Finite Group Databasebeta

← #9296

Subgroups — #9296

Order 256.

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

normal subgroups only

This group has 235 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#3229C211yesmaximal, characteristic
2#2327C211yesmaximal, characteristic
3#1995C211yesmaximal, characteristic
4#1546C211yesmaximal, characteristic
5#1518C211yesmaximal, characteristic
6#1286C211yesmaximal, characteristic
7#1274C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4×Q8—21no—
11#498C412yescomplemented
12#487C412yescomplemented
13#483—22no—
14#475—22no—
15#431C2211yescharacteristic
16#431C2211yescharacteristic
17#423C2211yescharacteristic
18#391—21no—
19#387—22no—
20#387—22no—
21#382C2211yescharacteristic
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—
40C4×Q8—42no—
41C4×Q8—42no—
42#117—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4D412yes—
47C2×C42—22no—
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51#106—24no—
52C4⋊C8—21no—
53C4⋊C8—21no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56#103—24no—
57C22⋊C8—21no—
58C22⋊C8—21no—
59C4×C8—21no—
60C4×C8C2×C412yes—
61#95—24no—
62#95—21no—
63#95—21no—
64#95—21no—
65#95D412yes—
66C2×Q8—22no—
67C2×Q8—22no—
68C2×Q8—21no—
69C2×Q8—21no—
70C2×Q8—41no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×Q8—41no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—22no—
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81M16—41no—
82M16—41no—
83C2×C8—22no—
84C2×C8—41no—
85C2×C8—41no—
86C4⋊C4—22no—
87C4⋊C4—22no—
88C4⋊C4—22no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—21no—
92C4⋊C4—21no—
93C4⋊C4—44no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—41no—
98C4⋊C4C22⋊C414yes—
99C22⋊C4—41no—
100C42—21no—
101C42—21no—
102C42—21no—
103C42—44no—
104C42—41no—
105C42SD1612yes—
106C42C22×C411yescharacteristic
107C42C2×D411yescharacteristic
108C42#4111yescharacteristic
109C42#4111yescharacteristic
110C23#12011yescharacteristic
111Q8—42no—
112Q8—42no—
113Q8—42no—
114Q8—42no—
115Q8—42no—
116Q8—81no—
117C2×C4—24no—
118C2×C4—22no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4C2×C22⋊C411yescharacteristic
134C2×C4C4×D411yescharacteristic
135C2×C4C4×D411yescharacteristic
136C2×C4#12111yescharacteristic
137C2×C4#12111yescharacteristic
138C2×C4#12311yescharacteristic
139C2×C4C2×SD1611yescharacteristic
140C2×C4#13511yescharacteristic
141C2×C4AGL(1,ℤ/8)11yescharacteristic
142C2×C4#13711yescharacteristic
143C8—42no—
144C8—81no—
145C8—81no—
146C22—21no—
147C22—21no—
148C22#38611yescharacteristic
149C22#45011yescharacteristic
150C22#45211yescharacteristic
151C4—21no—
152C4—21no—
153C4—21no—
154C4—21no—
155C4—42no—
156C4—44no—
157C4—82no—
158C4—81no—
159C4—81no—
160C4—81no—
161C4C4×SD1611yescharacteristic
162C4#44011yescharacteristic
163C4#46111yescharacteristic
164C4#46311yescharacteristic
165C2—21no—
166C2—21no—
167C2#187711yescharacteristic
168C2#190511yescharacteristic
169C2#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
11. #498C4semidirect
12. #487C4semidirect

Maximal subgroups

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

subgroupindex
#32292
#23272
#19952
#15462
#15182
#12862
#12742