Finite Group Databasebeta

← #25611

Subgroups — #25611

Order 256.

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

normal subgroups only

This group has 611 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×#540C211yesmaximal, characteristic, complemented
2C2×#496C211yesmaximal, characteristic, complemented
3C2×#488C211yesmaximal, characteristic, complemented
4#2620C211yesmaximal, characteristic, complemented
5#2000C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1992C211yesmaximal, characteristic, complemented
8#1957C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14#540—24no—
15#540—24no—
16C2×#127C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×#123—22no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121—22no—
22C2×#121C2212yes—
23C2×#121C2211yescharacteristic
24C2×C4×D4C2212yes—
25C2×#117C2211yescharacteristic
26#496—24no—
27#488—24no—
28C2×#103C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2212yes—
33C2×#97C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#396—22no—
36#394—22no—
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#384C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—22no—
45C22×D4—21no—
46C22×D4—22no—
47C22×D4C2311yescharacteristic
48C22×D4C2312yes—
49C23×C4—22no—
50C23×C4C2311yescharacteristic
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—42no—
54C2×D8—24no—
55C2×D8—24no—
56C2×D8—42no—
57C22×C8C2312yes—
58C22×C8C2312yes—
59#127—22no—
60#127D414yescomplemented
61#125—22no—
62#125D414yescomplemented
63#124—22no—
64#124—22no—
65#123—216no—
66C22⋊Q8—42no—
67#121—216no—
68#121—24no—
69#121—24no—
70#121—42no—
71C4×D4—24no—
72C4×D4—24no—
73#117—22no—
74#117—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4D412yes—
89C2×C42C2311yescharacteristic
90C2×C42C2311yescharacteristic
91#103—24no—
92#102—24no—
93#102—42no—
94#102—42no—
95C22⋊C8—24no—
96#97—22no—
97C4×C8—22no—
98#95—22no—
99C24—22no—
100C24—21no—
101C24C2×D411yescharacteristic
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×D4—28no—
107C2×D4—28no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—42no—
118C2×D4—82no—
119C2×D4—82no—
120C2×D4—82no—
121C2×D4—82no—
122C2×D4—82no—
123C2×D4C2×D414yes—
124C22×C4—22no—
125C22×C4—28no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—82no—
134C22×C4—82no—
135C22×C4C2×D412yes—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D412yes—
138C22×C4C2411yescharacteristic
139SD16—48no—
140D8—48no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8#4114yes—
147C4⋊C4—28no—
148C4⋊C4—24no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—82no—
154C4⋊C4C2×D414yes—
155C22⋊C4—28no—
156C22⋊C4—28no—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—44no—
161C22⋊C4—82no—
162C42—21no—
163C42—21no—
164C42C2×D412yes—
165C42C2×D412yes—
166C23—21no—
167C23—24no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23—22no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—42no—
182C23—42no—
183C23C22×D411yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—24no—
187D4—24no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—42no—
192D4—88no—
193D4—88no—
194D4—84no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—88no—
216C2×C4#12014yes—
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C2×C4C2×#4112yes—
220C2×C4H(5,𝔽2)12yes—
221C8—24no—
222C8—42no—
223C22—42no—
224C22—41no—
225C22—44no—
226C22—41no—
227C22—41no—
228C22—44no—
229C22—42no—
230C22—44no—
231C22—82no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—82no—
240C22C2×#12011yescharacteristic
241C22#53211yescharacteristic
242C22#53511yescharacteristic
243C22#57612yes—
244C22#57712yes—
245C4—24no—
246C4—41no—
247C4—42no—
248C4—44no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4#54612yes—
253C4#55012yes—
254C2—81no—
255C2—84no—
256C2—161no—
257C2#243111yescharacteristic
258C2#301212yes—
259C2#317912yes—
260C2#322011yescharacteristic
261C2#322111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#540C2semidirect
2. C2×#496C2semidirect
3. C2×#488C2semidirect
4. #2620C2semidirect
5. #2000C2semidirect
6. #1999C2semidirect
7. #1992C2semidirect
8. #1957C2semidirect
9. #1879C2semidirect
10. #1877C2semidirect
11. #1750C2semidirect
60. #127D4semidirect
62. #125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5402
C2×#4962
C2×#4882
#26202
#20002
#19992
#19922
#19572
#18792
#18772
#17502