Finite Group Databasebeta

← #32707

Subgroups — #32707

Order 256.

570 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 109 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#2847C212yesmaximal, complemented
2#2077C212yesmaximal, complemented
3#2076C211yesmaximal, characteristic, complemented
4#2034C212yesmaximal, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1999C212yesmaximal, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1779C211yesmaximal, characteristic, complemented
10C24×C4C2211yescharacteristic
11C22×D8C2211yescharacteristic, complemented
12C2×#121C2212yes—
13C2×#121C2212yes—
14C2×#121C2212yes—
15C2×#120—22no—
16C2×#107C2211yescharacteristic, complemented
17C2×#106C2212yescomplemented
18C2×#102C2212yescomplemented
19C2×#102C2212yescomplemented
20C2×#102C2212yescomplemented
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2212yes—
23C2×C22⋊C8C2212yes—
24C2×C22⋊C8C2211yescharacteristic
25#386—22no—
26#386—22no—
27#386—22no—
28#380C2212yes—
29#380C2212yes—
30#380C2212yes—
31#340C2212yescomplemented
32#340C2211yescharacteristic, complemented
33#340C2211yescharacteristic, complemented
34#340C2212yescomplemented
35C25D411yescharacteristic
36C22×D4—22no—
37C22×D4—22no—
38C22×D4—22no—
39C22×D4C2312yes—
40C23×C4—42no—
41C23×C4—41no—
42C23×C4—42no—
43C23×C4—41no—
44C23×C4D411yescharacteristic
45C23×C4D412yes—
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C2×D8—42no—
49C2×D8—41no—
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53#121—42no—
54#121—42no—
55#121—42no—
56#121—42no—
57#121—42no—
58#121—42no—
59#120—48no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69#107—41no—
70#106—42no—
71#102—42no—
72#102—42no—
73#102—42no—
74C22⋊C8—42no—
75C22⋊C8—41no—
76C22⋊C8—42no—
77C22⋊C8—41no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24—22no—
82C24—24no—
83C24—22no—
84C24—42no—
85C24—41no—
86C24—41no—
87C24C2×D411yescharacteristic
88C24C2×D412yes—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—44no—
96C2×D4—44no—
97C2×D4—44no—
98C2×D4—44no—
99C2×D4—44no—
100C2×D4—44no—
101C2×D4—44no—
102C2×D4—44no—
103C2×D4—44no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—42no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—48no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D412yes—
127C22×C4C2411yescharacteristic
128D8—44no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—22no—
135C2×C8—21no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C22⋊C4—44no—
143C22⋊C4—44no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C23—22no—
149C23—22no—
150C23—22no—
151C23—22no—
152C23—21no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—21no—
158C23—48no—
159C23—44no—
160C23—44no—
161C23—48no—
162C23—44no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—42no—
169C23—42no—
170C23#13512yes—
171C23AGL(1,ℤ/8)14yes—
172C23C22×D411yescharacteristic
173D4—42no—
174D4—42no—
175D4—42no—
176D4—42no—
177D4—88no—
178D4—88no—
179D4—88no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—48no—
186C2×C4—44no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4H(5,𝔽2)12yes—
204C2×C4H(5,𝔽2)11yescharacteristic
205C2×C4H(5,𝔽2)12yes—
206C2×C4H(5,𝔽2)11yescharacteristic
207C8—42no—
208C8—41no—
209C8—41no—
210C22—24no—
211C22—24no—
212C22—22no—
213C22—24no—
214C22—24no—
215C22—416no—
216C22—42no—
217C22—42no—
218C22—41no—
219C22—42no—
220C22—42no—
221C22—41no—
222C22—42no—
223C22—82no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22—82no—
229C22—82no—
230C22#53412yes—
231C22#53411yescharacteristic
232C22C2×#13511yescharacteristic
233C22C2×AGL(1,ℤ/8)12yes—
234C22#57511yescharacteristic
235C4—41no—
236C4—42no—
237C4—82no—
238C4—82no—
239C4—82no—
240C4#56012yes—
241C4#56011yescharacteristic
242C4H(3,𝔽4)11yescharacteristic
243C2—42no—
244C2—44no—
245C2—162no—
246C2#281211yescharacteristic
247C2#318812yes—
248C2#319111yescharacteristic
249C2#319211yescharacteristic
250C2#319412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2847C2semidirect
2. #2077C2semidirect
3. #2076C2semidirect
4. #2034C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1779C2semidirect
11. C22×D8C22semidirect
16. C2×#107C22semidirect
17. C2×#106C22semidirect
18. C2×#102C22semidirect
19. C2×#102C22semidirect
20. C2×#102C22semidirect
31. #340C22semidirect
32. #340C22semidirect
33. #340C22semidirect
34. #340C22semidirect

Maximal subgroups

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

subgroupindex
#28472
#20772
#20762
#20342
#20332
#19992
#17792