Finite Group Databasebeta

← #32980

Subgroups — #32980

Order 256.

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

normal subgroups only

This group has 754 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
1C22×#117C211yesmaximal, characteristic, complemented
2C2×#481C211yesmaximal, characteristic, complemented
3C2×#480C211yesmaximal, characteristic, complemented
4#2847C211yesmaximal, characteristic
5#1875C214yesmaximal, complemented
6#1854C214yesmaximal, complemented
7#1824C211yesmaximal, characteristic, complemented
8#1823C211yesmaximal, characteristic, complemented
9#1779C211yesmaximal, characteristic, complemented
10C24×C4C2211yescharacteristic
11C2×#121C2211yescharacteristic
12C2×#121C2212yes—
13C2×#120—21no—
14C2×#117—22no—
15C2×#117—24no—
16C2×#117—24no—
17C2×#117C2214yes—
18C22×C4⋊C4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4—21no—
21C22×C42—21no—
22#481—24no—
23#480—24no—
24C2×#107C2212yescomplemented
25C2×#106C2212yescomplemented
26C2×#102C2212yescomplemented
27C2×#102C2212yescomplemented
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2214yes—
31#386—22no—
32#386—21no—
33#380C2211yescharacteristic
34#380C2212yes—
35#340C2214yescomplemented
36#340C2214yescomplemented
37C25D411yescharacteristic
38C22×D4—22no—
39C22×D4—21no—
40C22×D4C2311yescharacteristic
41C23×C4—22no—
42C23×C4—22no—
43C23×C4—28no—
44C23×C4—21no—
45C23×C4—21no—
46C23×C4D412yes—
47C23×C4D411yescharacteristic
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C22×C8C2312yes—
51C22×C8C2312yes—
52#121—22no—
53#121—22no—
54#121—28no—
55#120—44no—
56#117—216no—
57#117—28no—
58#117—28no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4D414yescomplemented
64C2×C4⋊C4D414yescomplemented
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—28no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—22no—
75C2×C42—28no—
76C2×C42—24no—
77#107—24no—
78#106—24no—
79#102—24no—
80#102—24no—
81C22⋊C8—22no—
82C22⋊C8—22no—
83C22⋊C8—28no—
84#95—21no—
85#95—22no—
86C24—22no—
87C24—24no—
88C24—22no—
89C24—22no—
90C24—21no—
91C24—42no—
92C24C2×D412yes—
93C24C2×D411yescharacteristic
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—24no—
97C2×D4—44no—
98C2×D4—44no—
99C2×D4—42no—
100C2×D4—44no—
101C2×D4—42no—
102C2×D4—42no—
103C22×C4—216no—
104C22×C4—216no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—24no—
108C22×C4—24no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—24no—
114C22×C4—24no—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—44no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4#4118yes—
128C22×C4C2411yescharacteristic
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—24no—
132C2×C8—24no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—24no—
140C4⋊C4C2×D414yes—
141C4⋊C4C2×D414yes—
142C22⋊C4—216no—
143C22⋊C4—44no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C22⋊C4—44no—
147C42—216no—
148C23—22no—
149C23—22no—
150C23—28no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—48no—
158C23—48no—
159C23—44no—
160C23—44no—
161C23—41no—
162C23—42no—
163C23—41no—
164C23—41no—
165C23—42no—
166C23#12012yes—
167C23#12314yes—
168C23#13512yes—
169C23AGL(1,ℤ/8)12yes—
170C23C22×D411yescharacteristic
171D4—42no—
172D4—42no—
173D4—88no—
174D4—84no—
175C2×C4—216no—
176C2×C4—28no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—28no—
183C2×C4—28no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—48no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—44no—
190C2×C4—48no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4#12012yes—
198C2×C4#12314yes—
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4114yes—
202C8—42no—
203C8—42no—
204C22—24no—
205C22—24no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—24no—
210C22—24no—
211C22—416no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22—82no—
217C22—82no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22#44914yes—
222C22C2×#12011yescharacteristic
223C22C2×#12312yes—
224C22C2×#13511yescharacteristic
225C22C2×AGL(1,ℤ/8)11yescharacteristic
226C22#57512yes—
227C4—22no—
228C4—24no—
229C4—44no—
230C4—44no—
231C4—82no—
232C4—81no—
233C4#52512yes—
234C4#52512yes—
235C2—24no—
236C2—44no—
237C2—161no—
238C2#284911yescharacteristic
239C2C2×#44911yescharacteristic
240C2#300511yescharacteristic
241C2#309112yes—
242C2#309412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2847C2
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. C22×#117C2semidirect
2. C2×#481C2semidirect
3. C2×#480C2semidirect
5. #1875C2semidirect
6. #1854C2semidirect
7. #1824C2semidirect
8. #1823C2semidirect
9. #1779C2semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
26. C2×#102C22semidirect
27. C2×#102C22semidirect
35. #340C22semidirect
36. #340C22semidirect
63. C2×C4⋊C4D4semidirect
64. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#1172
C2×#4812
C2×#4802
#28472
#18752
#18542
#18242
#18232
#17792