Finite Group Databasebeta

← #9080

Subgroups — #9080

Order 256.

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

normal subgroups only

This group has 368 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
1C2×#431C211yesmaximal, characteristic, complemented
2C2×#388C211yesmaximal, characteristic
3#1817C211yesmaximal, characteristic, complemented
4C2×#323C212yesmaximal
5#1280C212yesmaximal, complemented
6C22×C4⋊C4C411yescharacteristic
7C22×C22⋊C4C411yescharacteristic
8C22×C22⋊C4C411yescharacteristic
9C22×C22⋊C4C2211yescharacteristic
10C22×C42C2211yescharacteristic
11#431—24no—
12C2×C4⋊C8—21no—
13C2×C22⋊C8C2212yes—
14C2×C22⋊C8C2212yes—
15C2×C4×C8—21no—
16#388—21no—
17#388—21no—
18#388—21no—
19#388—21no—
20C2×#95C411yescharacteristic
21C2×#95C2211yescharacteristic
22#336—21no—
23#336—21no—
24#323—24no—
25C25C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2311yescharacteristic
32C22×C8—22no—
33C22×C8—22no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—42no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4—42no—
47C2×C22⋊C4C814yescomplemented
48C2×C42—22no—
49C2×C42—22no—
50C2×C42D411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42D411yescharacteristic
54C4⋊C8—24no—
55C22⋊C8—22no—
56C22⋊C8—22no—
57C22⋊C8—22no—
58C22⋊C8D414yes—
59C4×C8—24no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95—21no—
64C24—21no—
65C24—21no—
66C24—22no—
67C24—22no—
68C24—41no—
69C24—41no—
70C24—41no—
71C24—41no—
72C24C2×C812yes—
73C24C22×C411yescharacteristic
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4—21no—
79C22×C4—22no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C2×C812yes—
98C22×C4C2×C812yes—
99C22×C4M1612yes—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D412yes—
103C22×C4#4112yes—
104C2×C8—24no—
105C2×C8—42no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—42no—
109C2×C8—42no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C22⋊C4—24no—
113C22⋊C4—24no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—41no—
117C22⋊C4—42no—
118C22⋊C4—41no—
119C42—21no—
120C42—21no—
121C42—22no—
122C42—22no—
123C42—21no—
124C42—21no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23—21no—
129C23—22no—
130C23—24no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23—42no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—82no—
143C23—82no—
144C23—82no—
145C23—82no—
146C23C2×C22⋊C411yescharacteristic
147C23C4×D412yes—
148C23C22×C811yescharacteristic
149C23C2×M1611yescharacteristic
150C23#13112yes—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4C22⋊C812yes—
183C2×C4C22⋊C812yes—
184C2×C4#10012yes—
185C2×C4#12011yescharacteristic
186C2×C4#12112yes—
187C2×C4#12311yescharacteristic
188C8—44no—
189C8—82no—
190C22—22no—
191C22—22no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—84no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22#38611yescharacteristic
208C22C2×C22⋊C811yescharacteristic
209C22#40811yescharacteristic
210C22#41011yescharacteristic
211C22#41011yescharacteristic
212C22C2×#10011yescharacteristic
213C22#41311yescharacteristic
214C22C8×D412yes—
215C22#43512yes—
216C4—22no—
217C4—22no—
218C4—21no—
219C4—22no—
220C4—21no—
221C4—22no—
222C4—41no—
223C4—42no—
224C4—81no—
225C4—81no—
226C2—22no—
227C2—22no—
228C2—82no—
229C2#145811yescharacteristic
230C2#146011yescharacteristic
231C2#146911yescharacteristic
232C2#147311yescharacteristic
233C2#186911yescharacteristic
234C2#189911yescharacteristic
235C2#190011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C22×C8
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
1. C2×#431C2semidirect
3. #1817C2semidirect
5. #1280C2semidirect
47. C2×C22⋊C4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#4312
C2×#3882
#18172
C2×#3232
#12802