Finite Group Databasebeta

← #23910

Subgroups — #23910

Order 256.

933 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 933 conjugacy classes of subgroups — too many to draw. Only its 147 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#3486C211yesmaximal, characteristic, complemented
2C22×#124C211yesmaximal, characteristic, complemented
3C2×#454C212yesmaximal, complemented
4#3003C212yesmaximal, complemented
5#2003C214yesmaximal, complemented
6#1896C212yesmaximal, complemented
7#1888C212yesmaximal, complemented
8#1784C211yesmaximal, characteristic, complemented
9C2×2−1+4C2211yescharacteristic, complemented
10C23×Q8C2211yescharacteristic
11C23×D4C2211yescharacteristic
12C2×AGL(1,ℤ/8)C2214yes—
13D4×Q8—24no—
14#548—24no—
15#536—22no—
16#536C2212yescomplemented
17C2×#124—22no—
18C2×#124—28no—
19C2×#124C2212yes—
20C2×#124C2212yes—
21#525—22no—
22#525C2212yes—
23#519C2211yescharacteristic, complemented
24C22×C22⋊C4—22no—
25C22×C42C2211yescharacteristic
26#454—28no—
27#450—24no—
28#447—24no—
29C2×#104C2214yescomplemented
30#418C2214yes—
31C2×#101C2212yescomplemented
32C2×#101C2212yescomplemented
33#407C2212yes—
34#339C2214yescomplemented
35C25—21no—
362−1+4—24no—
372−1+4—42no—
38C2×#41—22no—
39C2×#41—22no—
40C2×#41C2312yes—
41C22×Q8—24no—
42C22×Q8—22no—
43C22×Q8—24no—
44C22×Q8—42no—
45C22×Q8D412yes—
46C22×Q8C2311yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×D4—24no—
49C22×D4—21no—
50C22×D4—24no—
51C22×D4—42no—
52C22×D4C2312yes—
53C23×C4—21no—
54C23×C4D412yes—
55C23×C4C2311yescharacteristic
56AGL(1,ℤ/8)—28no—
57AGL(1,ℤ/8)—44no—
58C2×SD16—24no—
59C2×D8—24no—
60C2×M16C2314yes—
61C4⋊Q8—24no—
62C4⋊Q8—22no—
63#124—216no—
64#124—24no—
65#124—24no—
66#124—24no—
67#124—22no—
68#124—24no—
69#124D418yescomplemented
70#123—24no—
71#123—22no—
72C22⋊Q8—24no—
73C22⋊Q8—28no—
74C22⋊Q8—24no—
75C22⋊Q8D414yes—
76#121—24no—
77#121D414yes—
78#120—22no—
79C4×Q8—24no—
80C4×D4—28no—
81C4×D4—24no—
82#117—22no—
83#117C2312yes—
84C2×C22⋊C4—216no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C42—21no—
88C2×C42—24no—
89C2×C42C2312yes—
90#104—28no—
91#102—28no—
92#101—24no—
93#101—42no—
94C22⋊C8—24no—
95C24—24no—
96C24—22no—
97C24—42no—
98C24—48no—
99C24—42no—
100C24C2×D411yescharacteristic
101#41—24no—
102#41—24no—
103#41—44no—
104#41—48no—
105#41—44no—
106C2×Q8—22no—
107C2×Q8—24no—
108C2×Q8—24no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—42no—
115C2×Q8—416no—
116C2×Q8—48no—
117C2×Q8C2×D412yes—
118C2×Q8C2×D414yes—
119C2×D4—22no—
120C2×D4—24no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—24no—
124C2×D4—22no—
125C2×D4—24no—
126C2×D4—416no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4C2×D412yes—
130C2×D4C2×D414yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—42no—
140C22×C4—44no—
141C22×C4C2×D412yes—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D412yes—
144C22×C4C2×D414yes—
145C22×C4C2411yescharacteristic
146SD16—48no—
147D8—48no—
148M16—24no—
149M16—24no—
150C2×C8—24no—
151C4⋊C4—24no—
152C4⋊C4—28no—
153C4⋊C4—24no—
154C4⋊C4C2×D414yes—
155C22⋊C4—216no—
156C22⋊C4—28no—
157C22⋊C4—24no—
158C22⋊C4—22no—
159C22⋊C4—216no—
160C22⋊C4—24no—
161C42—24no—
162C42—24no—
163C42—22no—
164C42—22no—
165C42C2×D414yes—
166C23—21no—
167C23—21no—
168C23—24no—
169C23—24no—
170C23—22no—
171C23—48no—
172C23—416no—
173C23—48no—
174C23—42no—
175C23—42no—
176C23—88no—
177C23—84no—
178C23—84no—
179C23#12012yes—
180C23C22×D411yescharacteristic
181Q8—28no—
182Q8—24no—
183Q8—48no—
184Q8—44no—
185Q8—44no—
186D4—24no—
187D4—28no—
188D4—48no—
189D4—44no—
190D4—42no—
191D4—42no—
192D4—48no—
193D4—44no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—28no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—48no—
206C2×C4—48no—
207C2×C4—48no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4#12014yes—
211C2×C4#12012yes—
212C2×C4C22×D412yes—
213C2×C4C22×D412yes—
214C2×C4H(5,𝔽2)12yes—
215C8—44no—
216C22—24no—
217C22—24no—
218C22—22no—
219C22—22no—
220C22—22no—
221C22—48no—
222C22—42no—
223C22—44no—
224C22—44no—
225C22—84no—
226C22—816no—
227C22—84no—
228C22#45414yescomplemented
229C22C2×#12011yescharacteristic
230C22C2×#12011yescharacteristic
231C22#53411yescharacteristic
232C4—22no—
233C4—24no—
234C4—44no—
235C4—42no—
236C4—42no—
237C4—44no—
238C4D4214yes—
239C2—21no—
240C2—21no—
241C2—24no—
242C2—82no—
243C2—84no—
244C2#240311yescharacteristic
245C2C2×#45412yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
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
1. #3486C2semidirect
2. C22×#124C2semidirect
3. C2×#454C2semidirect
4. #3003C2semidirect
5. #2003C2semidirect
6. #1896C2semidirect
7. #1888C2semidirect
8. #1784C2semidirect
9. C2×2−1+4C22semidirect
16. #536C22semidirect
23. #519C22semidirect
29. C2×#104C22semidirect
31. C2×#101C22semidirect
32. C2×#101C22semidirect
34. #339C22semidirect
69. #124D4semidirect
228. C22#454semidirect

Maximal subgroups

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

subgroupindex
#34862
C22×#1242
C2×#4542
#30032
#20032
#18962
#18882
#17842