Finite Group Databasebeta

← #8715

Subgroups — #8715

Order 256.

434 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 77 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#2280C211yesmaximal, characteristic
2#1779C211yesmaximal, characteristic
3C22⋊#95C211yesmaximal, characteristic
4#1294C212yesmaximal, complemented
5#1294C212yesmaximal, complemented
6C24×C4C2211yescharacteristic
7C2×#117C411yescharacteristic, complemented
8C2×#117C411yescharacteristic, complemented
9C22×C22⋊C4—21no—
10C2×C22⋊C8—21no—
11C2×C22⋊C8—21no—
12C2×C22⋊C8C2212yes—
13#381—21no—
14#380C411yescharacteristic, complemented
15#380C411yescharacteristic, complemented
16#380C2211yescharacteristic
17#380C2211yescharacteristic
18#379—21no—
19C4×C22⋊C4—21no—
20C4×C22⋊C4—21no—
21C2×#95—21no—
22C2×#95C2212yes—
23C25D411yescharacteristic
24C23×C4—24no—
25C23×C4—22no—
26C23×C4—42no—
27C23×C4—42no—
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4D411yescharacteristic
31C23×C4D411yescharacteristic
32C23×C4D411yescharacteristic
33C23×C4C2311yescharacteristic
34C22×C8—22no—
35#117—42no—
36#117—42no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—24no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—42no—
49C2×C22⋊C4—42no—
50C2×C42—21no—
51C2×C42—21no—
52C22⋊C8—24no—
53C22⋊C8—42no—
54C22⋊C8—42no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—24no—
60#95—42no—
61C24—22no—
62C24—22no—
63C24—21no—
64C24—21no—
65C24—22no—
66C24—22no—
67C24—41no—
68C24—41no—
69C24C22⋊C411yescharacteristic
70C24C22⋊C411yescharacteristic
71C24C2×D411yescharacteristic
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—22no—
79C22×C4—22no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—24no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—416no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4C22⋊C411yescharacteristic
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22×C411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4#4114yes—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—42no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C22⋊C4—42no—
114C22⋊C4—48no—
115C22⋊C4—42no—
116C22⋊C4—42no—
117C22⋊C4—42no—
118C22⋊C4—42no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C42—42no—
122C42—42no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—24no—
128C23—24no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—21no—
134C23—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—44no—
142C23—44no—
143C23—44no—
144C23—44no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23#9912yes—
150C23#10412yes—
151C23C2×C22⋊C411yescharacteristic
152C23#12312yes—
153C23#13512yes—
154C23AGL(1,ℤ/8)11yescharacteristic
155C23#13711yescharacteristic
156C2×C4—28no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—44no—
167C2×C4—44no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4#9912yes—
191C2×C4#11712yes—
192C2×C4#12312yes—
193C8—82no—
194C22—22no—
195C22—22no—
196C22—22no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—48no—
209C22—48no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22#38111yescharacteristic
219C22C2×#9911yescharacteristic
220C22C2×#9911yescharacteristic
221C22#41611yescharacteristic
222C22#41711yescharacteristic
223C22C2×#10411yescharacteristic
224C22#42111yescharacteristic
225C22#48212yes—
226C22#48512yes—
227C4—22no—
228C4—22no—
229C4—22no—
230C4—41no—
231C4—84no—
232C4—81no—
233C4—81no—
234C4—81no—
235C4—81no—
236C2—22no—
237C2—22no—
238C2—42no—
239C2—42no—
240C2#150111yescharacteristic
241C2#150411yescharacteristic
242C2#151311yescharacteristic
243C2#151611yescharacteristic
244C2#181911yescharacteristic
245C2#182511yescharacteristic
246C2#182611yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1294C2semidirect
5. #1294C2semidirect
7. C2×#117C4semidirect
8. C2×#117C4semidirect
14. #380C4semidirect
15. #380C4semidirect

Maximal subgroups

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

subgroupindex
#22802
#17792
C22⋊#952
#12942