Finite Group Databasebeta

← #9384

Subgroups — #9384

Order 256.

320 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 75 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#2972C211yesmaximal, characteristic, complemented
2#2543C211yesmaximal, characteristic
3#1799C211yesmaximal, characteristic, complemented
4#1519C212yesmaximal, complemented
5#1294C212yesmaximal, complemented
6C2×#121—21no—
7C2×#121C2211yescharacteristic
8C2×C4×D4C2211yescharacteristic
9C8×D4—24no—
10#431—21no—
11C2×#99—21no—
12C2×C22⋊C8—21no—
13C2×C22⋊C8C2212yes—
14#398C411yescharacteristic, complemented
15#397C411yescharacteristic, complemented
16#394C411yescharacteristic, complemented
17#392C411yescharacteristic, complemented
18#388—21no—
19#386—21no—
20#386—21no—
21#380C2212yes—
22C2×#95C2211yescharacteristic
23#342—21no—
24#327—24no—
25C22×D4—21no—
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C23×C4D411yescharacteristic
31C23×C4C2311yescharacteristic
32C22×C8—22no—
33C22×C8—22no—
34#121—22no—
35#121—22no—
36#121—42no—
37#121—42no—
38C4×D4—22no—
39C4×D4—22no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C4⋊C4C2×C411yescharacteristic
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4C2×C411yescharacteristic
51C2×C22⋊C4D411yescharacteristic
52C2×C42D411yescharacteristic
53C4⋊C8—22no—
54#99—42no—
55C22⋊C8—22no—
56C22⋊C8—22no—
57C22⋊C8D414yescomplemented
58C4×C8—22no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—22no—
63#95C2×C411yescharacteristic
64#95C2×C411yescharacteristic
65C24—21no—
66C24C22⋊C411yescharacteristic
67C24C2×D411yescharacteristic
68C2×D4—22no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—42no—
72C2×D4—42no—
73C2×D4—42no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—42no—
77C2×D4—81no—
78C2×D4—81no—
79C2×D4D812yes—
80C2×D4SD1612yes—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—42no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—42no—
90C22×C4—81no—
91C22×C4—81no—
92C22×C4—81no—
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22×C411yescharacteristic
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4#4112yes—
100C2×C8—24no—
101C2×C8—82no—
102C2×C8—82no—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—42no—
106C4⋊C4—82no—
107C22⋊C4—22no—
108C22⋊C4—22no—
109C22⋊C4—42no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—42no—
115C22⋊C4—42no—
116C22⋊C4—81no—
117C42—21no—
118C42—21no—
119C23—21no—
120C23—21no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23#10212yes—
133C23C2×C22⋊C411yescharacteristic
134C23#12112yes—
135C23C2×D811yescharacteristic
136C23C2×SD1611yescharacteristic
137C23#13512yes—
138D4—24no—
139D4—24no—
140D4—82no—
141D4—84no—
142D4—82no—
143D4—82no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4#10212yes—
171C2×C4C4×D412yes—
172C2×C4#12011yescharacteristic
173C2×C4#12311yescharacteristic
174C8—44no—
175C22—21no—
176C22—22no—
177C22—21no—
178C22—22no—
179C22—22no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—42no—
184C22—41no—
185C22—41no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22#38611yescharacteristic
195C22C2×#10211yescharacteristic
196C22#41611yescharacteristic
197C22#46512yes—
198C22#46612yes—
199C4—22no—
200C4—21no—
201C4—21no—
202C4—22no—
203C4—42no—
204C4—41no—
205C4—41no—
206C4—81no—
207C4—81no—
208C4—162no—
209C2—21no—
210C2—22no—
211C2—21no—
212C2—22no—
213C2—42no—
214C2—161no—
215C2#188411yescharacteristic
216C2#189311yescharacteristic
217C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#2543C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2972C2semidirect
3. #1799C2semidirect
4. #1519C2semidirect
5. #1294C2semidirect
14. #398C4semidirect
15. #397C4semidirect
16. #394C4semidirect
17. #392C4semidirect
57. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
#29722
#25432
#17992
#15192
#12942