Finite Group Databasebeta

← #9009

Subgroups — #9009

Order 256.

392 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 63 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#2973C211yesmaximal, characteristic, complemented
2#2388C211yesmaximal, characteristic
3#1789C212yesmaximal, complemented
4C4×#99C211yesmaximal, characteristic, complemented
5#1297C212yesmaximal, complemented
6C22×#41C2211yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C2211yescharacteristic
9#435—24no—
10#432—21no—
11C2×#104—24no—
12C2×#99—21no—
13#407C2212yes—
14C2×C22⋊C8—21no—
15#395C411yescharacteristic, complemented
16#394—21no—
17#394C411yescharacteristic, complemented
18#393C411yescharacteristic, complemented
19#392—21no—
20#392C411yescharacteristic, complemented
21C4×C22⋊C4—21no—
22C4×C22⋊C4C2212yes—
23C4×C22⋊C4C2211yescharacteristic
24#342—21no—
25C2×#41—22no—
26C2×#41—22no—
27C2×#41—44no—
28C22×Q8C2×C411yescharacteristic
29C22×D4—21no—
30C22×D4—21no—
31C22×D4C2×C411yescharacteristic
32C22×D4D411yescharacteristic
33C23×C4—21no—
34C23×C4D411yescharacteristic
35C23×C4C2311yescharacteristic
36C2×M16D414yescomplemented
37C22×C8—22no—
38C4×D4—22no—
39C4×D4—22no—
40C4×D4—44no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C22⋊C4—22no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4C2×C411yescharacteristic
50C2×C22⋊C4D411yescharacteristic
51C2×C42—21no—
52C2×C42—22no—
53C2×C42D411yescharacteristic
54C4⋊C8—22no—
55#104—48no—
56#99—24no—
57C22⋊C8—22no—
58C22⋊C8—22no—
59C22⋊C8—22no—
60#97—22no—
61#95—21no—
62#95—21no—
63#95—22no—
64#95C2×C411yescharacteristic
65#95C2×C411yescharacteristic
66C24—21no—
67C24C22⋊C411yescharacteristic
68C24C2×D411yescharacteristic
69#41—28no—
70#41—44no—
71#41—44no—
72#41—44no—
73C2×Q8—22no—
74C2×Q8—42no—
75C2×D4—22no—
76C2×D4—22no—
77C2×D4—22no—
78C2×D4—41no—
79C2×D4—42no—
80C2×D4—42no—
81C2×D4—42no—
82C2×D4—42no—
83C2×D4—42no—
84C2×D4—42no—
85C2×D4—42no—
86C22×C4—24no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4C22⋊C411yescharacteristic
99C22×C4C22⋊C411yescharacteristic
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22×C411yescharacteristic
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4#4112yes—
105M16—44no—
106C2×C8—24no—
107C2×C8—82no—
108C4⋊C4—22no—
109C4⋊C4—42no—
110C22⋊C4—22no—
111C22⋊C4—22no—
112C22⋊C4—24no—
113C22⋊C4—42no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—42no—
117C22⋊C4—42no—
118C22⋊C4—42no—
119C42—21no—
120C42—21no—
121C42—42no—
122C42—44no—
123C23—21no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—22no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23#10412yescomplemented
139C23C2×C22⋊C411yescharacteristic
140C23#12112yes—
141Q8—24no—
142Q8—42no—
143D4—24no—
144D4—42no—
145D4—44no—
146D4—48no—
147D4—48no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—44no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—44no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4C4×D412yes—
176C2×C4#12011yescharacteristic
177C2×C4#12311yescharacteristic
178C8—44no—
179C22—21no—
180C22—21no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—41no—
189C22—41no—
190C22—42no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22#38611yescharacteristic
199C22C2×#10411yescharacteristic
200C22#42111yescharacteristic
201C4—21no—
202C4—22no—
203C4—42no—
204C4—41no—
205C4—41no—
206C4—42no—
207C4—41no—
208C4—81no—
209C4—84no—
210C4B(4,𝔽2)12yes—
211C2—21no—
212C2—21no—
213C2—42no—
214C2—42no—
215C2—82no—
216C2#187111yescharacteristic
217C2#188411yescharacteristic
218C2#188911yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
socleC22#386
prime radical#2388C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2973C2semidirect
3. #1789C2semidirect
4. C4×#99C2semidirect
5. #1297C2semidirect
15. #395C4semidirect
17. #394C4semidirect
18. #393C4semidirect
20. #392C4semidirect
36. C2×M16D4semidirect
138. C23#104semidirect

Maximal subgroups

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

subgroupindex
#29732
#23882
#17892
C4×#992
#12972