Finite Group Databasebeta

← #24022

Subgroups — #24022

Order 256.

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

normal subgroups only

This group has 725 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#3484C211yesmaximal, characteristic
2#3093C212yesmaximal, complemented
3#2309C211yesmaximal, characteristic, complemented
4#2037C214yesmaximal, complemented
5#1897C212yesmaximal, complemented
6#1856C212yesmaximal, complemented
7#1792C212yesmaximal, complemented
8#1783C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)C2211yescharacteristic
10C23×D4C2211yescharacteristic
11#546—24no—
12D42—24no—
13#535C2212yes—
14#534—22no—
15#525C2212yes—
16C2×#120—22no—
17#518C2211yescharacteristic
18C2×#117C2211yescharacteristic, complemented
19C2×#117C2211yescharacteristic, complemented
20#485—24no—
21#482—24no—
22#428C2214yescomplemented
23C2×#104C2214yescomplemented
24#417C2214yescomplemented
25C2×#100C2212yes—
26C2×#100C2212yescomplemented
27#407C2212yescomplemented
28#390—22no—
29#390C2212yes—
30#390C2212yescomplemented
31#379—22no—
32#339C2214yescomplemented
33C25D412yes—
34H(5,𝔽2)—24no—
35H(5,𝔽2)—42no—
36C2×#41—22no—
37C2×#41C2312yes—
38C22×D4—22no—
39C22×D4—24no—
40C22×D4—22no—
41C22×D4—24no—
42C22×D4—24no—
43C22×D4—42no—
44C22×D4D412yes—
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C2×M16C2314yes—
49#127—22no—
50#124—22no—
51#123—24no—
52#123—22no—
53C22⋊Q8D414yes—
54#121—28no—
55#121—24no—
56#121D414yes—
57#120—28no—
58#120—24no—
59#120—22no—
60#120—24no—
61C4×D4—24no—
62C4×D4—24no—
63#117—24no—
64#117—42no—
65#117C2312yes—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C42—21no—
75C2×C42C2312yes—
76#107—24no—
77#106—24no—
78#104—44no—
79#103—24no—
80#102—24no—
81#100—24no—
82#100—42no—
83C22⋊C8—24no—
84C24—22no—
85C24—24no—
86C24—24no—
87C24—42no—
88C24—84no—
89C24C2×D412yes—
90C24C2×D411yescharacteristic
91C24C2×D412yes—
92#41—24no—
93#41—24no—
94#41—48no—
95C2×Q8C2×D412yes—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—22no—
100C2×D4—28no—
101C2×D4—24no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—416no—
108C2×D4—42no—
109C2×D4—48no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—88no—
113C2×D4C2×D412yes—
114C2×D4#4114yes—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D414yes—
124C22×C4C2411yescharacteristic
125M16—24no—
126M16—24no—
127C2×C8—24no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—42no—
131C4⋊C4C2×D414yes—
132C22⋊C4—24no—
133C22⋊C4—28no—
134C22⋊C4—24no—
135C22⋊C4—22no—
136C22⋊C4—24no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—84no—
140C22⋊C4—84no—
141C42—22no—
142C42—24no—
143C42—42no—
144C23—22no—
145C23—21no—
146C23—21no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—42no—
153C23—48no—
154C23—44no—
155C23—44no—
156C23—48no—
157C23—44no—
158C23—84no—
159C23—88no—
160C23—84no—
161C23—88no—
162C23#12012yes—
163C23#12314yes—
164C23C22×D411yescharacteristic
165Q8—24no—
166D4—28no—
167D4—24no—
168D4—48no—
169D4—48no—
170D4—48no—
171D4—42no—
172D4—42no—
173D4—44no—
174D4—88no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—28no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—84no—
188C2×C4#12012yes—
189C2×C4C22×D412yes—
190C2×C4C2×#4112yes—
191C2×C4H(5,𝔽2)12yes—
192C8—44no—
193C22—22no—
194C22—22no—
195C22—24no—
196C22—22no—
197C22—22no—
198C22—44no—
199C22—42no—
200C22—42no—
201C22—44no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—44no—
206C22—44no—
207C22—82no—
208C22—82no—
209C22—816no—
210C22—164no—
211C22C2×#12011yescharacteristic
212C22C2×#12311yescharacteristic
213C22#53411yescharacteristic
214C4—22no—
215C4—42no—
216C4—44no—
217C4—82no—
218C4—82no—
219C4#54614yes—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—44no—
224C2—82no—
225C2—82no—
226C2#240511yescharacteristic
227C2#301912yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3484C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3093C2semidirect
3. #2309C2semidirect
4. #2037C2semidirect
5. #1897C2semidirect
6. #1856C2semidirect
7. #1792C2semidirect
8. #1783C2semidirect
18. C2×#117C22semidirect
19. C2×#117C22semidirect
22. #428C22semidirect
23. C2×#104C22semidirect
24. #417C22semidirect
26. C2×#100C22semidirect
27. #407C22semidirect
30. #390C22semidirect
32. #339C22semidirect

Maximal subgroups

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

subgroupindex
#34842
#30932
#23092
#20372
#18972
#18562
#17922
#17832