Finite Group Databasebeta

← #31270

Subgroups — #31270

Order 256.

528 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#2563C211yesmaximal, characteristic, complemented
2#2408C211yesmaximal, characteristic
3#2076C212yesmaximal, complemented
4#2033C212yesmaximal, complemented
5#2023C212yesmaximal, complemented
6#1925C212yesmaximal, complemented
7#1895C212yesmaximal, complemented
8#1870C211yesmaximal, characteristic, complemented
9#1786C212yesmaximal, complemented
10C23×D4C2211yescharacteristic
11C22×M16C2211yescharacteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic, complemented
13C2×#121C2212yes—
14C2×#120—21no—
15C2×C4×D4C2211yescharacteristic
16C2×#107C2212yescomplemented
17C2×#102C2212yes—
18C2×#102C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×C22⋊C8C2212yes—
22C2×C22⋊C8C2211yescharacteristic, complemented
23#394—21no—
24#393—21no—
25#393—21no—
26#392—21no—
27#390—21no—
28#390C2212yes—
29#382—21no—
30#380C2212yes—
31#380C2211yescharacteristic
32#380C2212yescomplemented
33#379—21no—
34C4×C4⋊C4C2211yescharacteristic, complemented
35#340C2212yescomplemented
36#340C2212yescomplemented
37#340C2212yescomplemented
38#336C2212yescomplemented
39C25—21no—
40C22×Q8—21no—
41C22×D4—24no—
42C22×D4—22no—
43C22×D4—21no—
44C22×D4—42no—
45C22×D4—41no—
46C22×D4—41no—
47C22×D4D411yescharacteristic
48C22×D4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C2×M16—42no—
52C2×M16—41no—
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—41no—
56C4⋊Q8—41no—
57#121—24no—
58#121—42no—
59#120—44no—
60C4×D4—41no—
61C4×D4—41no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4D411yescharacteristic
74C2×C22⋊C4D411yescharacteristic
75C2×C42—21no—
76C2×C42C2311yescharacteristic
77#107—42no—
78#102—24no—
79#102—42no—
80#102—42no—
81#102—42no—
82C22⋊C8—42no—
83C22⋊C8—41no—
84#95—21no—
85#95—22no—
86#95—21no—
87#95—22no—
88C24—22no—
89C24—44no—
90C24—42no—
91C24—42no—
92C24—42no—
93C24—82no—
94C24C2×D411yescharacteristic
95C24C2×D411yescharacteristic
96C2×Q8—42no—
97C2×Q8—82no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—41no—
106C2×D4—44no—
107C2×D4—42no—
108C2×D4—44no—
109C2×D4—48no—
110C2×D4—44no—
111C2×D4—44no—
112C2×D4—44no—
113C2×D4—82no—
114C2×D4#4114yes—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131M16—44no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—41no—
145C4⋊C4—42no—
146C22⋊C4—41no—
147C22⋊C4—41no—
148C22⋊C4—44no—
149C22⋊C4—42no—
150C22⋊C4—44no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C22⋊C4—82no—
154C42—21no—
155C42—21no—
156C42—42no—
157C23—21no—
158C23—22no—
159C23—21no—
160C23—22no—
161C23—21no—
162C23—24no—
163C23—22no—
164C23—21no—
165C23—21no—
166C23—22no—
167C23—48no—
168C23—42no—
169C23—42no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—88no—
175C23—84no—
176C23—88no—
177C23—84no—
178C23#13712yes—
179C23C22×D411yescharacteristic
180Q8—84no—
181D4—24no—
182D4—24no—
183D4—44no—
184D4—48no—
185D4—42no—
186D4—42no—
187D4—88no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4AGL(1,ℤ/8)12yes—
212C2×C4C2×#4112yes—
213C2×C4H(5,𝔽2)11yescharacteristic
214C2×C4H(5,𝔽2)12yes—
215C2×C4H(5,𝔽2)11yescharacteristic
216C8—42no—
217C8—42no—
218C22—22no—
219C22—24no—
220C22—41no—
221C22—41no—
222C22—42no—
223C22—41no—
224C22—41no—
225C22—44no—
226C22—41no—
227C22—42no—
228C22—44no—
229C22—42no—
230C22—44no—
231C22—816no—
232C22—82no—
233C22—82no—
234C22—82no—
235C22—82no—
236C22#53411yescharacteristic
237C22#54612yes—
238C22C2×AGL(1,ℤ/8)11yescharacteristic
239C22C2×#13711yescharacteristic
240C22#57612yes—
241C4—22no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—82no—
246C4—82no—
247C4—82no—
248C4#53812yes—
249C4#55111yescharacteristic
250C4H(3,𝔽4)11yescharacteristic
251C2—42no—
252C2—81no—
253C2—84no—
254C2#268411yescharacteristic
255C2#319311yescharacteristic
256C2#319911yescharacteristic
257C2#331412yes—
258C2#332512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2563C2semidirect
3. #2076C2semidirect
4. #2033C2semidirect
5. #2023C2semidirect
6. #1925C2semidirect
7. #1895C2semidirect
8. #1870C2semidirect
9. #1786C2semidirect
11. C22×M16C22semidirect
12. C2×C4⋊Q8C22semidirect
16. C2×#107C22semidirect
22. C2×C22⋊C8C22semidirect
32. #380C22semidirect
34. C4×C4⋊C4C22semidirect
35. #340C22semidirect
36. #340C22semidirect
37. #340C22semidirect
38. #336C22semidirect

Maximal subgroups

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

subgroupindex
#25632
#24082
#20762
#20332
#20232
#19252
#18952
#18702
#17862