Finite Group Databasebeta

← #13077

Subgroups — #13077

Order 256.

235 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 73 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#3407C211yesmaximal, characteristic, complemented
2#3060C211yesmaximal, characteristic
3#2145C212yesmaximal, complemented
4#1812C211yesmaximal, characteristic, complemented
5#1386C212yesmaximal, complemented
6C2×#135C2211yescharacteristic
7C2×#131C2211yescharacteristic
8#537—21no—
9#504—24no—
10C2×M32—22no—
11C2×M32C2212yes—
12#470C412yescomplemented
13#469C412yescomplemented
14#465—22no—
15#427C2212yes—
16C2×#106C2211yescharacteristic
17#417—21no—
18#410—21no—
19#360—24no—
20H(3,ℤ/4)—21no—
21C2×#41—21no—
22C2×#41D411yescharacteristic
23#135—22no—
24#135—22no—
25C2×Q16C2×C411yescharacteristic
26C2×SD16—21no—
27C2×D8C2×C411yescharacteristic
28#131—22no—
29#131—22no—
30C2×M16C2×C412yes—
31C2×M16D411yescharacteristic
32C22×C8D412yes—
33C22×C8C2311yescharacteristic
34C22⋊Q8—22no—
35C22⋊Q8—41no—
36#121—22no—
37#121—42no—
38#121—41no—
39#117—22no—
40C2×C4⋊C4—21no—
41M32—24no—
42M32—22no—
43M32D414yescomplemented
44C2×C16—24no—
45C2×C16—22no—
46#107—22no—
47#106—21no—
48#106—22no—
49#106C2×C412yes—
50#103—22no—
51#102—22no—
52#99—42no—
53#41—22no—
54#41—22no—
55#41—42no—
56#41—42no—
57C2×Q8—21no—
58C2×Q8SD1611yescharacteristic
59C2×D4—21no—
60C2×D4—41no—
61C2×D4—42no—
62C2×D4D812yes—
63C2×D4SD1611yescharacteristic
64C22×C4—41no—
65C22×C4—41no—
66C22×C4C22⋊C412yes—
67C22×C4C2×D411yescharacteristic
68Q16—22no—
69SD16—41no—
70SD16—41no—
71D8—22no—
72M16—22no—
73M16—24no—
74C2×C8—22no—
75C2×C8—22no—
76C2×C8C22⋊C412yes—
77C2×C8C22×C411yescharacteristic
78C2×C8C2×D412yes—
79C2×C8C2×D411yescharacteristic
80C2×C8#4112yes—
81C4⋊C4—22no—
82C4⋊C4—42no—
83C4⋊C4—41no—
84C4⋊C4—42no—
85C22⋊C4—42no—
86C22⋊C4—42no—
87C22⋊C4—42no—
88C42—42no—
89C16—24no—
90C23—41no—
91C23#10212yes—
92C23C2×D811yescharacteristic
93Q8—22no—
94Q8—41no—
95Q8—41no—
96D4—24no—
97D4—22no—
98D4—41no—
99D4—41no—
100D4—82no—
101D4—162no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—42no—
105C2×C4—41no—
106C2×C4—82no—
107C2×C4—82no—
108C2×C4—161no—
109C2×C4#10212yes—
110C2×C4C2×C22⋊C411yescharacteristic
111C2×C4#12112yes—
112C2×C4C2×SD1611yescharacteristic
113C2×C4#13512yes—
114C8—22no—
115C8C4×D412yes—
116C8#12011yescharacteristic
117C8#12311yescharacteristic
118C22—22no—
119C22—41no—
120C22—42no—
121C22—81no—
122C22—81no—
123C22—81no—
124C22C2×#10211yescharacteristic
125C22#46612yes—
126C4—22no—
127C4—82no—
128C4—81no—
129C4—162no—
130C4#38611yescharacteristic
131C4#41611yescharacteristic
132C4#46512yes—
133C2—21no—
134C2—22no—
135C2—42no—
136C2—161no—
137C2#189311yescharacteristic
subgroupG/H
center, socleC2#1893
commutatorC2×C8C22×C4
Frattini, cosocleC22×C8C23
prime radical#3060C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3407C2semidirect
3. #2145C2semidirect
4. #1812C2semidirect
5. #1386C2semidirect
12. #470C4semidirect
13. #469C4semidirect
43. M32D4semidirect

Maximal subgroups

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

subgroupindex
#34072
#30602
#21452
#18122
#13862