Finite Group Databasebeta

← #8155

Subgroups — #8155

Order 256.

355 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 99 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×#388C211yesmaximal, characteristic
2#1749C211yesmaximal, characteristic, complemented
3C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×#323C212yesmaximal
5#1280C212yesmaximal, complemented
6C22×C4⋊C4C411yescharacteristic
7C22×C22⋊C4C411yescharacteristic
8C22×C22⋊C4C411yescharacteristic
9C22×C22⋊C4C2211yescharacteristic
10C22×C42C2211yescharacteristic
11C2×C22⋊C8C2212yes—
12C2×C22⋊C8C2212yes—
13C2×#97—21no—
14C2×C4×C8—21no—
15#388—21no—
16#388—21no—
17#388—21no—
18#388—21no—
19C2×#95C411yescharacteristic
20C2×#95C2211yescharacteristic
21#336—21no—
22#336—21no—
23#323C418yescomplemented
24C25C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—42no—
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—42no—
44C2×C22⋊C4—42no—
45C2×C22⋊C4C2×C414yes—
46C2×C42—42no—
47C2×C42D411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C22⋊C8—22no—
52C22⋊C8—22no—
53C22⋊C8C2×C414yes—
54C22⋊C8C2×C414yes—
55#97—42no—
56C4×C8—42no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61C24—21no—
62C24—21no—
63C24—22no—
64C24—22no—
65C24—41no—
66C24—41no—
67C24—41no—
68C24—41no—
69C24C4212yes—
70C24C22×C411yescharacteristic
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4C4212yes—
92C22×C4C22⋊C411yescharacteristic
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22×C412yes—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4#4112yes—
100C2×C8—24no—
101C2×C8—24no—
102C2×C8—42no—
103C2×C8—42no—
104C2×C8—42no—
105C2×C8—42no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C22⋊C4—24no—
109C22⋊C4—24no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C42—22no—
115C42—22no—
116C42—41no—
117C42—41no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—22no—
123C23—24no—
124C23—42no—
125C23—42no—
126C23—42no—
127C23—42no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—42no—
135C23—82no—
136C23—82no—
137C23—82no—
138C23—82no—
139C23C2×C4211yescharacteristic
140C23C2×C22⋊C411yescharacteristic
141C23#11711yescharacteristic
142C23#13112yes—
143C23#13112yes—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—44no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4#9912yes—
176C2×C4#10012yes—
177C2×C4C4×D412yes—
178C2×C4C4×D412yes—
179C8—44no—
180C8—44no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22—84no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22C4×C22⋊C411yescharacteristic
199C22#40512yes—
200C22#40811yescharacteristic
201C22#40811yescharacteristic
202C22C2×#9911yescharacteristic
203C22#41011yescharacteristic
204C22C2×#10011yescharacteristic
205C22#41311yescharacteristic
206C22#43312yes—
207C4—22no—
208C4—22no—
209C4—22no—
210C4—22no—
211C4—41no—
212C4—41no—
213C4—42no—
214C4—81no—
215C4—81no—
216C2—22no—
217C2—22no—
218C2—82no—
219C2#146811yescharacteristic
220C2#146911yescharacteristic
221C2#147111yescharacteristic
222C2#147311yescharacteristic
223C2#175011yescharacteristic
224C2C4×#9911yescharacteristic
225C2C4×#10011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C2×C42
Frattini, cosocleC23×C4C23
prime radicalC25C2×C4
LCSG ▷ C23 ▷ 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. #1749C2semidirect
3. C4×C22⋊C8C2semidirect
5. #1280C2semidirect
23. #323C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3882
#17492
C4×C22⋊C82
C2×#3232
#12802