Finite Group Databasebeta

← #7671

Subgroups — #7671

Order 256.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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#2851C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3C4×C22⋊C8C211yesmaximal, characteristic
4#1271C212yesmaximal, complemented
5#1271C212yesmaximal, complemented
6C2×#125C411yescharacteristic
7C22×C42C2211yescharacteristic
8C2×C22⋊C8—21no—
9C2×C22⋊C8—21no—
10C2×C4×C8—21no—
11#389C2211yescharacteristic
12#389C2211yescharacteristic
13#387—21no—
14#387—21no—
15#385—21no—
16#385—21no—
17#382C411yescharacteristic
18#380C411yescharacteristic
19#380C411yescharacteristic
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C23×C4C2×C411yescharacteristic
24C23×C4D411yescharacteristic
25C23×C4D411yescharacteristic
26C22×C8—22no—
27C22×C8—22no—
28#125—42no—
29C2×C4⋊C4—21no—
30C2×C4⋊C4—21no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4C2×C411yescharacteristic
34C2×C4⋊C4C2×C411yescharacteristic
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C42—42no—
40C2×C42C2×C411yescharacteristic
41C2×C42D411yescharacteristic
42C2×C42D411yescharacteristic
43C2×C42C2311yescharacteristic
44C22⋊C8—24no—
45C22⋊C8—42no—
46C4×C8—42no—
47#95—21no—
48#95—21no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53C24C22⋊C411yescharacteristic
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4C22⋊C411yescharacteristic
68C22×C4C22⋊C411yescharacteristic
69C22×C4C22⋊C411yescharacteristic
70C22×C4C22×C411yescharacteristic
71C22×C4C2×D411yescharacteristic
72C22×C4C2×D411yescharacteristic
73C2×C8—24no—
74C2×C8—42no—
75C2×C8—42no—
76C2×C8—42no—
77C2×C8—42no—
78C2×C8—42no—
79C4⋊C4—22no—
80C4⋊C4—22no—
81C4⋊C4—41no—
82C4⋊C4—41no—
83C4⋊C4—42no—
84C4⋊C4—42no—
85C4⋊C4—42no—
86C4⋊C4—42no—
87C22⋊C4—42no—
88C22⋊C4—42no—
89C42—22no—
90C42—22no—
91C42—41no—
92C42—41no—
93C23—21no—
94C23—21no—
95C23—21no—
96C23—21no—
97C23—21no—
98C23—21no—
99C23#10412yes—
100C23C2×C22⋊C411yescharacteristic
101C2×C4—21no—
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—21no—
107C2×C4—21no—
108C2×C4—21no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—44no—
115C2×C4—44no—
116C2×C4—44no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—41no—
120C2×C4—41no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4#9912yes—
130C2×C4#10112yes—
131C2×C4#13112yes—
132C2×C4#13512yes—
133C2×C4#13512yes—
134C8—44no—
135C8—82no—
136C22—22no—
137C22—22no—
138C22—22no—
139C22—41no—
140C22—41no—
141C22—41no—
142C22—41no—
143C22#40811yescharacteristic
144C22C2×#9911yescharacteristic
145C22C2×#10111yescharacteristic
146C22#41611yescharacteristic
147C22#41611yescharacteristic
148C22C2×#10411yescharacteristic
149C22#42111yescharacteristic
150C4—22no—
151C4—22no—
152C4—22no—
153C4—22no—
154C4—22no—
155C4—41no—
156C4—81no—
157C4—81no—
158C4—81no—
159C4—81no—
160C2—42no—
161C2#147211yescharacteristic
162C2#148211yescharacteristic
163C2#149411yescharacteristic
164C2#150111yescharacteristic
165C2#151611yescharacteristic
166C2#152811yescharacteristic
167C2#154411yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1271C2semidirect
5. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#18172
C4×C22⋊C82
#12712