Finite Group Databasebeta

← #10163

Subgroups — #10163

Order 256.

244 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 65 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#3209C211yesmaximal, characteristic
2#3165C211yesmaximal, characteristic
3C2×#487C211yesmaximal, characteristic
4#1997C211yesmaximal, characteristic
5#1548C211yesmaximal, characteristic
6#1493C211yesmaximal, characteristic
7#1278C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#501—22no—
12#500—21no—
13#498—21no—
14#494—21no—
15#487—22no—
16#487—21no—
17#487—22no—
18#462—22no—
19#451—22no—
20#431C2211yescharacteristic
21#428—22no—
22#423C2211yescharacteristic
23C2×#103—22no—
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#333—22no—
28Q8⋊C8—22no—
29C22×Q8—21no—
30C22×Q8D412yes—
31C2×Q16—24no—
32C2×M16—21no—
33C2×M16D412yes—
34C22×C8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8D412yes—
43C4⋊Q8D411yescharacteristic
44C4⋊Q8D411yescharacteristic
45#125—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50#117—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C42C2311yescharacteristic
56#107—24no—
57#106—24no—
58C4⋊C8—21no—
59C4⋊C8—21no—
60C4⋊C8D412yes—
61#103—28no—
62#103—24no—
63C22⋊C8—21no—
64#97—21no—
65C4×C8—21no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8D412yes—
69C2×Q8—22no—
70C2×Q8—22no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—22no—
74C2×Q8—44no—
75C2×Q8—41no—
76C2×Q8—82no—
77C22×C4—22no—
78C22×C4—41no—
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4#4111yescharacteristic
82Q16—84no—
83M16—22no—
84M16—42no—
85M16Q1614yescomplemented
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—22no—
89C2×C8—42no—
90C2×C8—41no—
91C4⋊C4—22no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—22no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C22⋊C4—41no—
106C42—41no—
107C42C2×D411yescharacteristic
108C42C2×D411yescharacteristic
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C23#12111yescharacteristic
112Q8—44no—
113Q8—42no—
114Q8—44no—
115Q8—42no—
116Q8—82no—
117Q8—82no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—82no—
125C2×C4—82no—
126C2×C4#12011yescharacteristic
127C2×C4#12011yescharacteristic
128C2×C4#12011yescharacteristic
129C2×C4#12111yescharacteristic
130C2×C4#12111yescharacteristic
131C2×C4#12711yescharacteristic
132C2×C4C2×Q1612yes—
133C2×C4#13712yes—
134C8—24no—
135C8—42no—
136C8—42no—
137C8—81no—
138C22—21no—
139C22—21no—
140C22#39211yescharacteristic
141C22#46212yes—
142C4—21no—
143C4—21no—
144C4—82no—
145C4—81no—
146C4—82no—
147C4—81no—
148C4#45112yes—
149C4#45512yes—
150C4#47112yes—
151C4#49411yescharacteristic
152C4#49711yescharacteristic
153C2—22no—
154C2#200411yescharacteristic
155C2#201611yescharacteristic
156C2#201911yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
85. M16Q16semidirect

Maximal subgroups

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

subgroupindex
#32092
#31652
C2×#4872
#19972
#15482
#14932
#12782