Finite Group Databasebeta

← #7775

Subgroups — #7775

Order 256.

190 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 47 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#2810C211yesmaximal, characteristic
2#2058C211yesmaximal, characteristic
3#2057C211yesmaximal, characteristic
4#2043C211yesmaximal, characteristic
5#1271C211yesmaximal, characteristic
6#1271C211yesmaximal, characteristic
7#1271C211yesmaximal, characteristic
8C2×C4⋊Q8C2211yescharacteristic
9C2×#107—21no—
10C2×#106—21no—
11C2×#103—21no—
12C2×#103—21no—
13#400—21no—
14#400—21no—
15#400—21no—
16#389C2211yescharacteristic
17#389C2211yescharacteristic
18#389C2211yescharacteristic
19#387—21no—
20#387—21no—
21#387—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×C8—21no—
29C22×C8—21no—
30C22×C8—21no—
31C4⋊Q8—42no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4D411yescharacteristic
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4D411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C42—21no—
44C2×C42C2311yescharacteristic
45#107—42no—
46#106—42no—
47#103—42no—
48#103—42no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59C2×Q8—22no—
60C2×Q8—41no—
61C2×Q8—41no—
62C2×Q8—41no—
63C2×Q8—41no—
64C2×Q8—41no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—41no—
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75C2×C8—22no—
76C2×C8—22no—
77C2×C8—41no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C4⋊C4—22no—
85C4⋊C4—22no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C42—41no—
104C42—81no—
105C23#12011yescharacteristic
106Q8—42no—
107Q8—42no—
108Q8—81no—
109Q8—81no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—81no—
136C2×C4—81no—
137C2×C4#13511yescharacteristic
138C2×C4#13511yescharacteristic
139C2×C4#13511yescharacteristic
140C2×C4AGL(1,ℤ/8)11yescharacteristic
141C2×C4#13711yescharacteristic
142C2×C4#13711yescharacteristic
143C8—42no—
144C8—42no—
145C8—81no—
146C22#44911yescharacteristic
147C22#45211yescharacteristic
148C22#45211yescharacteristic
149C22#45411yescharacteristic
150C22#45611yescharacteristic
151C22#45611yescharacteristic
152C22B(4,𝔽2)11yescharacteristic
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—81no—
160C4—82no—
161C4—81no—
162C4—81no—
163C4—81no—
164C4—81no—
165C4—81no—
166C2#159711yescharacteristic
167C2#160311yescharacteristic
168C2#160311yescharacteristic
169C2#163811yescharacteristic
170C2#165011yescharacteristic
171C2#165211yescharacteristic
172C2#166311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#120
commutator, Frattini, cosocleC2×C42C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#28102
#20582
#20572
#20432
#12712