Finite Group Databasebeta

← #9439

Subgroups — #9439

Order 256.

245 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 59 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#3140C211yesmaximal, characteristic
2#2372C211yesmaximal, characteristic
3#1533C211yesmaximal, characteristic
4#1506C212yesmaximal, complemented
5#1300C212yesmaximal, complemented
6#533C2211yescharacteristic
7C2×#124C2211yescharacteristic
8#489C414yescomplemented
9#488—22no—
10#432C2212yes—
11#419—21no—
12#418—21no—
13C4×M16C2211yescharacteristic
14#391—21no—
15#388—22no—
16#386—21no—
17C4×C22⋊C4—22no—
18#376C2212yes—
19#329—22no—
20#328—22no—
21#328—22no—
22C22×Q8D411yescharacteristic
23C22×D4D411yescharacteristic
24C23×C4—21no—
25C2×M16—21no—
26C4⋊Q8C2×C412yescomplemented
27#125—21no—
28#124—22no—
29#124C2×C412yes—
30#124D412yes—
31C22⋊Q8—22no—
32C4×Q8—22no—
33#117—21no—
34C2×C4⋊C4—21no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—21no—
38C2×C42—22no—
39C2×C42—21no—
40C2×C42C2311yescharacteristic
41C4⋊C8—22no—
42#103—24no—
43#102—24no—
44C22⋊C8—22no—
45#97—22no—
46#97C2×C412yescomplemented
47C4×C8—21no—
48#95—22no—
49#95—22no—
50#95D412yes—
51#95D412yes—
52C24—21no—
53C2×Q8—21no—
54C2×Q8—22no—
55C2×Q8—41no—
56C2×Q8—41no—
57C2×Q8C22⋊C412yes—
58C2×D4—21no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4C22⋊C412yes—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—22no—
65C22×C4—44no—
66C22×C4—81no—
67C22×C4C2×D411yescharacteristic
68C22×C4C2×D412yes—
69M16—41no—
70M16—41no—
71C2×C8—22no—
72C2×C8—42no—
73C4⋊C4—22no—
74C4⋊C4—22no—
75C4⋊C4—42no—
76C4⋊C4—81no—
77C22⋊C4—24no—
78C22⋊C4—24no—
79C22⋊C4—44no—
80C22⋊C4—41no—
81C22⋊C4—81no—
82C42—44no—
83C42—41no—
84C42—81no—
85C42C22×C411yescharacteristic
86C42C2×D411yescharacteristic
87C42#4112yes—
88C23—22no—
89C23—42no—
90C23—42no—
91C23—41no—
92C23—41no—
93C23#12011yescharacteristic
94Q8—42no—
95Q8—81no—
96Q8—82no—
97D4—42no—
98D4—81no—
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—24no—
102C2×C4—22no—
103C2×C4—42no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—41no—
107C2×C4—41no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—84no—
113C2×C4C2×C22⋊C411yescharacteristic
114C2×C4C4×D412yes—
115C2×C4#12112yes—
116C2×C4#12311yescharacteristic
117C8—42no—
118C8—82no—
119C22—22no—
120C22—42no—
121C22—44no—
122C22—81no—
123C22—81no—
124C22—82no—
125C22#38611yescharacteristic
126C22#45511yescharacteristic
127C22#45511yescharacteristic
128C4—21no—
129C4—21no—
130C4—22no—
131C4—22no—
132C4—44no—
133C4—42no—
134C4—82no—
135C4—82no—
136C2—21no—
137C2—21no—
138C2—82no—
139C2#188511yescharacteristic
140C2#190612yes—
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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
4. #1506C2semidirect
5. #1300C2semidirect
8. #489C4semidirect
26. C4⋊Q8C2×C4semidirect
46. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
#31402
#23722
#15332
#15062
#13002