Finite Group Databasebeta

← #28332

Subgroups — #28332

Order 256.

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

normal subgroups only

This group has 672 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×#481C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C22×#102C211yesmaximal, characteristic, complemented
5#2502C211yesmaximal, characteristic, complemented
6#2433C211yesmaximal, characteristic
7#2094C212yesmaximal, complemented
8#2014C212yesmaximal, complemented
9#1968C212yesmaximal, complemented
10#1957C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C22×SD16C2211yescharacteristic, complemented
15C23×C8C2211yescharacteristic
16C2×#127C2212yes—
17C2×C22⋊Q8C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×#120—22no—
20C22×C4⋊C4C2211yescharacteristic
21#481—24no—
22#465—24no—
23#450—24no—
24C2×#106C2212yescomplemented
25C2×#106C2211yescharacteristic, complemented
26C2×#103C2211yescharacteristic, complemented
27C2×#102—24no—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yescomplemented
32C2×#102C2211yescharacteristic, complemented
33C2×C22⋊C8C2211yescharacteristic
34C2×C4×C8C2212yescomplemented
35#398—22no—
36#392—22no—
37#391C2212yescomplemented
38#390C2212yes—
39#385—22no—
40#384C2212yescomplemented
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—24no—
51C22×D4—42no—
52C22×D4C2311yescharacteristic
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4C2311yescharacteristic
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—22no—
59C2×SD16—42no—
60C2×SD16—42no—
61C22×C8—42no—
62C22×C8D414yescomplemented
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2312yes—
66#127—24no—
67#127—42no—
68C22⋊Q8—22no—
69C22⋊Q8—22no—
70#121—22no—
71#121—22no—
72#120—48no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C22⋊C4—21no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4D412yes—
85C2×C42C2312yes—
86#106—22no—
87#106—24no—
88#103—22no—
89#102—24no—
90#102—24no—
91#102—24no—
92#102—22no—
93#102—24no—
94C22⋊C8D414yescomplemented
95C4×C8—42no—
96#95—24no—
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—48no—
123C2×D4—42no—
124C2×D4—44no—
125C2×D4—42no—
126C2×D4—48no—
127C2×D4—84no—
128C2×D4#4114yes—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—42no—
136C22×C4—82no—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2×D412yes—
140C22×C4C2×D414yes—
141C22×C4C2411yescharacteristic
142SD16—48no—
143SD16—44no—
144C2×C8—22no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—44no—
149C2×C8C2×D414yes—
150C2×C8C2×D412yes—
151C2×C8C2×D412yes—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—84no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—48no—
164C22⋊C4SD1618yescomplemented
165C42—22no—
166C42—22no—
167C23—22no—
168C23—22no—
169C23—24no—
170C23—48no—
171C23—42no—
172C23—42no—
173C23—44no—
174C23—44no—
175C23—41no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—816no—
180C23—84no—
181C23—84no—
182C23#12712yes—
183C23C2×SD1614yes—
184C23C22×D411yescharacteristic
185Q8—41no—
186Q8—41no—
187Q8—42no—
188D4—24no—
189D4—24no—
190D4—48no—
191D4—41no—
192D4—42no—
193D4—41no—
194D4—42no—
195D4—42no—
196D4—816no—
197D4—84no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—84no—
213C2×C4#12712yes—
214C2×C4C2×SD1618yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C2×#4112yes—
218C2×C4H(5,𝔽2)12yes—
219C8—22no—
220C8—22no—
221C8—24no—
222C22—22no—
223C22—28no—
224C22—22no—
225C22—48no—
226C22—44no—
227C22—816no—
228C22—82no—
229C22—82no—
230C22—84no—
231C22—81no—
232C22—81no—
233C22—84no—
234C22—81no—
235C22#49214yes—
236C22C2×#12711yescharacteristic
237C22#54612yes—
238C22C22×SD1612yes—
239C22#57612yes—
240C4—28no—
241C4—22no—
242C4—81no—
243C4—81no—
244C4—82no—
245C4—81no—
246C4#52511yescharacteristic
247C4#53411yescharacteristic
248C4#54612yes—
249C2—24no—
250C2—84no—
251C2—161no—
252C2#253911yescharacteristic
253C2C2×#49211yescharacteristic
254C2#315511yescharacteristic
255C2#329514yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
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×#481C2semidirect
2. C2×#465C2semidirect
3. C2×#450C2semidirect
4. C22×#102C2semidirect
5. #2502C2semidirect
7. #2094C2semidirect
8. #2014C2semidirect
9. #1968C2semidirect
10. #1957C2semidirect
11. C8×C22⋊C4C2semidirect
14. C22×SD16C22semidirect
24. C2×#106C22semidirect
25. C2×#106C22semidirect
26. C2×#103C22semidirect
31. C2×#102C22semidirect
32. C2×#102C22semidirect
34. C2×C4×C8C22semidirect
37. #391C22semidirect
40. #384C22semidirect
42. #336C22semidirect
62. C22×C8D4semidirect
94. C22⋊C8D4semidirect
164. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4652
C2×#4502
C22×#1022
#25022
#24332
#20942
#20142
#19682
#19572
C8×C22⋊C42