Finite Group Databasebeta

← #25598

Subgroups — #25598

Order 256.

649 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 155 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×#533C211yesmaximal, characteristic, complemented
2C2×#492C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#2856C211yesmaximal, characteristic
5#2000C214yesmaximal, complemented
6#1990C211yesmaximal, characteristic, complemented
7#1956C211yesmaximal, characteristic, complemented
8#1880C214yesmaximal, complemented
9C4×C22⋊C8C211yesmaximal, characteristic, complemented
10C22×SD16C2214yes—
11#533—24no—
12#533—24no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×#121—21no—
20C2×#121—21no—
21C2×#121C2212yes—
22C2×C4×Q8—21no—
23C2×C4×Q8C2212yes—
24C2×#117—21no—
25C22×C42C2211yescharacteristic
26#492—24no—
27#486—24no—
28C2×#103C2214yes—
29C2×#102C2214yes—
30C2×C22⋊C8C2212yes—
31C2×C4×C8C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33#390—21no—
34#390—21no—
35#390C2212yes—
36#384C2211yescharacteristic
37#340C2214yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8C2312yes—
41C22×D4—22no—
42C22×D4—21no—
43C22×D4—21no—
44C22×D4C2312yes—
45C23×C4D411yescharacteristic
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C2×SD16—28no—
49C2×SD16—28no—
50C2×SD16—44no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8D414yescomplemented
56C4⋊Q8D414yescomplemented
57#127—22no—
58#127—22no—
59#125—28no—
60C22⋊Q8—28no—
61C22⋊Q8—24no—
62C22⋊Q8—24no—
63#121—44no—
64#121—44no—
65#121—44no—
66C4×Q8—28no—
67C4×Q8—24no—
68C4×Q8—24no—
69#117—28no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2312yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42D411yescharacteristic
87C2×C42D411yescharacteristic
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90#103—44no—
91#102—28no—
92C22⋊C8—24no—
93C4×C8—22no—
94C4×C8—22no—
95C24—22no—
96C24C2×D411yescharacteristic
97C2×Q8—24no—
98C2×Q8—22no—
99C2×Q8—24no—
100C2×Q8—24no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8C2×D414yes—
104C2×D4—22no—
105C2×D4—22no—
106C2×D4—24no—
107C2×D4—24no—
108C2×D4—44no—
109C2×D4—44no—
110C2×D4—44no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4—42no—
115C2×D4—42no—
116C2×D4—42no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4SD1614yescomplemented
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136SD16—416no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8#4114yes—
143C4⋊C4—28no—
144C4⋊C4—28no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4C2×D414yes—
153C22⋊C4—28no—
154C22⋊C4—44no—
155C22⋊C4—44no—
156C22⋊C4—44no—
157C42—28no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42C2×D412yes—
163C42C2×D412yes—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23C2×SD1612yes—
178C23C22×D411yescharacteristic
179Q8—28no—
180Q8—44no—
181Q8—48no—
182D4—44no—
183D4—44no—
184D4—88no—
185D4—84no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—28no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4#12012yes—
209C2×C4#12012yes—
210C2×C4C2×SD1612yes—
211C2×C4C2×SD1612yes—
212C2×C4#13512yes—
213C2×C4AGL(1,ℤ/8)12yes—
214C2×C4#13712yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4112yes—
218C2×C4H(5,𝔽2)12yes—
219C8—24no—
220C8—42no—
221C22—22no—
222C22—22no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—82no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22C2×#12011yescharacteristic
236C22#53211yescharacteristic
237C22#53511yescharacteristic
238C22C22×SD1611yescharacteristic
239C22C2×#13511yescharacteristic
240C22C2×AGL(1,ℤ/8)11yescharacteristic
241C22C2×#13711yescharacteristic
242C4—22no—
243C4—41no—
244C4—44no—
245C4—41no—
246C4—44no—
247C4—82no—
248C4#44814yes—
249C4#44914yes—
250C4#54612yes—
251C4#55012yes—
252C2—42no—
253C2—162no—
254C2#243111yescharacteristic
255C2C2×#44811yescharacteristic
256C2C2×#44911yescharacteristic
257C2#316711yescharacteristic
258C2#318211yescharacteristic
259C2#320811yescharacteristic
260C2#321011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2856C2
LCSG ▷ C22×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
1. C2×#533C2semidirect
2. C2×#492C2semidirect
3. C2×#486C2semidirect
5. #2000C2semidirect
6. #1990C2semidirect
7. #1956C2semidirect
8. #1880C2semidirect
9. C4×C22⋊C8C2semidirect
55. C4⋊Q8D4semidirect
56. C4⋊Q8D4semidirect
129. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4922
C2×#4862
#28562
#20002
#19902
#19562
#18802
C4×C22⋊C82