Finite Group Databasebeta

← #10127

Subgroups — #10127

Order 256.

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

normal subgroups only

This group has 210 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#3237C211yesmaximal, characteristic
2#3236C211yesmaximal, characteristic
3#3166C211yesmaximal, characteristic
4#1497C211yesmaximal, characteristic
5#1485C211yesmaximal, characteristic
6#1470C211yesmaximal, characteristic
7#1291C211yesmaximal, characteristic
8#533C2211yescharacteristic
9#533C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#501—21no—
13#501—21no—
14#501—21no—
15#499—22no—
16#491—21no—
17#491—21no—
18#485—22no—
19#484—21no—
20#483—21no—
21#479—22no—
22#477—21no—
23#475—21no—
24#432C2211yescharacteristic
25#432C2211yescharacteristic
26#431C2211yescharacteristic
27#428—21no—
28#428—21no—
29C4×M16C2211yescharacteristic
30Q8⋊C8—22no—
31Q8⋊C8—22no—
32#324—22no—
33C2×M16—21no—
34C4⋊Q8D411yescharacteristic
35C4⋊Q8D411yescharacteristic
36C4⋊Q8D411yescharacteristic
37C4⋊Q8D411yescharacteristic
38#125—21no—
39#125—21no—
40#125—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44C22⋊Q8—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52#117—21no—
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C42C2311yescharacteristic
59#107—22no—
60#107—22no—
61#107—22no—
62#107—22no—
63#106—22no—
64#106—22no—
65#106—22no—
66#106—22no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8D412yes—
70#103—22no—
71#103—22no—
72#103—22no—
73#103—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77#97—21no—
78#97—21no—
79#97—21no—
80C4×C8—21no—
81C4×C8D412yes—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4#4111yescharacteristic
91M16—24no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—41no—
96C4⋊C4—21no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C42—41no—
119C42—41no—
120C42C2×D411yescharacteristic
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C23#12111yescharacteristic
125Q8—42no—
126Q8—42no—
127Q8—42no—
128Q8—42no—
129C2×C4—22no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—81no—
140C2×C4#12011yescharacteristic
141C2×C4#12011yescharacteristic
142C2×C4#12011yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12711yescharacteristic
146C8—24no—
147C8—42no—
148C8—42no—
149C8—81no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22#39211yescharacteristic
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—81no—
160C4—81no—
161C4—81no—
162C4—81no—
163C4—81no—
164C4—81no—
165C4#45612yes—
166C2—41no—
167C2#200511yescharacteristic
168C2#200911yescharacteristic
169C2#201711yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC23#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#32372
#32362
#31662
#14972
#14852
#14702
#12912