Finite Group Databasebeta

← #10344

Subgroups — #10344

Order 256.

428 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 61 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#2646C211yesmaximal, characteristic, complemented
2#1786C212yesmaximal
3#1438C211yesmaximal, characteristic
4#1294C212yesmaximal, complemented
5#1280C211yesmaximal, characteristic, complemented
6C23×D4C2211yescharacteristic
7C22×C22⋊C4—22no—
8C2×#102—22no—
9C2×#102—22no—
10C2×C22⋊C8C2212yes—
11#400—22no—
12#398—22no—
13#385—22no—
14#380C2212yes—
15C2×#95C2211yescharacteristic
16C2×#95C2211yescharacteristic
17C25—21no—
18C22×D4—22no—
19C22×D4—22no—
20C22×D4—21no—
21C22×D4—44no—
22C22×D4Q812yes—
23C23×C4—21no—
24C23×C4D412yes—
25C23×C4C2311yescharacteristic
26C22×C8—22no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4D412yes—
30C2×C4⋊C4D412yes—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—44no—
33C2×C22⋊C4—44no—
34C2×C22⋊C4—44no—
35#102—44no—
36#102—44no—
37C22⋊C8—22no—
38C22⋊C8—22no—
39#95—22no—
40#95—22no—
41#95—22no—
42#95—22no—
43#95—22no—
44C24—24no—
45C24—22no—
46C24—42no—
47C24—42no—
48C24—42no—
49C24—41no—
50C24—41no—
51C24—42no—
52C24—42no—
53C24C2×D411yescharacteristic
54C2×D4—24no—
55C2×D4—24no—
56C2×D4—24no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—44no—
60C2×D4—48no—
61C2×D4—44no—
62C2×D4—42no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—42no—
76C22×C4C2×D412yes—
77C22×C4C2×Q811yescharacteristic
78C22×C4#4112yes—
79C22×C4#4111yescharacteristic
80C2×C8—42no—
81C2×C8—42no—
82C2×C8—42no—
83C4⋊C4—42no—
84C4⋊C4—42no—
85C4⋊C4—82no—
86C4⋊C4—82no—
87C22⋊C4—416no—
88C22⋊C4—42no—
89C22⋊C4—42no—
90C23—21no—
91C23—21no—
92C23—22no—
93C23—24no—
94C23—44no—
95C23—44no—
96C23—42no—
97C23—44no—
98C23—44no—
99C23—44no—
100C23—44no—
101C23—42no—
102C23—44no—
103C23—44no—
104C23—82no—
105C23—82no—
106C23—82no—
107C23—82no—
108C23—82no—
109C23—82no—
110C23—81no—
111C23—81no—
112C23—81no—
113C23—81no—
114C23#12011yescharacteristic
115C23C22⋊Q811yescharacteristic
116C23#12411yescharacteristic
117C23#13512yes—
118C23AGL(1,ℤ/8)12yes—
119D4—28no—
120D4—44no—
121D4—44no—
122D4—44no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—42no—
128C2×C4—44no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—82no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—82no—
144C2×C4C22⋊Q812yes—
145C2×C4#12412yes—
146C8—82no—
147C22—22no—
148C22—22no—
149C22—22no—
150C22—22no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—48no—
156C22—44no—
157C22—84no—
158C22—84no—
159C22—84no—
160C22—84no—
161C22—84no—
162C22—82no—
163C22—82no—
164C22#39311yescharacteristic
165C22#44912yes—
166C22#45311yescharacteristic
167C22#45411yescharacteristic
168C22B(4,𝔽2)12yes—
169C22#47812yes—
170C22#48811yescharacteristic
171C22#49011yescharacteristic
172C4—22no—
173C4—22no—
174C4—82no—
175C4—82no—
176C4—82no—
177C4—82no—
178C2—21no—
179C2—22no—
180C2—21no—
181C2—84no—
182C2#159712yes—
183C2#160012yes—
184C2#202511yescharacteristic
185C2#202711yescharacteristic
186C2#202911yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC23×D4C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2646C2semidirect
4. #1294C2semidirect
5. #1280C2semidirect

Maximal subgroups

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

subgroupindex
#26462
#17862
#14382
#12942
#12802