Finite Group Databasebeta

← #22081

Subgroups — #22081

Order 256.

484 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C2×#480C211yesmaximal, characteristic
2C4×#121C211yesmaximal, characteristic
3C2×#389C211yesmaximal, characteristic, complemented
4#1943C212yesmaximal, complemented
5#1936C212yesmaximal, complemented
6#1921C212yesmaximal, complemented
7#1874C211yesmaximal, characteristic
8#1864C212yesmaximal, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1823C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic, complemented
12C2×#121C2211yescharacteristic
13C2×C4×D4—21no—
14C2×C4×D4C2211yescharacteristic
15C22×C4⋊C4—21no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#480C418yescomplemented
20C2×#107C2212yes—
21C2×#106C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2212yes—
27#390C2211yescharacteristic
28#389—21no—
29#389—21no—
30#389C2212yes—
31#389C2212yes—
32#389C2211yescharacteristic
33#388—21no—
34#386—21no—
35#380C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4—21no—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2312yes—
54#121—22no—
55#121C2×C414yes—
56C4×D4—42no—
57C4×D4—44no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—44no—
66C2×C4⋊C4C2×C414yescomplemented
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C42—21no—
78C2×C42—42no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#107C2×C418yescomplemented
84#106—42no—
85#102—42no—
86#102C2×C418yescomplemented
87C22⋊C8—22no—
88C22⋊C8C2×C414yescomplemented
89#97—42no—
90#95—22no—
91#95—22no—
92#95—21no—
93#95—21no—
94#95—21no—
95C24—21no—
96C24C2×D411yescharacteristic
97C2×D4—24no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4C22×C412yes—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C22×C412yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4#4112yes—
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8C22×C414yes—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4C22×C414yes—
152C4⋊C4C22×C412yes—
153C22⋊C4—24no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C42—22no—
159C42—22no—
160C42—41no—
161C42—41no—
162C42—42no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23C4×D412yes—
175C23AGL(1,ℤ/8)12yes—
176C23AGL(1,ℤ/8)12yes—
177C23C22×D411yescharacteristic
178D4—24no—
179D4—48no—
180D4—42no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—44no—
199C2×C4—44no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—48no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4C4×D412yes—
217C2×C4#13712yes—
218C2×C4C23×C411yescharacteristic
219C2×C4C2×#4111yescharacteristic
220C2×C4H(5,𝔽2)12yes—
221C2×C42−1+412yes—
222C8—24no—
223C8—42no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22#44214yes—
238C22C2×C4×D411yescharacteristic
239C22#53411yescharacteristic
240C22#53611yescharacteristic
241C22C2×AGL(1,ℤ/8)11yescharacteristic
242C22C2×AGL(1,ℤ/8)11yescharacteristic
243C22C2×#13711yescharacteristic
244C22#57511yescharacteristic
245C4—22no—
246C4—22no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—44no—
251C4—42no—
252C4—81no—
253C4—82no—
254C4—81no—
255C4#52012yes—
256C4#53912yes—
257C2—24no—
258C2—82no—
259C2#232211yescharacteristic
260C2C2×#44211yescharacteristic
261C2C4×#13711yescharacteristic
262C2#319311yescharacteristic
263C2#319411yescharacteristic
264C2#323411yescharacteristic
265C2#323811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#389C2semidirect
4. #1943C2semidirect
5. #1936C2semidirect
6. #1921C2semidirect
8. #1864C2semidirect
9. #1853C2semidirect
10. #1823C2semidirect
11. #1749C2semidirect
19. #480C4semidirect
66. C2×C4⋊C4C2×C4semidirect
83. #107C2×C4semidirect
86. #102C2×C4semidirect
88. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C4×#1212
C2×#3892
#19432
#19362
#19212
#18742
#18642
#18532
#18232
#17492