Finite Group Databasebeta

← #25341

Subgroups — #25341

Order 256.

447 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 139 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×#475C211yesmaximal, characteristic, complemented
2C2×C8×D4C211yesmaximal, characteristic
3#2547C211yesmaximal, characteristic
4#2538C211yesmaximal, characteristic
5#2077C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1990C211yesmaximal, characteristic, complemented
8#1948C211yesmaximal, characteristic, complemented
9#1892C212yesmaximal, complemented
10#1807C212yesmaximal, complemented
11C4×#103C211yesmaximal, characteristic, complemented
12C23×C8C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#124—21no—
15C2×C22⋊Q8C2212yes—
16C2×C4×Q8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18#475—24no—
19C8×D4—24no—
20C2×#106C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×C22⋊C8C2212yes—
27C2×C4×C8C2211yescharacteristic
28#397—22no—
29#396—22no—
30#391—21no—
31#390—21no—
32#389—21no—
33#384C2211yescharacteristic
34#380C2212yes—
35#380C2212yes—
36#380C2212yes—
37C4×C4⋊C4C2211yescharacteristic
38#340C2211yescharacteristic
39#340C2212yes—
40#340C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4D411yescharacteristic
46C23×C4C2312yes—
47C22×C8—24no—
48C22×C8—24no—
49C22×C8—42no—
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55#124—44no—
56C22⋊Q8—24no—
57C22⋊Q8—42no—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×D4—22no—
61C4×D4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4D411yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2312yes—
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2311yescharacteristic
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—22no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4D412yes—
78C2×C42—21no—
79C2×C42—21no—
80C2×C42C2311yescharacteristic
81#106—24no—
82#106—42no—
83C4⋊C8D414yes—
84#103—22no—
85#103—24no—
86#103—22no—
87C22⋊C8—24no—
88C4×C8—22no—
89#95—21no—
90#95—22no—
91#95—22no—
92#95—22no—
93C24C2×D412yes—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8#4112yes—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4SD1614yes—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4C2×D412yes—
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2411yescharacteristic
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—48no—
130C2×C8C2×D414yes—
131C2×C8#4112yes—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—81no—
148C22⋊C4—24no—
149C22⋊C4—44no—
150C22⋊C4—44no—
151C22⋊C4—44no—
152C22⋊C4—82no—
153C42—42no—
154C42—42no—
155C42C2×D412yes—
156C42#4112yes—
157C23—22no—
158C23—22no—
159C23—22no—
160C23—22no—
161C23—22no—
162C23—22no—
163C23C2×SD1614yes—
164C23C22×D411yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—41no—
168Q8—41no—
169Q8—84no—
170D4—28no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4#12112yes—
199C2×C4#12112yes—
200C2×C4C2×SD1612yes—
201C2×C4#13512yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4C2×#4111yescharacteristic
205C2×C4C2×#4111yescharacteristic
206C2×C4H(5,𝔽2)11yescharacteristic
207C2×C42−1+411yescharacteristic
208C8—22no—
209C8—24no—
210C8—41no—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C2×#12111yescharacteristic
219C22#54611yescharacteristic
220C22#54711yescharacteristic
221C22C22×SD1611yescharacteristic
222C22C2×#13511yescharacteristic
223C22#57711yescharacteristic
224C22#57811yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—81no—
229C4—81no—
230C4—82no—
231C4—82no—
232C4—81no—
233C4#46514yes—
234C4#52911yescharacteristic
235C4#54211yescharacteristic
236C4#54611yescharacteristic
237C4#54811yescharacteristic
238C2—44no—
239C2#242711yescharacteristic
240C2C2×#46511yescharacteristic
241C2#306311yescharacteristic
242C2#329811yescharacteristic
243C2#330411yescharacteristic
244C2#333511yescharacteristic
245C2#334211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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. C2×#475C2semidirect
5. #2077C2semidirect
6. #2035C2semidirect
7. #1990C2semidirect
8. #1948C2semidirect
9. #1892C2semidirect
10. #1807C2semidirect
11. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4752
C2×C8×D42
#25472
#25382
#20772
#20352
#19902
#19482
#18922
#18072
C4×#1032