Finite Group Databasebeta

← #11898

Subgroups — #11898

Order 256.

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

normal subgroups only

This group has 290 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#2409C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1460C211yesmaximal, characteristic
4#1351C214yesmaximal, complemented
5C23×Q8C411yescharacteristic, complemented
6C2×C22⋊Q8—21no—
7C2×C4×D4C2211yescharacteristic
8#431—21no—
9C2×#99—21no—
10#395C412yescomplemented
11#393C2212yes—
12#391—21no—
13#386—21no—
14#385—21no—
15#380—21no—
16#380C411yescharacteristic, complemented
17#342C2212yes—
18#323—21no—
19#323C2212yes—
20C22×Q8—21no—
21C22×Q8—41no—
22C22×Q8—41no—
23C22×Q8—44no—
24C22×Q8C2×C412yes—
25C22×D4D411yescharacteristic
26C23×C4—21no—
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C22⋊Q8—44no—
30C4×D4—41no—
31C4×D4—41no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—41no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4D411yescharacteristic
40C2×C22⋊C4C2311yescharacteristic
41C2×C42D411yescharacteristic
42C4⋊C8—41no—
43#99—42no—
44C22⋊C8—22no—
45C4×C8—41no—
46#95—22no—
47#95—22no—
48#95—41no—
49#95C2×C412yes—
50C24C22⋊C411yescharacteristic
51C24C2×D411yescharacteristic
52C2×Q8—24no—
53C2×Q8—44no—
54C2×Q8—42no—
55C2×Q8—42no—
56C2×Q8—44no—
57C2×Q8—48no—
58C2×Q8—44no—
59C2×Q8—41no—
60C2×Q8—41no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4SD1612yes—
64C2×D4Q1612yes—
65C22×C4—22no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—22no—
71C22×C4—44no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C2×C8—42no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C4⋊C4—44no—
85C4⋊C4—82no—
86C22⋊C4—41no—
87C22⋊C4—41no—
88C22⋊C4—44no—
89C22⋊C4—42no—
90C22⋊C4—42no—
91C22⋊C4—42no—
92C42—21no—
93C42—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—41no—
99C23—41no—
100C23—41no—
101C23—41no—
102C23#10312yes—
103C23C2×C22⋊C411yescharacteristic
104C23C2×SD1611yescharacteristic
105C23C2×Q1611yescharacteristic
106Q8—44no—
107Q8—44no—
108Q8—44no—
109Q8—44no—
110D4—42no—
111D4—42no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—22no—
115C2×C4—24no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—48no—
122C2×C4—42no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—81no—
132C2×C4—82no—
133C2×C4—82no—
134C2×C4—82no—
135C2×C4#9911yescharacteristic
136C2×C4#9911yescharacteristic
137C2×C4#10312yes—
138C8—82no—
139C22—21no—
140C22—21no—
141C22—21no—
142C22—22no—
143C22—21no—
144C22—42no—
145C22—41no—
146C22—41no—
147C22—42no—
148C22—41no—
149C22—41no—
150C22—41no—
151C22—41no—
152C22—81no—
153C22C2×#9911yescharacteristic
154C22C2×#10311yescharacteristic
155C22#42111yescharacteristic
156C4—21no—
157C4—22no—
158C4—44no—
159C4—41no—
160C4—41no—
161C4—41no—
162C4—82no—
163C4—162no—
164C4AΣL(1,𝔽16)12yes—
165C2—21no—
166C2—21no—
167C2—41no—
168C2—42no—
169C2—41no—
170C2#152011yescharacteristic
171C2C2×AΣL(1,𝔽16)11yescharacteristic
172C2#212911yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1351C2semidirect
5. C23×Q8C4semidirect
10. #395C4semidirect
16. #380C4semidirect

Maximal subgroups

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

subgroupindex
#24092
#17992
#14602
#13512