Finite Group Databasebeta

← #23913

Subgroups — #23913

Order 256.

949 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 949 conjugacy classes of subgroups — too many to draw. Only its 143 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#3485C211yesmaximal, characteristic, complemented
2C2×#535C211yesmaximal, characteristic, complemented
3C2×#453C212yesmaximal, complemented
4#3003C212yesmaximal, complemented
5#2003C214yesmaximal, complemented
6#1897C212yesmaximal, complemented
7#1887C212yesmaximal, complemented
8#1785C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)C2211yescharacteristic, complemented
10C22×#41C2211yescharacteristic
11C23×D4C2211yescharacteristic, complemented
12C2×AGL(1,ℤ/8)C2214yes—
13#547—24no—
14#546—24no—
15#536C2212yescomplemented
16#535—28no—
17#534—22no—
18C2×#127C2212yes—
19C2×#124C2212yescomplemented
20#525C2212yes—
21C2×#123—22no—
22C2×#121—22no—
23C2×#120—22no—
24#518C2211yescharacteristic, complemented
25C2×#117C2211yescharacteristic, complemented
26#453—28no—
27#450—24no—
28#447—24no—
29C2×#104C2214yescomplemented
30#418C2214yescomplemented
31C2×#101C2212yescomplemented
32C2×#100C2212yescomplemented
33#407C2212yescomplemented
34#339C2214yescomplemented
35C25—21no—
36H(5,𝔽2)—24no—
37H(5,𝔽2)—42no—
38C2×#41—22no—
39C2×#41—24no—
40C2×#41—24no—
41C2×#41—42no—
42C2×#41C2312yes—
43C22×Q8C2311yescharacteristic
44C22×D4—22no—
45C22×D4—24no—
46C22×D4—24no—
47C22×D4—22no—
48C22×D4—21no—
49C22×D4—24no—
50C22×D4—42no—
51C22×D4D412yes—
52C22×D4C2312yes—
53C22×D4C2311yescharacteristic
54C23×C4D412yes—
55C23×C4C2311yescharacteristic
56AGL(1,ℤ/8)—28no—
57AGL(1,ℤ/8)—44no—
58C2×SD16—24no—
59C2×D8—24no—
60C2×M16C2314yes—
61C4⋊Q8—22no—
62#127—24no—
63#127D418yescomplemented
64#124—24no—
65#124—22no—
66#124—24no—
67#123—28no—
68#123—24no—
69#123—22no—
70#123—24no—
71C22⋊Q8—24no—
72C22⋊Q8D414yescomplemented
73#121—216no—
74#121—28no—
75#121D414yescomplemented
76#120—216no—
77#120—24no—
78#120—22no—
79C4×D4—24no—
80C4×D4—24no—
81#117—24no—
82#117C2312yes—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—21no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—24no—
89C2×C42C2312yes—
90#104—28no—
91#102—28no—
92#101—42no—
93#100—24no—
94C22⋊C8—24no—
95C24—22no—
96C24—24no—
97C24—22no—
98C24—42no—
99C24—42no—
100C24—84no—
101C24C2×D411yescharacteristic
102C24C2×D412yes—
103#41—24no—
104#41—24no—
105#41—416no—
106#41—48no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8C2×D412yes—
111C2×D4—24no—
112C2×D4—24no—
113C2×D4—24no—
114C2×D4—24no—
115C2×D4—22no—
116C2×D4—24no—
117C2×D4—216no—
118C2×D4—22no—
119C2×D4—24no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—24no—
123C2×D4—416no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—48no—
127C2×D4—44no—
128C2×D4—44no—
129C2×D4—42no—
130C2×D4—88no—
131C2×D4C2×D412yes—
132C2×D4C2×D414yes—
133C2×D4C2×D414yes—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—42no—
143C22×C4C2×D412yes—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D414yes—
146C22×C4C2411yescharacteristic
147SD16—48no—
148D8—48no—
149M16—24no—
150M16—24no—
151C2×C8—24no—
152C4⋊C4—24no—
153C4⋊C4—28no—
154C4⋊C4—24no—
155C4⋊C4C2×D414yes—
156C22⋊C4—24no—
157C22⋊C4—24no—
158C22⋊C4—216no—
159C22⋊C4—28no—
160C22⋊C4—24no—
161C22⋊C4—22no—
162C22⋊C4—24no—
163C42—22no—
164C42—22no—
165C42C2×D414yes—
166C23—21no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23—24no—
171C23—22no—
172C23—42no—
173C23—44no—
174C23—48no—
175C23—48no—
176C23—44no—
177C23—42no—
178C23—42no—
179C23—88no—
180C23—88no—
181C23—84no—
182C23—88no—
183C23#12012yes—
184C23#12014yes—
185C23C22×D411yescharacteristic
186Q8—24no—
187Q8—44no—
188D4—28no—
189D4—24no—
190D4—28no—
191D4—48no—
192D4—48no—
193D4—48no—
194D4—42no—
195D4—42no—
196D4—44no—
197D4—84no—
198D4—816no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—28no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—48no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—84no—
213C2×C4#12012yes—
214C2×C4C22×D412yes—
215C2×C4C22×D412yes—
216C2×C4H(5,𝔽2)12yes—
217C8—44no—
218C22—22no—
219C22—22no—
220C22—24no—
221C22—24no—
222C22—44no—
223C22—42no—
224C22—42no—
225C22—48no—
226C22—44no—
227C22—42no—
228C22—44no—
229C22—44no—
230C22—82no—
231C22—84no—
232C22—816no—
233C22—84no—
234C22—164no—
235C22C2×#12011yescharacteristic
236C22C2×#12011yescharacteristic
237C22#53411yescharacteristic
238C4—22no—
239C4—44no—
240C4—42no—
241C4—42no—
242C4—44no—
243C4D4214yes—
244C2—21no—
245C2—21no—
246C2—42no—
247C2—44no—
248C2—84no—
249C2—82no—
250C2#240311yescharacteristic
251C2#302112yes—
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. #3485C2semidirect
2. C2×#535C2semidirect
3. C2×#453C2semidirect
4. #3003C2semidirect
5. #2003C2semidirect
6. #1897C2semidirect
7. #1887C2semidirect
8. #1785C2semidirect
9. C2×H(5,𝔽2)C22semidirect
11. C23×D4C22semidirect
15. #536C22semidirect
19. C2×#124C22semidirect
24. #518C22semidirect
25. C2×#117C22semidirect
29. C2×#104C22semidirect
30. #418C22semidirect
31. C2×#101C22semidirect
32. C2×#100C22semidirect
33. #407C22semidirect
34. #339C22semidirect
63. #127D4semidirect
72. C22⋊Q8D4semidirect
75. #121D4semidirect

Maximal subgroups

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

subgroupindex
#34852
C2×#5352
C2×#4532
#30032
#20032
#18972
#18872
#17852