Finite Group Databasebeta

← #18132

Subgroups — #18132

Order 256.

827 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 827 conjugacy classes of subgroups — too many to draw. Only its 117 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×#534C211yesmaximal, characteristic, complemented
2#3006C212yesmaximal, complemented
3#2404C211yesmaximal, characteristic, complemented
4#1601C214yesmaximal, complemented
5#1595C214yesmaximal, complemented
6#1510C212yesmaximal, complemented
7#1462C211yesmaximal, characteristic, complemented
8C22×#41C2211yescharacteristic, complemented
9C23×D4—21no—
10C2×AGL(1,ℤ/8)—22no—
11#534—24no—
12#534—22no—
13#534C2214yes—
14C2×#123—22no—
15C2×#121—21no—
16C2×#121C2212yes—
17C2×#120—22no—
18C2×#120—21no—
19C22×C22⋊C4C2211yescharacteristic
20#449—24no—
21#448C2214yescomplemented
22#447C2214yescomplemented
23#417—22no—
24#407C2212yescomplemented
25#393C2212yescomplemented
26#392C2212yescomplemented
27#386—21no—
28#386—22no—
29#381C2211yescharacteristic, complemented
30#327C2218yescomplemented
31#323C2214yescomplemented
32C25—21no—
33C25D411yescharacteristic
34C2×#41—22no—
35C2×#41—22no—
36C2×#41—44no—
37C22×Q8C2311yescharacteristic, complemented
38C22×D4—22no—
39C22×D4—24no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4—24no—
43C22×D4—21no—
44C22×D4—22no—
45C22×D4—48no—
46C22×D4D412yescomplemented
47C22×D4C2311yescharacteristic, complemented
48C23×C4—21no—
49C23×C4—21no—
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52AGL(1,ℤ/8)—48no—
53C2×SD16—24no—
54C2×D8—24no—
55C2×M16—22no—
56#123—28no—
57#123—28no—
58#121—24no—
59#121—24no—
60#121—22no—
61#121—22no—
62#121D414yescomplemented
63#121C2314yes—
64#120—28no—
65#120—28no—
66#120—44no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4D412yescomplemented
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4D412yes—
77C2×C22⋊C4C2312yes—
78#103—24no—
79#102—24no—
80C22⋊C8—22no—
81C22⋊C8C2314yescomplemented
82#95—21no—
83#95—22no—
84#95C2312yescomplemented
85C24—21no—
86C24—21no—
87C24—24no—
88C24—24no—
89C24—22no—
90C24—21no—
91C24—44no—
92C24—42no—
93C24—81no—
94C24—82no—
95C24—82no—
96C24C2×D411yescharacteristic
97C24C2×D412yes—
98#41—28no—
99#41—44no—
100#41—44no—
101#41—44no—
102C2×Q8—22no—
103C2×Q8—42no—
104C2×D4—28no—
105C2×D4—24no—
106C2×D4—24no—
107C2×D4—24no—
108C2×D4—24no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—44no—
112C2×D4—42no—
113C2×D4—48no—
114C2×D4—416no—
115C2×D4—48no—
116C2×D4—48no—
117C2×D4—42no—
118C2×D4—41no—
119C2×D4—41no—
120C2×D4—48no—
121C2×D4—44no—
122C2×D4—44no—
123C2×D4—82no—
124C2×D4—82no—
125C2×D4C2×D414yes—
126C22×C4—24no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—44no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—81no—
138C22×C4C2×D412yes—
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2411yescharacteristic
143SD16—48no—
144D8—48no—
145M16—44no—
146C2×C8—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4C2×D414yes—
150C22⋊C4—28no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C22⋊C4—24no—
154C22⋊C4—24no—
155C22⋊C4—44no—
156C22⋊C4—42no—
157C22⋊C4—82no—
158C23—24no—
159C23—24no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23—22no—
164C23—22no—
165C23—24no—
166C23—41no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—41no—
172C23—44no—
173C23—48no—
174C23—44no—
175C23—42no—
176C23—41no—
177C23—42no—
178C23—44no—
179C23—81no—
180C23—81no—
181C23—88no—
182C23—84no—
183C23—84no—
184C23—816no—
185C23#12011yescharacteristic
186C23#12011yescharacteristic
187C23AGL(1,ℤ/8)14yescomplemented
188C23C22×D411yescharacteristic
189Q8—24no—
190Q8—42no—
191D4—24no—
192D4—44no—
193D4—42no—
194D4—416no—
195D4—48no—
196D4—44no—
197D4—48no—
198D4—44no—
199D4—82no—
200D4—84no—
201D4—88no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—48no—
210C2×C4—44no—
211C2×C4—48no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4#12012yes—
221C2×C4C22×D412yes—
222C8—44no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—24no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—44no—
232C22—44no—
233C22—48no—
234C22—44no—
235C22—42no—
236C22—42no—
237C22—44no—
238C22—44no—
239C22—42no—
240C22—44no—
241C22—82no—
242C22—82no—
243C22—88no—
244C22—84no—
245C22—816no—
246C22—81no—
247C22—81no—
248C22B(4,𝔽2)12yes—
249C22C2×#12011yescharacteristic
250C22C2×AGL(1,ℤ/8)12yes—
251C4—22no—
252C4—22no—
253C4—44no—
254C4—42no—
255C4—42no—
256C4—82no—
257C2—22no—
258C2—22no—
259C2—41no—
260C2—44no—
261C2—82no—
262C2—84no—
263C2#300311yescharacteristic
264C2#301911yescharacteristic
265C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#534C2semidirect
2. #3006C2semidirect
3. #2404C2semidirect
4. #1601C2semidirect
5. #1595C2semidirect
6. #1510C2semidirect
7. #1462C2semidirect
8. C22×#41C22semidirect
21. #448C22semidirect
22. #447C22semidirect
24. #407C22semidirect
25. #393C22semidirect
26. #392C22semidirect
29. #381C22semidirect
30. #327C22semidirect
31. #323C22semidirect
37. C22×Q8C23semidirect
46. C22×D4D4semidirect
47. C22×D4C23semidirect
62. #121D4semidirect
68. C2×C4⋊C4D4semidirect
81. C22⋊C8C23semidirect
84. #95C23semidirect
187. C23AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C2×#5342
#30062
#24042
#16012
#15952
#15102
#14622