Finite Group Databasebeta

← #8542

Subgroups — #8542

Order 256.

789 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 789 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#121C211yesmaximal, characteristic
2#1819C212yesmaximal, complemented
3#1781C212yesmaximal, complemented
4#1781C212yesmaximal, complemented
5C23×D4C411yescharacteristic, complemented
6C23×D4C2211yescharacteristic
7C23×D4C2211yescharacteristic
8C24×C4C411yescharacteristic, complemented
9C2×#121—24no—
10C2×#121—24no—
11C2×#121—24no—
12C22×C4⋊C4C411yescharacteristic, complemented
13C22×C22⋊C4C411yescharacteristic, complemented
14C22×C22⋊C4C2211yescharacteristic
15C2×#99—24no—
16C2×#99—24no—
17#379C2214yes—
18#342—24no—
19C25C2×C411yescharacteristic
20C25C2×C411yescharacteristic
21C25C2311yescharacteristic
22C22×D4—22no—
23C22×D4—22no—
24C22×D4—22no—
25C22×D4—22no—
26C22×D4—22no—
27C22×D4—22no—
28C22×D4—22no—
29C22×D4—44no—
30C22×D4—44no—
31C22×D4—44no—
32C22×D4D414yes—
33C22×D4D414yes—
34C23×C4—22no—
35C23×C4—22no—
36C23×C4—22no—
37C23×C4—44no—
38C23×C4C2×C411yescharacteristic
39C23×C4C2×C411yescharacteristic
40C23×C4C2×C411yescharacteristic
41C23×C4C2×C411yescharacteristic
42#121—232no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4—22no—
51C2×C22⋊C4—24no—
52C2×C22⋊C4—24no—
53C2×C22⋊C4—24no—
54C2×C22⋊C4D414yes—
55#99—48no—
56#99—48no—
57C24—24no—
58C24—21no—
59C24—21no—
60C24—21no—
61C24—21no—
62C24—42no—
63C24—42no—
64C24—42no—
65C24—42no—
66C24—42no—
67C24—42no—
68C24—42no—
69C24—42no—
70C24C22⋊C412yes—
71C24C22⋊C412yes—
72C24C22×C411yescharacteristic
73C24C2×D412yes—
74C24C2×D412yes—
75C24C2×D412yes—
76C2×D4—28no—
77C2×D4—28no—
78C2×D4—28no—
79C2×D4—28no—
80C2×D4—28no—
81C2×D4—44no—
82C2×D4—44no—
83C2×D4—48no—
84C2×D4—48no—
85C2×D4—48no—
86C2×D4—48no—
87C2×D4—48no—
88C2×D4—44no—
89C2×D4—48no—
90C2×D4—44no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—24no—
94C22×C4—28no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—42no—
102C22×C4—48no—
103C22×C4—48no—
104C22×C4—44no—
105C22×C4—44no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4—44no—
109C22×C4C22⋊C412yes—
110C22×C4C22⋊C412yes—
111C22×C4C22⋊C412yes—
112C22×C4C22⋊C412yes—
113C4⋊C4—28no—
114C22⋊C4—28no—
115C22⋊C4—28no—
116C22⋊C4—48no—
117C22⋊C4—48no—
118C22⋊C4—48no—
119C23—22no—
120C23—22no—
121C23—22no—
122C23—24no—
123C23—24no—
124C23—22no—
125C23—22no—
126C23—24no—
127C23—24no—
128C23—24no—
129C23—22no—
130C23—24no—
131C23—42no—
132C23—42no—
133C23—42no—
134C23—42no—
135C23—42no—
136C23—42no—
137C23—42no—
138C23—42no—
139C23—42no—
140C23—416no—
141C23—42no—
142C23—42no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—88no—
153C23—88no—
154C23#9914yescomplemented
155C23C2×C22⋊C411yescharacteristic
156C23C2×C22⋊C411yescharacteristic
157C23C2×C22⋊C411yescharacteristic
158C23#12014yes—
159D4—416no—
160D4—416no—
161D4—416no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—416no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—84no—
180C2×C4—84no—
181C2×C4—84no—
182C2×C4#9914yescomplemented
183C22—28no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—24no—
191C22—24no—
192C22—24no—
193C22—24no—
194C22—44no—
195C22—48no—
196C22—44no—
197C22—48no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—84no—
204C22—84no—
205C22—84no—
206C22—84no—
207C22—84no—
208C22—84no—
209C22#37911yescharacteristic
210C22C2×#9912yes—
211C22C2×#9912yes—
212C22#41012yes—
213C4—24no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—164no—
218C2—22no—
219C2—22no—
220C2—22no—
221C2—44no—
222C2—44no—
223C2—82no—
224C2#178111yescharacteristic
225C2#178111yescharacteristic
226C2#178211yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC24C22×C4
Frattini, cosocleC25C23
prime radicalC22×#121C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1819C2semidirect
3. #1781C2semidirect
4. #1781C2semidirect
5. C23×D4C4semidirect
8. C24×C4C4semidirect
12. C22×C4⋊C4C4semidirect
13. C22×C22⋊C4C4semidirect
154. C23#99semidirect
182. C2×C4#99semidirect

Maximal subgroups

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

subgroupindex
C22×#1212
#18192
#17812