Finite Group Databasebeta

← #10505

Subgroups — #10505

Order 256.

201 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 201 conjugacy classes of subgroups — too many to draw. Only its 57 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#1997C211yesmaximal, characteristic
3#1986C211yesmaximal, characteristic
4#1981C211yesmaximal, characteristic
5#1565C211yesmaximal, characteristic
6#1550C211yesmaximal, characteristic
7#1303C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#494—21no—
11#487—21no—
12#462—21no—
13#462—21no—
14#451—22no—
15#431C2211yescharacteristic
16#423C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18C2×#103—22no—
19C4×M16C2211yescharacteristic
20C4×C4⋊C4C2211yescharacteristic
21#343—21no—
22#343—21no—
23#339—21no—
24#339—21no—
25#335—21no—
26#334—21no—
27#333—21no—
28#332—21no—
29C22×Q8D412yes—
30C2×Q16—22no—
31C2×Q16—22no—
32C2×M16—21no—
33C2×M16D411yescharacteristic
34C2×M16D411yescharacteristic
35C22×C8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#125—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47#117—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8D411yescharacteristic
58C4⋊C8D411yescharacteristic
59#103—22no—
60#103—22no—
61#103—44no—
62C22⋊C8—21no—
63#97—21no—
64C4×C8—21no—
65C4×C8—21no—
66#95—21no—
67C2×Q8—22no—
68C2×Q8—21no—
69C2×Q8—22no—
70C2×Q8—21no—
71C2×Q8—42no—
72C2×Q8—42no—
73C22×C4—22no—
74C22×C4—41no—
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77C22×C4#4111yescharacteristic
78Q16—82no—
79Q16—82no—
80M16—21no—
81M16—21no—
82M16—21no—
83M16—41no—
84M16—41no—
85M16Q1612yescomplemented
86C2×C8—21no—
87C2×C8—21no—
88C2×C8—21no—
89C2×C8—21no—
90C2×C8—42no—
91C2×C8—41no—
92C2×C8—41no—
93C4⋊C4—22no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—21no—
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—
107C42—41no—
108C42—41no—
109C42C2×D411yescharacteristic
110C42C2×D411yescharacteristic
111C42#4111yescharacteristic
112C42#4111yescharacteristic
113C23#12111yescharacteristic
114Q8—44no—
115Q8—41no—
116Q8—42no—
117Q8—41no—
118Q8—42no—
119Q8—81no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—81no—
128C2×C4—81no—
129C2×C4—82no—
130C2×C4#12011yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12111yescharacteristic
133C2×C4#12311yescharacteristic
134C2×C4#12311yescharacteristic
135C2×C4#12411yescharacteristic
136C2×C4C2×Q1611yescharacteristic
137C2×C4#13511yescharacteristic
138C2×C4#13711yescharacteristic
139C2×C4#13711yescharacteristic
140C8—22no—
141C8—41no—
142C8—41no—
143C8—41no—
144C8—81no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22#39411yescharacteristic
149C22#46211yescharacteristic
150C22#46411yescharacteristic
151C4—21no—
152C4—21no—
153C4—21no—
154C4—82no—
155C4—81no—
156C4—82no—
157C4—81no—
158C4#45311yescharacteristic
159C4#45511yescharacteristic
160C4#46211yescharacteristic
161C4#46411yescharacteristic
162C4#46711yescharacteristic
163C4#47011yescharacteristic
164C2—21no—
165C2—21no—
166C2#203711yescharacteristic
167C2#204911yescharacteristic
168C2#205611yescharacteristic
subgroupG/H
center, socleC22#394
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
#19972
#19862
#19812
#15652
#15502
#13032