Finite Group Databasebeta

← #7835

Subgroups — #7835

Order 256.

175 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 175 conjugacy classes of subgroups — too many to draw. Only its 49 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#2805C211yesmaximal, characteristic
2#2100C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic
4#2095C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic
6#1271C211yesmaximal, characteristic
7#1271C211yesmaximal, characteristic
8C2×#107—21no—
9C2×#103—21no—
10#400—21no—
11#400—21no—
12#395—21no—
13#391C2211yescharacteristic
14#389—21no—
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×Q8D411yescharacteristic
28C22×C8—21no—
29C22×C8—21no—
30C22×C8—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4D411yescharacteristic
41C2×C4⋊C4D411yescharacteristic
42C2×C4⋊C4D411yescharacteristic
43C2×C42—21no—
44C2×C42C2311yescharacteristic
45#107—42no—
46#103—42no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56C2×Q8—21no—
57C2×Q8—21no—
58C2×Q8—41no—
59C2×Q8—41no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—41no—
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D411yescharacteristic
69C22×C4C2×D411yescharacteristic
70C2×C8—22no—
71C2×C8—41no—
72C2×C8—41no—
73C2×C8—41no—
74C2×C8—41no—
75C2×C8—41no—
76C2×C8—41no—
77C2×C8—41no—
78C2×C8—41no—
79C4⋊C4—21no—
80C4⋊C4—21no—
81C4⋊C4—21no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—42no—
88C4⋊C4—42no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C4⋊C4—81no—
95C4⋊C4Q1612yes—
96C42—41no—
97C42—81no—
98C23#12011yescharacteristic
99Q8—42no—
100Q8—81no—
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—22no—
108C2×C4—41no—
109C2×C4—41no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
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—81no—
126C2×C4—81no—
127C2×C4—81no—
128C2×C4C2×Q1611yescharacteristic
129C2×C4#13511yescharacteristic
130C2×C4#13511yescharacteristic
131C2×C4#13711yescharacteristic
132C2×C4#13711yescharacteristic
133C2×C4#13711yescharacteristic
134C8—42no—
135C8—81no—
136C8—81no—
137C22#45111yescharacteristic
138C22#45211yescharacteristic
139C22#45211yescharacteristic
140C22#45311yescharacteristic
141C22#45411yescharacteristic
142C22#45411yescharacteristic
143C22#45811yescharacteristic
144C4—21no—
145C4—21no—
146C4—21no—
147C4—21no—
148C4—21no—
149C4—21no—
150C4—42no—
151C4—81no—
152C4—81no—
153C4—81no—
154C4—81no—
155C4—81no—
156C4—161no—
157C2#161511yescharacteristic
158C2#161711yescharacteristic
159C2#161811yescharacteristic
160C2#163211yescharacteristic
161C2#163611yescharacteristic
162C2#163811yescharacteristic
163C2#165711yescharacteristic
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
#28052
#21002
#20952
#12722
#12712