Finite Group Databasebeta

← #7794

Subgroups — #7794

Order 256.

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

normal subgroups only

This group has 197 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
1#2734C211yesmaximal, characteristic
2#2099C211yesmaximal, characteristic
3#2083C211yesmaximal, characteristic
4#2072C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic
6#1271C211yesmaximal, characteristic
7#1271C211yesmaximal, characteristic
8C2×C4⋊Q8—21no—
9C2×#107—21no—
10C2×#103—21no—
11#398—21no—
12#395—21no—
13#395—21no—
14#391C2211yescharacteristic
15#389C2211yescharacteristic
16#389C2211yescharacteristic
17#387—21no—
18#387—21no—
19#384C2211yescharacteristic
20#340—21no—
21#340—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×Q8D411yescharacteristic
29C22×C8—21no—
30C22×C8—21no—
31C22×C8—21no—
32C4⋊Q8—42no—
33C4⋊Q8—42no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C42—21no—
48C2×C42C2311yescharacteristic
49#107—42no—
50#103—42no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58C2×Q8—21no—
59C2×Q8—42no—
60C2×Q8—41no—
61C2×Q8—41no—
62C2×Q8—81no—
63C2×Q8—81no—
64C2×Q8D812yes—
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—41no—
77C2×C8—41no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—42no—
90C4⋊C4—42no—
91C4⋊C4—42no—
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—81no—
101C4⋊C4SD1612yes—
102C4⋊C4Q1612yes—
103C42—41no—
104C42—42no—
105C23#12011yescharacteristic
106Q8—42no—
107Q8—81no—
108Q8—84no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
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—81no—
134C2×C4—81no—
135C2×C4—81no—
136C2×C4C2×D811yescharacteristic
137C2×C4C2×SD1611yescharacteristic
138C2×C4C2×Q1611yescharacteristic
139C2×C4AGL(1,ℤ/8)11yescharacteristic
140C2×C4#13711yescharacteristic
141C2×C4#13711yescharacteristic
142C8—42no—
143C8—81no—
144C8—81no—
145C22#44711yescharacteristic
146C22#44811yescharacteristic
147C22#45111yescharacteristic
148C22#45411yescharacteristic
149C22#45511yescharacteristic
150C22#45611yescharacteristic
151C22#45811yescharacteristic
152C4—21no—
153C4—21no—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—42no—
159C4—42no—
160C4—42no—
161C4—81no—
162C4—81no—
163C4—81no—
164C4—161no—
165C2#160711yescharacteristic
166C2#161111yescharacteristic
167C2#161711yescharacteristic
168C2#162711yescharacteristic
169C2#164511yescharacteristic
170C2#164911yescharacteristic
171C2#166411yescharacteristic
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
#27342
#20992
#20832
#20722
#12722
#12712