Finite Group Databasebeta

← #25015

Subgroups — #25015

Order 256.

413 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 131 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×#485C212yesmaximal, complemented
2#2538C211yesmaximal, characteristic
3#2354C211yesmaximal, characteristic
4#2077C212yesmaximal, complemented
5#2035C212yesmaximal, complemented
6#1954C211yesmaximal, characteristic
7#1947C211yesmaximal, characteristic, complemented
8#1854C212yesmaximal, complemented
9#1844C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11C4×#103C211yesmaximal, characteristic, complemented
12C2×C22⋊Q8C2212yes—
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#117C2212yes—
16#485—28no—
17C2×#107C2212yes—
18C2×#106C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×#103C2212yes—
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×C22⋊C8C2212yes—
25C2×C22⋊C8C2212yes—
26C2×C4×C8C2211yescharacteristic
27#396—22no—
28#390—21no—
29#390—21no—
30#389—21no—
31#388—22no—
32#384C2211yescharacteristic
33#384C2211yescharacteristic
34#380C2212yes—
35#380C2212yes—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41#340C2211yescharacteristic
42C22×Q8C2311yescharacteristic
43C22×D4D411yescharacteristic
44C23×C4C2312yes—
45C22×C8C2312yes—
46C22×C8C2311yescharacteristic
47C22×C8C2311yescharacteristic
48C22⋊Q8—24no—
49C22⋊Q8—42no—
50C4×Q8—22no—
51C4×Q8—22no—
52C4×D4—22no—
53C4×D4—22no—
54#117—24no—
55#117—42no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4D412yes—
73C2×C42—22no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77#107—24no—
78#106—24no—
79#106—42no—
80C4⋊C8—22no—
81#103—24no—
82#103—22no—
83#103D414yescomplemented
84C22⋊C8—24no—
85C22⋊C8—24no—
86C4×C8—22no—
87#95—22no—
88#95—21no—
89#95—22no—
90C24C2×D412yes—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8C2×D412yes—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4C2×D412yes—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2411yescharacteristic
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8C2×D412yes—
124C2×C8#4112yes—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—44no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4—81no—
140C4⋊C4—81no—
141C4⋊C4C2×D412yes—
142C22⋊C4—24no—
143C22⋊C4—44no—
144C22⋊C4—44no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C42—44no—
148C42—42no—
149C42—42no—
150C42#4112yes—
151C42#4112yes—
152C23—22no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—22no—
157C23—22no—
158C23—22no—
159C23C22×D411yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164D4—44no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4#12312yes—
193C2×C4#12312yes—
194C2×C4#13512yes—
195C2×C4#13712yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4111yescharacteristic
198C2×C4C2×#4111yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4H(5,𝔽2)11yescharacteristic
201C2×C42−1+411yescharacteristic
202C8—22no—
203C8—22no—
204C8—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22C2×#12311yescharacteristic
213C22D4211yescharacteristic
214C22#54711yescharacteristic
215C22C2×#13511yescharacteristic
216C22C2×#13711yescharacteristic
217C22#57711yescharacteristic
218C22#57711yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—42no—
223C4—81no—
224C4—81no—
225C4—82no—
226C4—82no—
227C4#48514yescomplemented
228C4#53211yescharacteristic
229C4#54211yescharacteristic
230C4#54711yescharacteristic
231C4#55311yescharacteristic
232C2—82no—
233C2#242211yescharacteristic
234C2C2×#48511yescharacteristic
235C2#309911yescharacteristic
236C2#327811yescharacteristic
237C2#328411yescharacteristic
238C2#334011yescharacteristic
239C2#335611yescharacteristic
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×#485C2semidirect
4. #2077C2semidirect
5. #2035C2semidirect
7. #1947C2semidirect
8. #1854C2semidirect
9. #1844C2semidirect
10. #1803C2semidirect
11. C4×#103C2semidirect
83. #103D4semidirect
227. C4#485semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
#25382
#23542
#20772
#20352
#19542
#19472
#18542
#18442
#18032
C4×#1032