Finite Group Databasebeta

← #8940

Subgroups — #8940

Order 256.

356 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 101 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
2C4×#121C211yesmaximal, characteristic
3C4×C22⋊C8C211yesmaximal, characteristic, complemented
4C2×D4⋊C8C212yesmaximal, complemented
5#1279C212yesmaximal, complemented
6C2×#121C411yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C2211yescharacteristic
9C22×C42C2211yescharacteristic
10#431—24no—
11C2×C4⋊C8C2212yes—
12C2×C22⋊C8—21no—
13C2×C22⋊C8—21no—
14C2×C4×C8—21no—
15C2×C4×C8C2212yes—
16#390C411yescharacteristic
17#389C411yescharacteristic
18#388—21no—
19#386—21no—
20#380C411yescharacteristic
21C4×C4⋊C4C2211yescharacteristic
22C4×C22⋊C4—21no—
23#336—21no—
24D4⋊C8—24no—
25C22×D4—21no—
26C22×D4C2×C411yescharacteristic
27C23×C4—21no—
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4D411yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33#121—22no—
34#121C814yescomplemented
35C4×D4—22no—
36C4×D4—22no—
37C4×D4—44no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42—22no—
48C2×C42—22no—
49C2×C42—21no—
50C2×C42C2×C411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42C2311yescharacteristic
54C4⋊C8—22no—
55C4⋊C8D414yescomplemented
56C22⋊C8—24no—
57C22⋊C8—24no—
58C4×C8—24no—
59C4×C8—22no—
60C4×C8—22no—
61#95—21no—
62#95—21no—
63#95—21no—
64C24—21no—
65C24C22⋊C411yescharacteristic
66C2×D4—24no—
67C2×D4—22no—
68C2×D4—21no—
69C2×D4—22no—
70C2×D4—42no—
71C2×D4—42no—
72C2×D4C2×C812yes—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—21no—
78C22×C4—22no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C2×C812yes—
93C22×C4M1612yes—
94C22×C4C22×C411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C2×C8—24no—
98C2×C8—24no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—42no—
103C4⋊C4—21no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—42no—
107C4⋊C4C2×C812yes—
108C22⋊C4—24no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C42—22no—
114C42—21no—
115C42—22no—
116C42—21no—
117C42—42no—
118C42C2×D412yes—
119C42#4112yes—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—22no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23C22⋊C812yes—
134C23C2×C22⋊C411yescharacteristic
135D4—24no—
136D4—48no—
137D4—42no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—24no—
145C2×C4—24no—
146C2×C4—21no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—44no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4C22⋊C812yes—
164C2×C4#10412yes—
165C2×C4C4×D412yes—
166C2×C4#12011yescharacteristic
167C2×C4#12112yes—
168C2×C4#12311yescharacteristic
169C2×C4C22×C811yescharacteristic
170C2×C4C2×M1611yescharacteristic
171C2×C4#13112yes—
172C2×C4#13512yes—
173C2×C4AGL(1,ℤ/8)12yes—
174C8—44no—
175C8—44no—
176C22—22no—
177C22—22no—
178C22—22no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22#38611yescharacteristic
191C22C2×C22⋊C811yescharacteristic
192C22#40811yescharacteristic
193C22#41611yescharacteristic
194C22#41811yescharacteristic
195C22C2×#10411yescharacteristic
196C22#42111yescharacteristic
197C4—22no—
198C4—22no—
199C4—22no—
200C4—41no—
201C4—42no—
202C4—81no—
203C4—81no—
204C4C8×D412yes—
205C4#43512yes—
206C4#44912yes—
207C4#48212yes—
208C2—42no—
209C2—82no—
210C2#147611yescharacteristic
211C2#147811yescharacteristic
212C2#148211yescharacteristic
213C2#148411yescharacteristic
214C2#186911yescharacteristic
215C2#187111yescharacteristic
216C2#187511yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
socleC23C2×C22⋊C4
prime radicalC2×#121C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#431C2semidirect
3. C4×C22⋊C8C2semidirect
4. C2×D4⋊C8C2semidirect
5. #1279C2semidirect
34. #121C8semidirect
55. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4312
C4×#1212
C4×C22⋊C82
C2×D4⋊C82
#12792