Finite Group Databasebeta

← #12237

Subgroups — #12237

Order 256.

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

normal subgroups only

This group has 167 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#2789C211yesmaximal, characteristic
2#2061C211yesmaximal, characteristic
3#1611C211yesmaximal, characteristic
4#1605C211yesmaximal, characteristic
5#1354C211yesmaximal, characteristic
6#1346C211yesmaximal, characteristic
7#1324C211yesmaximal, characteristic
8#544C2211yescharacteristic
9#483—21no—
10#451—21no—
11#428—21no—
12#401—21no—
13#400—21no—
14#398C2211yescharacteristic
15#395C2211yescharacteristic
16#391—21no—
17#387—21no—
18#387—21no—
19#387—21no—
20#382C2211yescharacteristic
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C2×Q16—41no—
29C2×M16—21no—
30C4⋊Q8—41no—
31#125—41no—
32C22⋊Q8—21no—
33C22⋊Q8—21no—
34#117—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4D411yescharacteristic
39C2×C4⋊C4D411yescharacteristic
40C2×C4⋊C4C2311yescharacteristic
41C2×C42—21no—
42C2×C42D411yescharacteristic
43#107—41no—
44#106—41no—
45#106—41no—
46#103—41no—
47#103—41no—
48C22⋊C8—21no—
49C22⋊C8—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—
59#95—21no—
60#95D411yescharacteristic
61#95D411yescharacteristic
62C2×Q8—21no—
63C2×Q8—21no—
64C2×Q8—42no—
65C2×Q8—41no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4C2×D411yescharacteristic
73C22×C4C2×D411yescharacteristic
74C22×C4C2×D411yescharacteristic
75Q16—82no—
76M16—22no—
77M16—41no—
78C2×C8—41no—
79C2×C8—41no—
80C2×C8—41no—
81C4⋊C4—21no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—42no—
89C4⋊C4—42no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C22⋊C4—41no—
95C42—21no—
96C42—21no—
97C42—41no—
98C42—81no—
99C23#12011yescharacteristic
100Q8—42no—
101Q8—81no—
102Q8—81no—
103Q8—81no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—22no—
107C2×C4—21no—
108C2×C4—21no—
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—42no—
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#13511yescharacteristic
130C2×C4#13711yescharacteristic
131C8—42no—
132C8—81no—
133C8—81no—
134C22—21no—
135C22—21no—
136C22#45211yescharacteristic
137C22#45611yescharacteristic
138C22B(4,𝔽2)11yescharacteristic
139C4—21no—
140C4—21no—
141C4—41no—
142C4—41no—
143C4—42no—
144C4—81no—
145C4—81no—
146C4—81no—
147C4—81no—
148C4—81no—
149C4—81no—
150C4—81no—
151C2—21no—
152C2—21no—
153C2#160311yescharacteristic
154C2#219711yescharacteristic
155C2#220311yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23
prime radicalC23#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#27892
#20612
#16112
#16052
#13542
#13462
#13242