Finite Group Databasebeta

← #27588

Subgroups — #27588

Order 256.

542 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 125 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×#390C211yesmaximal, characteristic
2#2283C211yesmaximal, characteristic, complemented
3#2092C212yesmaximal, complemented
4#2073C212yesmaximal, complemented
5#1926C212yesmaximal, complemented
6#1925C212yesmaximal, complemented
7#1853C211yesmaximal, characteristic, complemented
8#1852C211yesmaximal, characteristic, complemented
9#1817C211yesmaximal, characteristic, complemented
10#1786C211yesmaximal, characteristic
11#1786C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C22⋊C4—21no—
15C22×C22⋊C4—21no—
16C22×C42C2211yescharacteristic
17C2×#107C2212yescomplemented
18C2×#106C2212yescomplemented
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25#390—21no—
26#390—21no—
27#390C2212yes—
28#390C2212yes—
29#383—21no—
30#382C2211yescharacteristic, complemented
31#382C2211yescharacteristic, complemented
32#381—21no—
33#380C2211yescharacteristic
34#380C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic, complemented
36C4×C4⋊C4C2211yescharacteristic, complemented
37C4×C22⋊C4—21no—
38#376—21no—
39#340C2212yescomplemented
40#340C2212yescomplemented
41#336C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×D4—24no—
45C22×D4—21no—
46C22×D4—22no—
47C22×D4—22no—
48C22×D4—42no—
49C22×D4C2312yes—
50C23×C4—21no—
51C23×C4D411yescharacteristic
52C23×C4D411yescharacteristic
53C23×C4C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2312yes—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—44no—
70C2×C22⋊C4—44no—
71C2×C42—22no—
72C2×C42—22no—
73C2×C42—21no—
74C2×C42—21no—
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42C2311yescharacteristic
78C2×C42C2311yescharacteristic
79#107—42no—
80#106—42no—
81#102—24no—
82#102—24no—
83#102—42no—
84#102—42no—
85C22⋊C8—22no—
86C22⋊C8—22no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94#95—21no—
95C24—24no—
96C24—22no—
97C24—42no—
98C24—42no—
99C24—42no—
100C24—42no—
101C24—42no—
102C24—42no—
103C24C2×D411yescharacteristic
104C2×D4—24no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—48no—
111C2×D4—48no—
112C2×D4—42no—
113C2×D4—42no—
114C2×D4#4114yes—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4#4112yes—
137C22×C4#4112yes—
138C22×C4C2411yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—42no—
152C22⋊C4—44no—
153C22⋊C4—44no—
154C22⋊C4—44no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—44no—
158C42—22no—
159C42—22no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—42no—
165C42—42no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—24no—
171C23—48no—
172C23—48no—
173C23—48no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—84no—
181C23—84no—
182C23—84no—
183C23—84no—
184C23#12312yes—
185C23#13512yes—
186C23AGL(1,ℤ/8)12yes—
187C23C22×D411yescharacteristic
188D4—24no—
189D4—24no—
190D4—48no—
191D4—42no—
192D4—42no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—84no—
226C2×C4#12312yes—
227C2×C4AGL(1,ℤ/8)12yes—
228C2×C4C2×#4111yescharacteristic
229C2×C4C2×#4111yescharacteristic
230C2×C4C2×#4112yes—
231C2×C4H(5,𝔽2)12yes—
232C8—42no—
233C8—42no—
234C22—22no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—44no—
241C22—44no—
242C22—44no—
243C22—816no—
244C22—82no—
245C22—82no—
246C22—82no—
247C22—82no—
248C22#48214yes—
249C22C2×#12311yescharacteristic
250C22#52511yescharacteristic
251C22#53511yescharacteristic
252C22C2×#13511yescharacteristic
253C22C2×AGL(1,ℤ/8)11yescharacteristic
254C22C2×AGL(1,ℤ/8)11yescharacteristic
255C22#57511yescharacteristic
256C4—22no—
257C4—22no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—81no—
265C4—82no—
266C4#55112yes—
267C4#55112yes—
268C2—24no—
269C2—84no—
270C2#248211yescharacteristic
271C2C2×#48211yescharacteristic
272C2#309411yescharacteristic
273C2#310511yescharacteristic
274C2#311411yescharacteristic
275C2#321111yescharacteristic
276C2#321711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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
2. #2283C2semidirect
3. #2092C2semidirect
4. #2073C2semidirect
5. #1926C2semidirect
6. #1925C2semidirect
7. #1853C2semidirect
8. #1852C2semidirect
9. #1817C2semidirect
17. C2×#107C22semidirect
18. C2×#106C22semidirect
30. #382C22semidirect
31. #382C22semidirect
35. C4×C4⋊C4C22semidirect
36. C4×C4⋊C4C22semidirect
39. #340C22semidirect
40. #340C22semidirect
41. #336C22semidirect
42. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#3902
#22832
#20922
#20732
#19262
#19252
#18532
#18522
#18172
#17862