Finite Group Databasebeta

← #13608

Subgroups — #13608

Order 256.

154 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 43 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#3223C211yesmaximal, characteristic
2#2133C211yesmaximal, characteristic
3#2129C211yesmaximal, characteristic
4#1414C212yesmaximal, complemented
5#1407C212yesmaximal, complemented
6#541C411yescharacteristic, complemented
7#536C2211yescharacteristic
8#497C2211yescharacteristic
9#494C411yescharacteristic, complemented
10#491C411yescharacteristic, complemented
11#488C2211yescharacteristic
12#464—21no—
13#462—21no—
14#452—21no—
15#451—21no—
16#433C411yescharacteristic, complemented
17C2×#101—21no—
18#410—21no—
19#356—21no—
20#355C2212yes—
21#354—21no—
22#354C2212yes—
23C2×#41D411yescharacteristic
24C22×Q8D411yescharacteristic
25C2×Q16—21no—
26C2×Q16—21no—
27C2×Q16—21no—
28C2×SD16—21no—
29C2×M16—41no—
30C4⋊Q8C2×C411yescharacteristic
31C4⋊Q8C2×C411yescharacteristic
32C4⋊Q8D411yescharacteristic
33#126—41no—
34#125—41no—
35#125C2×C411yescharacteristic
36#124D411yescharacteristic
37#124C2311yescharacteristic
38#123—41no—
39C22⋊Q8—21no—
40C22⋊Q8—41no—
41C4×Q8—21no—
42#117—41no—
43#117C2×C411yescharacteristic
44C4⋊C8—21no—
45#103—21no—
46#103—21no—
47#103—21no—
48#102—21no—
49#101—22no—
50#101—41no—
51#99—22no—
52#99—41no—
53C22⋊C8—21no—
54#97C2×C411yescharacteristic
55C4×C8C2×C411yescharacteristic
56#41—81no—
57C2×Q8—21no—
58C2×Q8—21no—
59C2×Q8—81no—
60C2×Q8C22⋊C411yescharacteristic
61C2×Q8C22⋊C411yescharacteristic
62C2×Q8C2×D411yescharacteristic
63C2×D4—21no—
64C2×D4C2×D411yescharacteristic
65C22×C4—21no—
66C22×C4—21no—
67C22×C4C22⋊C411yescharacteristic
68Q16—81no—
69Q16—81no—
70Q16—81no—
71SD16—81no—
72M16—42no—
73M16—81no—
74C2×C8—21no—
75C2×C8—21no—
76C2×C8—81no—
77C4⋊C4—21no—
78C4⋊C4—21no—
79C4⋊C4—41no—
80C4⋊C4—41no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—41no—
84C4⋊C4—41no—
85C4⋊C4—81no—
86C22⋊C4—21no—
87C22⋊C4—41no—
88C22⋊C4—41no—
89C22⋊C4—41no—
90C22⋊C4—42no—
91C42—41no—
92C42—81no—
93C42C22⋊C411yescharacteristic
94C42C22×C411yescharacteristic
95C23—21no—
96C23#9911yescharacteristic
97Q8—41no—
98Q8—41no—
99Q8—41no—
100Q8—41no—
101Q8—41no—
102Q8—81no—
103D4—41no—
104D4—81no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—41no—
111C2×C4—41no—
112C2×C4—41no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—82no—
118C2×C4#9911yescharacteristic
119C2×C4C2×C22⋊C411yescharacteristic
120C8—41no—
121C8—41no—
122C8—82no—
123C22—21no—
124C22—41no—
125C22—41no—
126C22—81no—
127C22C2×#9911yescharacteristic
128C4—21no—
129C4—21no—
130C4—41no—
131C4—41no—
132C4—41no—
133C4—41no—
134C4—41no—
135C4—81no—
136C4—81no—
137C4—162no—
138C2—21no—
139C2—41no—
140C2—81no—
141C2#212711yescharacteristic
subgroupG/H
center, socleC2#2127
commutatorC42C22×C4
Frattini, cosocle#124C23
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ G
LECSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1414C2semidirect
5. #1407C2semidirect
6. #541C4semidirect
9. #494C4semidirect
10. #491C4semidirect
16. #433C4semidirect

Maximal subgroups

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

subgroupindex
#32232
#21332
#21292
#14142
#14072