Finite Group Databasebeta

← #8537

Subgroups — #8537

Order 256.

397 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 77 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#3501C211yesmaximal, characteristic
2#1901C212yesmaximal, complemented
3#1870C211yesmaximal, characteristic
4#1472C212yesmaximal, complemented
5#1465C211yesmaximal, characteristic
6C22×M16C2211yescharacteristic
7#554—24no—
8#554—22no—
9D4×Q8—22no—
10#544—21no—
11#543—21no—
12#543C412yes—
13#533—21no—
14C2×C22⋊Q8—21no—
15C2×C22⋊Q8C2212yes—
16#518C412yes—
17C2×C4×D4C2211yescharacteristic
18C2×#101—22no—
19#407C2212yes—
20C2×C22⋊C8—21no—
21C2×C22⋊C8C2211yescharacteristic
22#343—22no—
23#336—21no—
24#324—21no—
25#324—24no—
26#323—21no—
27C22×Q8—21no—
28C22×Q8D412yes—
29C22×D4C2×C411yescharacteristic
30C23×C4D411yescharacteristic
31C23×C4C2311yescharacteristic
32C2×M16—22no—
33C2×M16—22no—
34C2×M16—42no—
35C22×C8—21no—
36C22×C8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—24no—
40#125—22no—
41#125—24no—
42C22⋊Q8—22no—
43C22⋊Q8—22no—
44C22⋊Q8—24no—
45C22⋊Q8—24no—
46C22⋊Q8—22no—
47C22⋊Q8—22no—
48C22⋊Q8—24no—
49C22⋊Q8—24no—
50C22⋊Q8—22no—
51C22⋊Q8—22no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×D4—24no—
57C4×D4—24no—
58C4×D4—22no—
59C4×D4—22no—
60#117—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D414yes—
65C2×C4⋊C4D412yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4C2×C411yescharacteristic
68C2×C22⋊C4C2×C414yes—
69C2×C22⋊C4D411yescharacteristic
70C2×C42D411yescharacteristic
71#101—44no—
72C22⋊C8—22no—
73C22⋊C8—22no—
74C22⋊C8—22no—
75C22⋊C8—42no—
76C24C22⋊C411yescharacteristic
77C24C22×C411yescharacteristic
78C2×Q8—22no—
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8—42no—
83C2×Q8—42no—
84C2×Q8—81no—
85C2×Q8—81no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4C22⋊C411yescharacteristic
98C22×C4C22⋊C411yescharacteristic
99C22×C4C22⋊C414yes—
100C22×C4C22⋊C414yes—
101C22×C4C22⋊C411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106M16—24no—
107M16—42no—
108M16—42no—
109M16—42no—
110C2×C8—22no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—42no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—24no—
122C4⋊C4—22no—
123C4⋊C4—24no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—24no—
128C22⋊C4—24no—
129C22⋊C4—24no—
130C22⋊C4—22no—
131C22⋊C4—24no—
132C22⋊C4—22no—
133C42—24no—
134C42—21no—
135C42—21no—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23#10112yescomplemented
145C23C2×C22⋊C412yes—
146C23C2×C22⋊C411yescharacteristic
147Q8—42no—
148Q8—42no—
149Q8—82no—
150Q8—82no—
151Q8—82no—
152D4—28no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—21no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—44no—
169C2×C4—41no—
170C2×C4—44no—
171C2×C4—81no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4#10112yescomplemented
175C2×C4#12011yescharacteristic
176C2×C4#12012yes—
177C2×C4#12011yescharacteristic
178C8—42no—
179C8—81no—
180C8—82no—
181C22—21no—
182C22—21no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—42no—
188C22—41no—
189C22—41no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22#37911yescharacteristic
194C22C2×#10111yescharacteristic
195C22C2×#10111yescharacteristic
196C4—22no—
197C4—21no—
198C4—22no—
199C4—41no—
200C4—44no—
201C4—44no—
202C4—41no—
203C4B(4,𝔽2)11yescharacteristic
204C4#45811yescharacteristic
205C2—21no—
206C2—21no—
207C2—42no—
208C2—42no—
209C2#178411yescharacteristic
210C2#179011yescharacteristic
211C2#179111yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1901C2semidirect
4. #1472C2semidirect
144. C23#101semidirect
174. C2×C4#101semidirect

Maximal subgroups

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

subgroupindex
#35012
#19012
#18702
#14722
#14652