Finite Group Databasebeta

← #24267

Subgroups — #24267

Order 256.

901 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 901 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C23×D8C211yesmaximal, characteristic, complemented
2C2×#466C212yesmaximal, complemented
3C2×C8×D4C211yesmaximal, characteristic, complemented
4#2408C212yesmaximal, complemented
5#2054C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1970C212yesmaximal, complemented
8#1808C211yesmaximal, characteristic, complemented
9#1786C212yesmaximal, complemented
10C23×D4C2212yes—
11C22×D8—28no—
12C22×D8—24no—
13C22×D8C2212yes—
14C22×D8C2212yes—
15C23×C8C2211yescharacteristic
16C23×C8C2211yescharacteristic, complemented
17C2×#121C2214yescomplemented
18C2×#120—22no—
19C2×C4×D4C2211yescharacteristic
20#466—28no—
21C8×D4—24no—
22C2×#107C2212yescomplemented
23C2×C4⋊C8C2211yescharacteristic, complemented
24C2×#102C2214yescomplemented
25C2×#102C2214yescomplemented
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic, complemented
28C2×C4×C8C2211yescharacteristic, complemented
29#394—22no—
30#392—22no—
31#390C2214yescomplemented
32#380C2212yes—
33#379—22no—
34#340C2212yescomplemented
35C25—22no—
36C22×D4—28no—
37C22×D4—24no—
38C22×D4—24no—
39C22×D4—22no—
40C22×D4—24no—
41C22×D4—44no—
42C22×D4D411yescharacteristic
43C22×D4C2314yes—
44C23×C4C2311yescharacteristic
45C23×C4C2311yescharacteristic
46C2×D8—232no—
47C2×D8—28no—
48C2×D8—24no—
49C2×D8—24no—
50C2×D8—24no—
51C2×D8D418yescomplemented
52C22×C8—24no—
53C22×C8—22no—
54C22×C8—24no—
55C22×C8D414yescomplemented
56C22×C8C2311yescharacteristic
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60#121—28no—
61#121—44no—
62#120—48no—
63C4×D4—22no—
64C4×D4—22no—
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4C2314yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4D411yescharacteristic
72C2×C22⋊C4D411yescharacteristic
73C2×C42C2311yescharacteristic
74#107—24no—
75C4⋊C8—22no—
76#102—28no—
77#102—44no—
78C22⋊C8—22no—
79C22⋊C8—22no—
80C4×C8—22no—
81#95—22no—
82C24—24no—
83C24—48no—
84C24—44no—
85C24—44no—
86C24—44no—
87C24—84no—
88C24C2×D411yescharacteristic
89C24C2×D411yescharacteristic
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—216no—
93C2×D4—24no—
94C2×D4—24no—
95C2×D4—24no—
96C2×D4—24no—
97C2×D4—24no—
98C2×D4—416no—
99C2×D4—48no—
100C2×D4—48no—
101C2×D4—48no—
102C2×D4—48no—
103C2×D4—44no—
104C2×D4—84no—
105C2×D4D814yescomplemented
106C2×D4C2×D418yes—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4C2×D411yescharacteristic
119C22×C4#4112yes—
120C22×C4C2411yescharacteristic
121D8—216no—
122D8—216no—
123C2×C8—22no—
124C2×C8—28no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—44no—
130C2×C8C2×D412yes—
131C2×C8C2×D412yes—
132C2×C8C2×D414yes—
133C4⋊C4—22no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—48no—
139C22⋊C4—44no—
140C22⋊C4—48no—
141C22⋊C4—84no—
142C42—21no—
143C42—21no—
144C23—22no—
145C23—22no—
146C23—21no—
147C23—22no—
148C23—21no—
149C23—22no—
150C23—28no—
151C23—416no—
152C23—44no—
153C23—44no—
154C23—44no—
155C23—44no—
156C23—44no—
157C23—44no—
158C23—816no—
159C23—816no—
160C23—88no—
161C23—88no—
162C23#12112yes—
163C23C2×D812yes—
164C23C2×D812yescomplemented
165C23#13512yes—
166C23C22×D411yescharacteristic
167D4—28no—
168D4—28no—
169D4—28no—
170D4—416no—
171D4—44no—
172D4—44no—
173D4—816no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—44no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4#12112yes—
191C2×C4C2×D812yescomplemented
192C2×C4C22×D411yescharacteristic
193C2×C4C22×D412yes—
194C2×C4C2×#4111yescharacteristic
195C2×C4H(5,𝔽2)12yes—
196C8—22no—
197C8—41no—
198C8—41no—
199C8#12014yescomplemented
200C22—22no—
201C22—24no—
202C22—24no—
203C22—22no—
204C22—24no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—48no—
210C22—48no—
211C22—48no—
212C22—832no—
213C22—84no—
214C22—84no—
215C22—84no—
216C22—84no—
217C22#46614yescomplemented
218C22C2×#12111yescharacteristic
219C22D4212yes—
220C22C22×D811yescharacteristic
221C22C2×#13511yescharacteristic
222C22#57612yes—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—84no—
228C4C2×#12011yescharacteristic
229C4#53511yescharacteristic
230C4#54612yes—
231C2—24no—
232C2—42no—
233C2—88no—
234C2#240811yescharacteristic
235C2C2×#46611yescharacteristic
236C2#305811yescharacteristic
237C2D4×D812yes—
238C2#328012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×D8C2semidirect
2. C2×#466C2semidirect
3. C2×C8×D4C2semidirect
4. #2408C2semidirect
5. #2054C2semidirect
6. #1999C2semidirect
7. #1970C2semidirect
8. #1808C2semidirect
9. #1786C2semidirect
16. C23×C8C22semidirect
17. C2×#121C22semidirect
22. C2×#107C22semidirect
23. C2×C4⋊C8C22semidirect
24. C2×#102C22semidirect
25. C2×#102C22semidirect
27. C2×C22⋊C8C22semidirect
28. C2×C4×C8C22semidirect
31. #390C22semidirect
34. #340C22semidirect
51. C2×D8D4semidirect
55. C22×C8D4semidirect
105. C2×D4D8semidirect
164. C23C2×D8semidirect
191. C2×C4C2×D8semidirect
199. C8#120semidirect
217. C22#466semidirect

Maximal subgroups

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

subgroupindex
C23×D82
C2×#4662
C2×C8×D42
#24082
#20542
#19992
#19702
#18082
#17862