Finite Group Databasebeta

← #8524

Subgroups — #8524

Order 256.

455 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#536C211yesmaximal, characteristic
2#1558C212yesmaximal, complemented
3#1533C212yesmaximal, complemented
4#1464C212yesmaximal, complemented
5C22×#41C411yescharacteristic
6C23×Q8C411yescharacteristic
7#536—24no—
8#536—24no—
9C2×C4⋊Q8C411yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#124C411yescharacteristic
12C2×#124C2211yescharacteristic
13C2×#123—22no—
14C2×C22⋊Q8—22no—
15C2×#117C2211yescharacteristic
16#432C2214yes—
17#333—24no—
18#329—24no—
19#324—22no—
20#323—22no—
21C2×#41—22no—
22C2×#41—22no—
23C2×#41—42no—
24C2×#41—42no—
25C22×Q8—22no—
26C22×Q8—22no—
27C22×Q8—21no—
28C22×Q8—42no—
29C22×Q8—42no—
30C22×Q8C2×C411yescharacteristic
31C22×Q8C2×C411yescharacteristic
32C22×Q8C2×C411yescharacteristic
33C22×D4—21no—
34C22×D4C2×C411yescharacteristic
35C23×C4—21no—
36C23×C4—21no—
37C23×C4C2×C411yescharacteristic
38C4⋊Q8—22no—
39C4⋊Q8—22no—
40C4⋊Q8—22no—
41C4⋊Q8D414yes—
42#124—22no—
43#124—22no—
44#124—22no—
45#124D414yes—
46#123—216no—
47C22⋊Q8—216no—
48#117—22no—
49#117—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4D412yes—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C22⋊C4D412yes—
56C2×C42C2×C411yescharacteristic
57C2×C42C2311yescharacteristic
58C4⋊C8—24no—
59C22⋊C8—24no—
60#97—24no—
61C24—21no—
62C24C22⋊C411yescharacteristic
63#41—48no—
64#41—48no—
65C2×Q8—24no—
66C2×Q8—21no—
67C2×Q8—21no—
68C2×Q8—21no—
69C2×Q8—48no—
70C2×Q8—48no—
71C2×Q8—42no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8—42no—
79C2×Q8C22⋊C412yes—
80C2×Q8C22⋊C412yes—
81C2×Q8C22⋊C412yes—
82C2×D4—24no—
83C2×D4—21no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—42no—
87C2×D4—42no—
88C2×D4C22⋊C412yes—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—24no—
94C22×C4—24no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—42no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4C22⋊C411yescharacteristic
103C22×C4C22⋊C412yes—
104C22×C4C22×C411yescharacteristic
105C22×C4C2×D412yes—
106C2×C8—44no—
107C4⋊C4—28no—
108C4⋊C4—28no—
109C4⋊C4—24no—
110C22⋊C4—28no—
111C22⋊C4—28no—
112C22⋊C4—24no—
113C42—21no—
114C42—21no—
115C42C2×D412yes—
116C42C2×D412yes—
117C23—21no—
118C23—21no—
119C23—22no—
120C23—42no—
121C23—44no—
122C23—41no—
123C23—41no—
124C23#10012yes—
125C23C2×C22⋊C411yescharacteristic
126Q8—42no—
127Q8—42no—
128Q8—42no—
129Q8—48no—
130Q8—42no—
131Q8—42no—
132Q8—42no—
133D4—42no—
134D4—42no—
135D4—48no—
136C2×C4—24no—
137C2×C4—24no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—42no—
147C2×C4—48no—
148C2×C4—48no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4#10012yes—
157C2×C4C2×C22⋊C411yescharacteristic
158C2×C4C2×C22⋊C411yescharacteristic
159C2×C4#12014yes—
160C8—84no—
161C22—22no—
162C22—22no—
163C22—42no—
164C22—42no—
165C22—42no—
166C22—44no—
167C22—42no—
168C22—82no—
169C22—82no—
170C22#37911yescharacteristic
171C22C2×#10011yescharacteristic
172C22C2×#10011yescharacteristic
173C4—24no—
174C4—22no—
175C4—21no—
176C4—21no—
177C4—42no—
178C4—42no—
179C4—42no—
180C4—42no—
181C2—21no—
182C2—21no—
183C2—42no—
184C2—82no—
185C2#178311yescharacteristic
186C2#179612yes—
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×#41C4
LCSG ▷ C22×C4 ▷ 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
2. #1558C2semidirect
3. #1533C2semidirect
4. #1464C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
#15582
#15332
#14642