Finite Group Databasebeta

← #8491

Subgroups — #8491

Order 256.

410 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 71 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#3549C211yesmaximal, characteristic
2#1963C211yesmaximal, characteristic, complemented
3#1557C212yesmaximal, complemented
4#1536C211yesmaximal, characteristic, complemented
5#1505C212yesmaximal, complemented
6H(3,𝔽4)—22no—
7#555—22no—
8#555C412yescomplemented
9D42—22no—
10D42C412yescomplemented
11#535—22no—
12#535—24no—
13#532C2211yescharacteristic
14#532C2212yes—
15#432C2212yescomplemented
16#423C2211yescharacteristic, complemented
17C2×#104—22no—
18#376C2211yescharacteristic, complemented
19#332—22no—
20#331—22no—
21#329—22no—
22#327—24no—
23C2×#41—22no—
24C2×#41D412yes—
25C22×D4—22no—
26C22×D4—24no—
27C2×M16—21no—
28C4⋊Q8C2×C411yescharacteristic
29C4⋊Q8D412yes—
30#127—24no—
31#127C2×C411yescharacteristic
32#127D412yes—
33#124—28no—
34#124—24no—
35#124—22no—
36#124D412yes—
37#121—24no—
38#121—24no—
39#121—22no—
40#121—22no—
41#121—24no—
42#121D414yes—
43#120—24no—
44#120—28no—
45#120—24no—
46C4×D4—22no—
47C4×D4—22no—
48C4×D4C2×C414yes—
49#117—22no—
50#117—24no—
51C2×C42—21no—
52C2×C42C2311yescharacteristic
53C4⋊C8—22no—
54C4⋊C8—22no—
55#104—44no—
56C22⋊C8—22no—
57#97—22no—
58#95—22no—
59C24—24no—
60#41—42no—
61#41—42no—
62#41—84no—
63C2×Q8—22no—
64C2×Q8C22⋊C412yes—
65C2×D4—24no—
66C2×D4—28no—
67C2×D4—22no—
68C2×D4—24no—
69C2×D4—22no—
70C2×D4—22no—
71C2×D4—22no—
72C2×D4—88no—
73C2×D4—82no—
74C2×D4—82no—
75C2×D4C22⋊C414yes—
76C2×D4C22⋊C412yes—
77C22×C4—24no—
78C22×C4—22no—
79C22×C4—41no—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82M16—42no—
83C2×C8—42no—
84C2×C8—41no—
85C4⋊C4—22no—
86C4⋊C4—22no—
87C4⋊C4C22⋊C414yes—
88C22⋊C4—28no—
89C22⋊C4—24no—
90C22⋊C4—24no—
91C22⋊C4—24no—
92C22⋊C4—24no—
93C42—24no—
94C42—22no—
95C42—41no—
96C42C22×C411yescharacteristic
97C42C2×D411yescharacteristic
98C42C2×D412yes—
99C23—22no—
100C23—24no—
101C23—84no—
102C23—88no—
103C23#12011yescharacteristic
104Q8—42no—
105Q8—82no—
106D4—44no—
107D4—42no—
108D4—44no—
109D4—42no—
110D4—42no—
111D4—82no—
112D4—82no—
113D4—84no—
114D4—84no—
115D4—88no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—24no—
120C2×C4—42no—
121C2×C4—48no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—82no—
126C2×C4C2×C22⋊C411yescharacteristic
127C2×C4C2×C22⋊C412yes—
128C2×C4#12012yes—
129C2×C4#12011yescharacteristic
130C8—82no—
131C8—81no—
132C22—21no—
133C22—22no—
134C22—42no—
135C22—44no—
136C22—44no—
137C22—44no—
138C22—44no—
139C22—168no—
140C22#37911yescharacteristic
141C4—21no—
142C4—22no—
143C4—22no—
144C4—42no—
145C4—44no—
146C4—82no—
147C4#45311yescharacteristic
148C4#45511yescharacteristic
149C2—41no—
150C2—82no—
151C2—84no—
152C2#179211yescharacteristic
153C2#179512yes—
subgroupG/H
center, socleC22#379
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#3549C2
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
2. #1963C2semidirect
3. #1557C2semidirect
4. #1536C2semidirect
5. #1505C2semidirect
8. #555C4semidirect
10. D42C4semidirect
15. #432C22semidirect
16. #423C22semidirect
18. #376C22semidirect

Maximal subgroups

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

subgroupindex
#35492
#19632
#15572
#15362
#15052