Finite Group Databasebeta

← #24272

Subgroups — #24272

Order 256.

888 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 151 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×#469C212yesmaximal, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4#2408C212yesmaximal, complemented
5#2054C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1972C212yesmaximal, complemented
8#1807C211yesmaximal, characteristic, complemented
9#1786C212yesmaximal, complemented
10C23×D4C2212yes—
11C22×D8—28no—
12C22×D8—24no—
13C22×D8C2212yes—
14C22×D8C2212yes—
15C22×M16C2211yescharacteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#121C2214yescomplemented
18C2×#120—22no—
19C2×C4×D4C2211yescharacteristic
20#469—28no—
21#435—24no—
22C2×#106C2212yescomplemented
23C2×C4⋊C8C2211yescharacteristic, complemented
24C2×#102C2214yescomplemented
25C2×#102C2214yescomplemented
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic, complemented
28C2×#97C2211yescharacteristic, 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
52C2×M16—22no—
53C2×M16—24no—
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#106—24no—
75C4⋊C8—22no—
76#102—28no—
77#102—44no—
78C22⋊C8—22no—
79C22⋊C8—22no—
80#97—22no—
81#95—22no—
82C24—24no—
83C24—48no—
84C24—44no—
85C24—44no—
86C24—44no—
87C24—84no—
88C24C2×D411yescharacteristic
89C24C2×D411yescharacteristic
90C2×D4—216no—
91C2×D4—22no—
92C2×D4—24no—
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×D4C2×D418yes—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D412yes—
118C22×C4#4112yes—
119C22×C4C2411yescharacteristic
120D8—216no—
121D8—216no—
122M16—44no—
123C2×C8—22no—
124C2×C8—28no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8C2×D412yes—
130C2×C8C2×D412yes—
131C2×C8C2×D414yes—
132C4⋊C4—22no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—48no—
138C22⋊C4—48no—
139C22⋊C4—44no—
140C22⋊C4—84no—
141C42—21no—
142C42—21no—
143C23—22no—
144C23—22no—
145C23—21no—
146C23—28no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—416no—
151C23—44no—
152C23—44no—
153C23—44no—
154C23—44no—
155C23—44no—
156C23—44no—
157C23—816no—
158C23—816no—
159C23—88no—
160C23—88no—
161C23#12112yes—
162C23AGL(1,ℤ/8)14yes—
163C23C22×D411yescharacteristic
164D4—28no—
165D4—28no—
166D4—416no—
167D4—44no—
168D4—44no—
169D4—44no—
170D4—816no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—44no—
187C2×C4#12112yes—
188C2×C4C22×D411yescharacteristic
189C2×C4C22×D412yes—
190C2×C4C2×#4111yescharacteristic
191C2×C4H(5,𝔽2)12yes—
192C8—22no—
193C8—41no—
194C8—41no—
195C8#12014yescomplemented
196C22—22no—
197C22—28no—
198C22—22no—
199C22—41no—
200C22—48no—
201C22—44no—
202C22—48no—
203C22—48no—
204C22—41no—
205C22—41no—
206C22—832no—
207C22—84no—
208C22—84no—
209C22—84no—
210C22—84no—
211C22#46914yescomplemented
212C22C2×#12111yescharacteristic
213C22D4212yes—
214C22C2×AGL(1,ℤ/8)12yes—
215C22#57612yes—
216C4—22no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—84no—
221C4C2×#12011yescharacteristic
222C4#53511yescharacteristic
223C4#54612yes—
224C2—24no—
225C2—81no—
226C2—88no—
227C2#240811yescharacteristic
228C2C2×#46911yescharacteristic
229C2#305811yescharacteristic
230C2#326414yes—
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×#469C2semidirect
3. C2×#435C2semidirect
4. #2408C2semidirect
5. #2054C2semidirect
6. #1999C2semidirect
7. #1972C2semidirect
8. #1807C2semidirect
9. #1786C2semidirect
15. C22×M16C22semidirect
17. C2×#121C22semidirect
22. C2×#106C22semidirect
23. C2×C4⋊C8C22semidirect
24. C2×#102C22semidirect
25. C2×#102C22semidirect
27. C2×C22⋊C8C22semidirect
28. C2×#97C22semidirect
31. #390C22semidirect
34. #340C22semidirect
51. C2×D8D4semidirect
55. C22×C8D4semidirect
195. C8#120semidirect
211. C22#469semidirect

Maximal subgroups

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

subgroupindex
C23×D82
C2×#4692
C2×#4352
#24082
#20542
#19992
#19722
#18072
#17862