Finite Group Databasebeta

← #10155

Subgroups — #10155

Order 256.

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

normal subgroups only

This group has 276 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×#494C211yesmaximal, characteristic
4#1997C211yesmaximal, characteristic
5#1549C211yesmaximal, characteristic
6#1493C211yesmaximal, characteristic
7#1278C211yesmaximal, characteristic
8C22×Q16—22no—
9#533C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12#501—22no—
13#500—21no—
14#498—21no—
15#494—22no—
16#494—21no—
17#494—22no—
18#487—21no—
19#462—22no—
20#451—22no—
21#431C2211yescharacteristic
22#428—22no—
23#423C2211yescharacteristic
24C4×M16C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#343—22no—
27#332—22no—
28Q8⋊C8—22no—
29C22×Q8—21no—
30C22×Q8D412yes—
31C2×Q16—28no—
32C2×Q16—24no—
33C2×Q16—48no—
34C2×M16—21no—
35C2×M16D412yes—
36C22×C8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8D412yes—
45C4⋊Q8D411yescharacteristic
46C4⋊Q8D411yescharacteristic
47#125—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52#117—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C42C2311yescharacteristic
58#107—24no—
59#106—24no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8D412yes—
63#103—24no—
64C22⋊C8—21no—
65#97—21no—
66C4×C8—21no—
67C4×C8—21no—
68C4×C8—21no—
69C4×C8D412yes—
70C2×Q8—22no—
71C2×Q8—21no—
72C2×Q8—22no—
73C2×Q8—22no—
74C2×Q8—21no—
75C2×Q8—44no—
76C2×Q8—44no—
77C2×Q8—41no—
78C22×C4—22no—
79C22×C4—41no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4#4111yescharacteristic
83Q16—416no—
84Q16—84no—
85M16—22no—
86M16—42no—
87M16Q1614yescomplemented
88C2×C8—24no—
89C2×C8—22no—
90C2×C8—22no—
91C2×C8—22no—
92C2×C8—41no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—22no—
96C4⋊C4—21no—
97C4⋊C4—22no—
98C4⋊C4—21no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C22⋊C4—41no—
108C42—41no—
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42C2×D411yescharacteristic
112C42C2×D411yescharacteristic
113C23#12111yescharacteristic
114Q8—44no—
115Q8—44no—
116Q8—42no—
117Q8—44no—
118Q8—42no—
119Q8—82no—
120C2×C4—22no—
121C2×C4—22no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—82no—
127C2×C4—82no—
128C2×C4#12011yescharacteristic
129C2×C4#12011yescharacteristic
130C2×C4#12011yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12711yescharacteristic
134C2×C4C2×Q1612yes—
135C2×C4#13712yes—
136C8—24no—
137C8—24no—
138C8—42no—
139C8—81no—
140C22—21no—
141C22—21no—
142C22#39211yescharacteristic
143C22#46212yes—
144C4—21no—
145C4—21no—
146C4—82no—
147C4—82no—
148C4—81no—
149C4—81no—
150C4#45112yes—
151C4#45312yes—
152C4#47312yes—
153C4#49411yescharacteristic
154C4#49711yescharacteristic
155C2—22no—
156C2#200211yescharacteristic
157C2#201611yescharacteristic
158C2#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

Complements

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

NH ≅ G/Ntype
87. M16Q16semidirect

Maximal subgroups

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

subgroupindex
#32092
#31652
C2×#4942
#19972
#15492
#14932
#12782