#12711yescharacteristic
161C8—44no—
162C8—82no—
163C22—22no—
164C22—22no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—44no—
170C22—44no—
171C22—44no—
172C22—88no—
173C22—88no—
174C22—82no—
175C22—82no—
176C22—82no—
177C22—82no—
178C22#39011yescharacteristic
179C22#40711yescharacteristic
180C22#40811yescharacteristic
181C22C2×#9911yescharacteristic
182C22#41011yescharacteristic
183C22C2×#10011yescharacteristic
184C22C2×#10011yescharacteristic
185C22#43514yes—
186C4—22no—
187C4—21no—
188C4—21no—
189C4—22no—
190C4—22no—
191C4—22no—
192C4—41no—
193C4—82no—
194C2—24no—
195C2—84no—
196C2#146211yescharacteristic
197C2#146611yescharacteristic
198C2#146911yescharacteristic
199C2#147111yescharacteristic
200C2#195511yescharacteristic
201C2#196411yescharacteristic
202C2#196511yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC23×D4C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1797C2semidirect
3. #1749C2semidirect
4. C2×#323C2semidirect
51. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
#17972
#17492
C2×#3232