Finite Group Databasebeta

← #28342

Subgroups — #28342

Order 256.

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

normal subgroups only

This group has 601 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×#484C211yesmaximal, characteristic, complemented
2C2×#467C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic
4C22×#102C211yesmaximal, characteristic
5#2502C211yesmaximal, characteristic, complemented
6#2435C211yesmaximal, characteristic
7#2093C212yesmaximal, complemented
8#2015C212yesmaximal, complemented
9#1973C212yesmaximal, complemented
10#1959C212yesmaximal, complemented
11#1752C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×Q16C2211yescharacteristic, complemented
14C22×SD16C2212yes—
15C23×C8C2211yescharacteristic
16C2×#124C2212yes—
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20#484—24no—
21#467—24no—
22#450—24no—
23C2×#107C2212yescomplemented
24C2×#107C2211yescharacteristic, complemented
25C2×#103C2212yescomplemented
26C2×#103C2211yescharacteristic, complemented
27C2×#103C2211yescharacteristic, complemented
28C2×#102—24no—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2212yescomplemented
34#398—22no—
35#394—22no—
36#391C2212yescomplemented
37#390C2212yes—
38#385—22no—
39#384C2212yescomplemented
40#379—22no—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yescomplemented
43C25—21no—
44C22×Q8C2311yescharacteristic
45C22×Q8C2311yescharacteristic
46C22×D4—24no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—42no—
51C22×D4C2312yes—
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×Q16—22no—
55C2×Q16—42no—
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—42no—
59C22×C8—42no—
60C22×C8D414yes—
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2312yes—
64#124—24no—
65#124—42no—
66C22⋊Q8—22no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4D412yes—
84C2×C42C2312yes—
85#107—22no—
86#107—24no—
87#103—22no—
88#103—22no—
89#103—24no—
90#102—24no—
91#102—24no—
92#102—24no—
93C22⋊C8D414yes—
94#97—42no—
95#95—24no—
96#95D412yes—
97C24—22no—
98C24—44no—
99C24—42no—
100C24—44no—
101C24—82no—
102C24C2×D411yescharacteristic
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×D4—28no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—48no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—42no—
121C2×D4#4114yes—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—42no—
130C22×C4—82no—
131C22×C4C2×D414yes—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2411yescharacteristic
136Q16—44no—
137SD16—48no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—44no—
143C2×C8C2×D414yes—
144C2×C8C2×D412yes—
145C2×C8C2×D412yes—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—84no—
156C22⋊C4—24no—
157C22⋊C4—48no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—84no—
161C42—22no—
162C42—22no—
163C23—22no—
164C23—22no—
165C23—24no—
166C23—48no—
167C23—42no—
168C23—42no—
169C23—44no—
170C23—44no—
171C23—816no—
172C23—84no—
173C23—84no—
174C23#12712yes—
175C23#13714yes—
176C23C22×D411yescharacteristic
177Q8—41no—
178Q8—41no—
179Q8—41no—
180Q8—41no—
181Q8—42no—
182Q8—42no—
183D4—24no—
184D4—24no—
185D4—48no—
186D4—42no—
187D4—42no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—84no—
206C2×C4#12712yes—
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C2×C4C2×#4112yes—
210C2×C4H(5,𝔽2)12yes—
211C8—22no—
212C8—22no—
213C8—24no—
214C22—22no—
215C22—22no—
216C22—28no—
217C22—44no—
218C22—48no—
219C22—816no—
220C22—82no—
221C22—82no—
222C22—84no—
223C22#49714yes—
224C22C2×#12711yescharacteristic
225C22#54612yes—
226C22C2×#13712yes—
227C22#57612yes—
228C4—22no—
229C4—44no—
230C4—81no—
231C4—81no—
232C4—82no—
233C4—81no—
234C4—81no—
235C4#52511yescharacteristic
236C4#53411yescharacteristic
237C4#54612yes—
238C2—24no—
239C2—84no—
240C2#253911yescharacteristic
241C2C2×#49711yescharacteristic
242C2#315511yescharacteristic
243C2#331414yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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×#484C2semidirect
2. C2×#467C2semidirect
5. #2502C2semidirect
7. #2093C2semidirect
8. #2015C2semidirect
9. #1973C2semidirect
10. #1959C2semidirect
11. #1752C2semidirect
13. C22×Q16C22semidirect
23. C2×#107C22semidirect
24. C2×#107C22semidirect
25. C2×#103C22semidirect
26. C2×#103C22semidirect
27. C2×#103C22semidirect
33. C2×#97C22semidirect
36. #391C22semidirect
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×#4842
C2×#4672
C2×#4502
C22×#1022
#25022
#24352
#20932
#20152
#19732
#19592
#17522