Finite Group Databasebeta

← #12277

Subgroups — #12277

Order 256.

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

normal subgroups only

This group has 159 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#2833C211yesmaximal, characteristic
2#2028C211yesmaximal, characteristic
3#1617C211yesmaximal, characteristic
4#1611C211yesmaximal, characteristic
5#1353C211yesmaximal, characteristic
6#1346C211yesmaximal, characteristic
7#1319C211yesmaximal, characteristic
8#544C2211yescharacteristic
9#484—21no—
10#483—21no—
11#419—21no—
12#401—21no—
13#400—21no—
14#400—21no—
15#400C2211yescharacteristic
16#398—21no—
17#398C2211yescharacteristic
18#395—21no—
19#391C2211yescharacteristic
20#387—21no—
21#343C2211yescharacteristic
22#339—21no—
23#328—21no—
24#328—21no—
25#324C2211yescharacteristic
26#324C2211yescharacteristic
27C22×Q8D411yescharacteristic
28C2×M16—21no—
29C4⋊Q8—41no—
30#125—41no—
31C22⋊Q8—21no—
32C22⋊Q8—21no—
33#117—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⋊C4C2311yescharacteristic
41C2×C42D411yescharacteristic
42#107—41no—
43#106—41no—
44#103—41no—
45#103—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—21no—
65C2×Q8—21no—
66C2×Q8—41no—
67C2×Q8—41no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77M16—41no—
78M16—41no—
79C2×C8—41no—
80C2×C8—41no—
81C2×C8—41no—
82C4⋊C4—21no—
83C4⋊C4—21no—
84C4⋊C4—21no—
85C4⋊C4—21no—
86C4⋊C4—41no—
87C4⋊C4—41no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—81no—
94C4⋊C4—81no—
95C4⋊C4—81no—
96C22⋊C4—41no—
97C42—41no—
98C42—41no—
99C23#12011yescharacteristic
100Q8—42no—
101Q8—81no—
102Q8—81no—
103Q8—81no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
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—42no—
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#13511yescharacteristic
130C2×C4#13711yescharacteristic
131C8—81no—
132C8—81no—
133C8—81no—
134C22—21no—
135C22—21no—
136C22#45211yescharacteristic
137C22#45511yescharacteristic
138C22B(4,𝔽2)11yescharacteristic
139C4—21no—
140C4—21no—
141C4—41no—
142C4—41no—
143C4—81no—
144C4—81no—
145C4—81no—
146C4—81no—
147C4—81no—
148C4—81no—
149C4—81no—
150C4—81no—
151C2—21no—
152C2—21no—
153C2#160611yescharacteristic
154C2#220111yescharacteristic
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
#28332
#20282
#16172
#16112
#13532
#13462
#13192