Finite Group Databasebeta

← #23985

Subgroups — #23985

Order 256.

737 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 737 conjugacy classes of subgroups — too many to draw. Only its 127 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#3490C211yesmaximal, characteristic
2#2849C212yesmaximal, complemented
3C2×#381C211yesmaximal, characteristic, complemented
4#2030C212yesmaximal, complemented
5#2022C212yesmaximal, complemented
6#1898C212yesmaximal, complemented
7#1890C214yesmaximal, complemented
8#1819C211yesmaximal, characteristic, complemented
9C24×C4C2211yescharacteristic
10#551—24no—
11#546—24no—
12#543C2211yescharacteristic
13#538—22no—
14#534C2211yescharacteristic
15#525—22no—
16C2×#123C2214yes—
17C2×#120C2212yes—
18#518C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21C2×#99C2214yes—
22#386—24no—
23#385—24no—
24#381—22no—
25#381C2214yes—
26#381C2212yes—
27#381C2212yes—
28#379C2212yes—
29C2×#95—22no—
30#342C2212yes—
31#342C2212yes—
32#342C2214yes—
33C25C2311yescharacteristic
34C2×#41—22no—
35C22×D4D414yes—
36C22×D4C2312yes—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4—24no—
40C23×C4—21no—
41C23×C4—24no—
42C23×C4—42no—
43C23×C4D412yes—
44C23×C4C2312yes—
45#126—24no—
46#124—24no—
47#123—24no—
48#123—24no—
49#123—22no—
50#123—28no—
51#123—44no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—24no—
55#121—22no—
56#121—24no—
57#121—22no—
58#120—22no—
59#120—22no—
60#120—24no—
61#120—22no—
62C4×D4—24no—
63C4×D4—24no—
64C4×D4—24no—
65#117—22no—
66#117—22no—
67C2×C4⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4D412yes—
75C2×C22⋊C4D414yes—
76C2×C22⋊C4C2312yes—
77C2×C22⋊C4C2314yes—
78C2×C22⋊C4C2311yescharacteristic
79C2×C22⋊C4C2311yescharacteristic
80C2×C22⋊C4C2312yes—
81#99—44no—
82#95—24no—
83#95—24no—
84#95—28no—
85C24—22no—
86C24—22no—
87C24—44no—
88C24C2×D411yescharacteristic
89C24C2×D414yes—
90C24C2×D412yes—
91C24C2×D411yescharacteristic
92C24C2411yescharacteristic
93#41—84no—
94C2×Q8—22no—
95C2×D4—24no—
96C2×D4—22no—
97C2×D4—22no—
98C2×D4—24no—
99C2×D4—24no—
100C2×D4—22no—
101C2×D4—24no—
102C2×D4—48no—
103C22×C4—24no—
104C22×C4—24no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—24no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—416no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—44no—
120C22×C4—42no—
121C22×C4—44no—
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×D414yes—
125C22×C4C2×D412yes—
126C22×C4#4114yescomplemented
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—24no—
131C4⋊C4—48no—
132C22⋊C4—24no—
133C22⋊C4—24no—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—24no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—24no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—48no—
145C22⋊C4—48no—
146C22⋊C4—42no—
147C42—24no—
148C23—21no—
149C23—21no—
150C23—24no—
151C23—21no—
152C23—21no—
153C23—28no—
154C23—28no—
155C23—22no—
156C23—24no—
157C23—42no—
158C23—44no—
159C23—42no—
160C23—48no—
161C23—48no—
162C23—44no—
163C23#12012yes—
164C23#12012yes—
165C23C22×D411yescharacteristic
166C23C22×D412yes—
167C23C2×#4112yes—
168C23H(5,𝔽2)12yes—
169Q8—82no—
170D4—42no—
171D4—42no—
172D4—44no—
173D4—48no—
174D4—84no—
175D4—82no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—28no—
182C2×C4—24no—
183C2×C4—48no—
184C2×C4—48no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—44no—
192C2×C4—48no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—82no—
197C2×C4—84no—
198C2×C4—84no—
199C22—24no—
200C22—24no—
201C22—22no—
202C22—22no—
203C22—24no—
204C22—22no—
205C22—22no—
206C22—24no—
207C22—42no—
208C22—42no—
209C22—416no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—44no—
214C22—84no—
215C22#45814yes—
216C22C2×#12011yescharacteristic
217C22#52511yescharacteristic
218C22#53511yescharacteristic
219C22#54614yes—
220C4—44no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—44no—
225C4—84no—
226C4—82no—
227C2—21no—
228C2—24no—
229C2—21no—
230C2—24no—
231C2—42no—
232C2—82no—
233C2#240411yescharacteristic
234C2C2×#45812yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC24C24
prime radical#3490C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2849C2semidirect
3. C2×#381C2semidirect
4. #2030C2semidirect
5. #2022C2semidirect
6. #1898C2semidirect
7. #1890C2semidirect
8. #1819C2semidirect
126. C22×C4#41semidirect

Maximal subgroups

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

subgroupindex
#34902
#28492
C2×#3812
#20302
#20222
#18982
#18902
#18192