Finite Group Databasebeta

← #12265

Subgroups — #12265

Order 256.

181 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 181 conjugacy classes of subgroups — too many to draw. Only its 35 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#2771C211yesmaximal, characteristic
2#2030C211yesmaximal, characteristic
3#1614C211yesmaximal, characteristic
4#1608C211yesmaximal, characteristic
5#1352C211yesmaximal, characteristic
6#1344C211yesmaximal, characteristic
7#1317C211yesmaximal, characteristic
8#543C2211yescharacteristic
9#484—21no—
10#483—21no—
11#400—21no—
12#400—21no—
13#399C2211yescharacteristic
14#397C2211yescharacteristic
15#389—21no—
16#388—21no—
17#387—21no—
18#385—21no—
19#381—21no—
20#381C2211yescharacteristic
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C23×C4D411yescharacteristic
28C22⋊Q8—21no—
29C22⋊Q8—21no—
30C22⋊Q8—41no—
31C22⋊Q8—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4D411yescharacteristic
37C2×C4⋊C4D411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4D411yescharacteristic
43C2×C22⋊C4C2311yescharacteristic
44C2×C42—21no—
45#107—41no—
46#106—41no—
47#103—41no—
48#103—41no—
49C22⋊C8—21no—
50C22⋊C8—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—41no—
59#95—41no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C24C2×D411yescharacteristic
63C2×Q8—21no—
64C2×Q8—21no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C2×C8—41no—
81C2×C8—41no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—41no—
85C4⋊C4—41no—
86C4⋊C4—42no—
87C4⋊C4—42no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—81no—
91C4⋊C4—81no—
92C22⋊C4—41no—
93C22⋊C4—42no—
94C22⋊C4—41no—
95C22⋊C4—41no—
96C22⋊C4—41no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C42—81no—
102C23—21no—
103C23—21no—
104C23—41no—
105C23—41no—
106C23#12011yescharacteristic
107C23#13511yescharacteristic
108C23#13711yescharacteristic
109Q8—81no—
110Q8—81no—
111C2×C4—22no—
112C2×C4—21no—
113C2×C4—21no—
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—42no—
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—41no—
135C2×C4—41no—
136C2×C4—81no—
137C2×C4—81no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—81no—
142C8—81no—
143C8—81no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—41no—
149C22—41no—
150C22—41no—
151C22—41no—
152C22—81no—
153C22#45211yescharacteristic
154C22#45511yescharacteristic
155C22B(4,𝔽2)11yescharacteristic
156C4—42no—
157C4—41no—
158C4—41no—
159C4—81no—
160C4—81no—
161C4—81no—
162C4—81no—
163C4—81no—
164C4—81no—
165C4—81no—
166C4—161no—
167C2—21no—
168C2—21no—
169C2—41no—
170C2—41no—
171C2#160611yescharacteristic
172C2#219811yescharacteristic
173C2#220111yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radicalC24C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#27712
#20302
#16142
#16082
#13522
#13442
#13172