Finite Group Databasebeta

← #9298

Subgroups — #9298

Order 256.

221 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 221 conjugacy classes of subgroups — too many to draw. Only its 69 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#3230C211yesmaximal, characteristic
2#2331C211yesmaximal, characteristic
3#1995C211yesmaximal, characteristic
4#1549C211yesmaximal, characteristic
5#1518C211yesmaximal, characteristic
6#1286C211yesmaximal, characteristic
7#1274C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10#500C412yescomplemented
11#487C412yescomplemented
12#484—22no—
13#477—22no—
14#431C2211yescharacteristic
15#431C2211yescharacteristic
16#423C2211yescharacteristic
17#391—22no—
18#389—21no—
19#387—22no—
20#382C2211yescharacteristic
21C4×C4⋊C4—21no—
22#376C2211yescharacteristic
23#339—21no—
24#339—21no—
25#332—22no—
26#328—22no—
27C22×Q8—21no—
28C2×M16—21no—
29C4⋊Q8—21no—
30C4⋊Q8—21no—
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8C2×C411yescharacteristic
33C4⋊Q8C2×C412yes—
34C4⋊Q8D412yes—
35#125—21no—
36C22⋊Q8—21no—
37C22⋊Q8—21no—
38C4×Q8—21no—
39C4×Q8—21no—
40#117—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4D412yes—
45C2×C42—22no—
46C2×C42D411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42C2311yescharacteristic
49#107—24no—
50C4⋊C8—21no—
51C4⋊C8—21no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54#103—24no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57C4×C8—21no—
58C4×C8C2×C412yes—
59#95—22no—
60#95—22no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95D412yes—
65C2×Q8—22no—
66C2×Q8—22no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—82no—
70C22×C4—21no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4C2×D411yescharacteristic
75C22×C4C2×D411yescharacteristic
76C22×C4C2×D411yescharacteristic
77M16—41no—
78M16—41no—
79C2×C8—22no—
80C2×C8—41no—
81C2×C8—41no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—22no—
85C4⋊C4—21no—
86C4⋊C4—21no—
87C4⋊C4—21no—
88C4⋊C4—21no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—82no—
94C4⋊C4C22⋊C414yes—
95C22⋊C4—41no—
96C42—21no—
97C42—21no—
98C42—21no—
99C42—41no—
100C42—82no—
101C42Q1612yes—
102C42C22×C411yescharacteristic
103C42C2×D411yescharacteristic
104C42#4111yescharacteristic
105C42#4111yescharacteristic
106C23#12011yescharacteristic
107Q8—42no—
108Q8—82no—
109Q8—82no—
110Q8—81no—
111C2×C4—24no—
112C2×C4—22no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—41no—
119C2×C4—42no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—42no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4C2×C22⋊C411yescharacteristic
128C2×C4C4×D411yescharacteristic
129C2×C4C4×D411yescharacteristic
130C2×C4#12111yescharacteristic
131C2×C4#12111yescharacteristic
132C2×C4#12311yescharacteristic
133C2×C4C2×Q1611yescharacteristic
134C2×C4#13511yescharacteristic
135C2×C4#13711yescharacteristic
136C2×C4#13711yescharacteristic
137C8—42no—
138C8—81no—
139C8—81no—
140C22—21no—
141C22—21no—
142C22#38611yescharacteristic
143C22#45111yescharacteristic
144C22#45211yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—21no—
149C4—42no—
150C4—44no—
151C4—82no—
152C4—81no—
153C4—81no—
154C4—81no—
155C4C4×Q1611yescharacteristic
156C4#44111yescharacteristic
157C4#46211yescharacteristic
158C4#46411yescharacteristic
159C2—21no—
160C2—21no—
161C2#187811yescharacteristic
162C2#190411yescharacteristic
163C2#191211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#120
LCSG ▷ 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
10. #500C4semidirect
11. #487C4semidirect

Maximal subgroups

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

subgroupindex
#32302
#23312
#19952
#15492
#15182
#12862
#12742