Finite Group Databasebeta

← #28343

Subgroups — #28343

Order 256.

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

normal subgroups only

This group has 642 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×#483C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4C22×#102C211yesmaximal, characteristic, complemented
5#2501C211yesmaximal, characteristic, complemented
6#2435C211yesmaximal, characteristic, complemented
7#2094C212yesmaximal, complemented
8#2012C212yesmaximal, complemented
9#1972C212yesmaximal, complemented
10#1959C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2211yescharacteristic, complemented
14C22×D8C2212yes—
15C23×C8C2211yescharacteristic
16C2×#124C2212yes—
17C2×#123—22no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#483—24no—
22#465—24no—
23#447—24no—
24C2×#106C2212yescomplemented
25C2×#106C2211yescharacteristic, complemented
26C2×#103C2212yescomplemented
27C2×#103C2211yescharacteristic, complemented
28C2×#102—24no—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2211yescharacteristic, complemented
33C2×C22⋊C8C2211yescharacteristic
34C2×#97C2212yescomplemented
35#399—22no—
36#394—22no—
37#390C2212yes—
38#390C2212yescomplemented
39#384C2212yescomplemented
40#379—22no—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yescomplemented
43C25—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—24no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—42no—
51C22×D4C2311yescharacteristic
52C22×D4C2312yes—
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C2×SD16—22no—
56C2×SD16—42no—
57C2×D8—24no—
58C2×D8—24no—
59C2×D8—42no—
60C22×C8—42no—
61C22×C8D414yes—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2312yes—
65#124—24no—
66#124—42no—
67#123—48no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70#121—22no—
71#121—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—24no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4D412yes—
86C2×C42C2312yes—
87#106—22no—
88#106—24no—
89#103—22no—
90#103—24no—
91#102—24no—
92#102—24no—
93#102—24no—
94#102—22no—
95C22⋊C8D414yes—
96#97—42no—
97#95D412yes—
98C24—22no—
99C24—21no—
100C24—44no—
101C24—42no—
102C24—44no—
103C24—82no—
104C24C2×D411yescharacteristic
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×D4—28no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—48no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—42no—
123C2×D4—44no—
124C2×D4—42no—
125C2×D4#4114yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—42no—
133C22×C4—44no—
134C22×C4C2×D414yes—
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D412yes—
137C22×C4C2×D412yes—
138C22×C4C2411yescharacteristic
139SD16—44no—
140D8—48no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—44no—
146C2×C8C2×D414yes—
147C2×C8C2×D412yes—
148C2×C8C2×D412yes—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—48no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C22⋊C4—24no—
159C22⋊C4—48no—
160C22⋊C4—48no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—84no—
164C42—22no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—24no—
169C23—48no—
170C23—42no—
171C23—42no—
172C23—44no—
173C23—44no—
174C23—41no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—816no—
179C23—84no—
180C23—84no—
181C23#12712yes—
182C23AGL(1,ℤ/8)14yes—
183C23C22×D411yescharacteristic
184Q8—41no—
185Q8—41no—
186Q8—42no—
187D4—24no—
188D4—24no—
189D4—48no—
190D4—41no—
191D4—42no—
192D4—41no—
193D4—42no—
194D4—42no—
195D4—84no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—84no—
212C2×C4#12712yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C2×#4112yes—
216C2×C4H(5,𝔽2)12yes—
217C8—22no—
218C8—22no—
219C8—24no—
220C22—22no—
221C22—28no—
222C22—22no—
223C22—48no—
224C22—44no—
225C22—816no—
226C22—82no—
227C22—82no—
228C22—84no—
229C22—81no—
230C22—81no—
231C22—84no—
232C22—81no—
233C22#49614yes—
234C22C2×#12711yescharacteristic
235C22#54612yes—
236C22C2×AGL(1,ℤ/8)12yes—
237C22#57712yes—
238C4—22no—
239C4—44no—
240C4—82no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4#52511yescharacteristic
245C4#53411yescharacteristic
246C4#54612yes—
247C2—24no—
248C2—84no—
249C2—161no—
250C2#253911yescharacteristic
251C2C2×#49611yescharacteristic
252C2#315411yescharacteristic
253C2#330614yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#483C2semidirect
2. C2×#465C2semidirect
4. C22×#102C2semidirect
5. #2501C2semidirect
6. #2435C2semidirect
7. #2094C2semidirect
8. #2012C2semidirect
9. #1972C2semidirect
10. #1959C2semidirect
11. #1752C2semidirect
13. C22×SD16C22semidirect
24. C2×#106C22semidirect
25. C2×#106C22semidirect
26. C2×#103C22semidirect
27. C2×#103C22semidirect
32. C2×#102C22semidirect
34. C2×#97C22semidirect
38. #390C22semidirect
39. #384C22semidirect
42. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4652
C2×#4472
C22×#1022
#25012
#24352
#20942
#20122
#19722
#19592
#17522