Finite Group Databasebeta

← #11802

Subgroups — #11802

Order 256.

379 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 53 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×#394C211yesmaximal, characteristic
2#1821C211yesmaximal, characteristic, complemented
3#1463C211yesmaximal, characteristic, complemented
4#1345C214yesmaximal, complemented
5C23×D4C411yescharacteristic
6C22×C4⋊C4C2211yescharacteristic
7C22×C22⋊C4—21no—
8C22×C22⋊C4—21no—
9#407—21no—
10#407—21no—
11#394—21no—
12#394—21no—
13#394C412yes—
14#394C2212yes—
15C2×#95C411yescharacteristic
16#343—21no—
17#343C2212yescomplemented
18#324—21no—
19#324C2212yescomplemented
20C25—21no—
21C22×D4—21no—
22C22×D4—41no—
23C22×D4—41no—
24C22×D4—44no—
25C22×D4C2×C412yes—
26C23×C4—21no—
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—22no—
31C2×M16—41no—
32C2×C4⋊C4—41no—
33C2×C4⋊C4—41no—
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4D411yescharacteristic
36C2×C4⋊C4D411yescharacteristic
37C2×C4⋊C4C2311yescharacteristic
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—44no—
43C2×C22⋊C4—44no—
44C22⋊C8—22no—
45C22⋊C8—41no—
46C22⋊C8—41no—
47C22⋊C8—41no—
48#95—21no—
49#95C2×C412yes—
50C24—24no—
51C24—22no—
52C24—42no—
53C24—42no—
54C24—42no—
55C24—42no—
56C24—42no—
57C24—41no—
58C24—41no—
59C24C22⋊C411yescharacteristic
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—44no—
63C2×D4—44no—
64C2×D4—48no—
65C2×D4—44no—
66C2×D4—44no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—22no—
75C22×C4—41no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22×C411yescharacteristic
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87M16—42no—
88M16—42no—
89M16—82no—
90C2×C8—42no—
91C2×C8—42no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C22⋊C4—44no—
97C22⋊C4—44no—
98C22⋊C4—44no—
99C22⋊C4—44no—
100C23—22no—
101C23—22no—
102C23—21no—
103C23—21no—
104C23—24no—
105C23—42no—
106C23—44no—
107C23—48no—
108C23—48no—
109C23—42no—
110C23—42no—
111C23—44no—
112C23—42no—
113C23—42no—
114C23—42no—
115C23—84no—
116C23—84no—
117C23—82no—
118C23—82no—
119C23—82no—
120C23—82no—
121C23#9911yescharacteristic
122C23#9911yescharacteristic
123C23C2×C22⋊C411yescharacteristic
124D4—44no—
125D4—44no—
126D4—48no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—81no—
149C2×C4—84no—
150C2×C4AGL(1,ℤ/8)12yes—
151C8—82no—
152C8—82no—
153C22—24no—
154C22—24no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—44no—
162C22—44no—
163C22—44no—
164C22—82no—
165C22—82no—
166C22—88no—
167C22—88no—
168C22—82no—
169C22—82no—
170C22AΣL(1,𝔽16)12yes—
171C22#35512yes—
172C22C2×#9911yescharacteristic
173C22#41811yescharacteristic
174C22#42111yescharacteristic
175C4—22no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4—41no—
180C4—41no—
181C4—82no—
182C2—21no—
183C2—21no—
184C2—22no—
185C2—22no—
186C2—84no—
187C2#151011yescharacteristic
188C2C2×AΣL(1,𝔽16)11yescharacteristic
189C2C2×#35511yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C4⋊C4C23
prime radicalC23×D4C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1821C2semidirect
3. #1463C2semidirect
4. #1345C2semidirect
17. #343C22semidirect
19. #324C22semidirect

Maximal subgroups

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

subgroupindex
C2×#3942
#18212
#14632
#13452