Finite Group Databasebeta

← #10139

Subgroups — #10139

Order 256.

223 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 51 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#3236C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic
3C2×#501C211yesmaximal, characteristic
4#1556C211yesmaximal, characteristic
5#1531C211yesmaximal, characteristic
6#1470C211yesmaximal, characteristic
7#1302C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—22no—
12#501—21no—
13#501—22no—
14#494—21no—
15#491—21no—
16#487—21no—
17#484—21no—
18#483—21no—
19#479—22no—
20#462—22no—
21#451—22no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25C2×#107—21no—
26C2×#106—21no—
27C2×#97C2211yescharacteristic
28#332—22no—
29#330—22no—
30#324—22no—
31C22×Q8—21no—
32C2×Q16—24no—
33C22×C8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8D412yes—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#125—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44#125—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49#117—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C42C2311yescharacteristic
57#107—24no—
58#107—22no—
59#106—24no—
60#106—22no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64#103—24no—
65#103—22no—
66#103—22no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8D412yes—
70#97—21no—
71#97—21no—
72#97—21no—
73#97D412yes—
74C4×C8—21no—
75C2×Q8—21no—
76C2×Q8—22no—
77C2×Q8—21no—
78C2×Q8—41no—
79C2×Q8—82no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—41no—
83C22×C4C2×D411yescharacteristic
84C22×C4C2×D411yescharacteristic
85C22×C4#4111yescharacteristic
86Q16—84no—
87C2×C8—24no—
88C2×C8—22no—
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—41no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—22no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C22⋊C4—41no—
111C42—41no—
112C42C2×D411yescharacteristic
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C23#12111yescharacteristic
117Q8—42no—
118Q8—44no—
119Q8—42no—
120Q8—82no—
121C2×C4—22no—
122C2×C4—22no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—82no—
131C2×C4—81no—
132C2×C4#12011yescharacteristic
133C2×C4#12011yescharacteristic
134C2×C4#12011yescharacteristic
135C2×C4#12111yescharacteristic
136C2×C4#12111yescharacteristic
137C2×C4#12711yescharacteristic
138C8—24no—
139C8—42no—
140C8—42no—
141C8—81no—
142C22—21no—
143C22—21no—
144C22#39211yescharacteristic
145C22#47312yes—
146C4—21no—
147C4—21no—
148C4—21no—
149C4—21no—
150C4—21no—
151C4—81no—
152C4—81no—
153C4—82no—
154C4—81no—
155C4—81no—
156C4#45612yes—
157C2—22no—
158C2#200511yescharacteristic
159C2#200811yescharacteristic
160C2#202011yescharacteristic
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
#32362
#32092
C2×#5012
#15562
#15312
#14702
#13022