Finite Group Databasebeta

← #28312

Subgroups — #28312

Order 256.

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

normal subgroups only

This group has 662 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×#480C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic
5#2850C211yesmaximal, characteristic, complemented
6C2×#390C211yesmaximal, characteristic, complemented
7#2092C212yesmaximal, complemented
8#2012C212yesmaximal, complemented
9#1973C212yesmaximal, complemented
10#1972C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C22×D8C2212yes—
15C2×#124C2211yescharacteristic, complemented
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C22⋊C4—22no—
20C22×C42C2211yescharacteristic
21#483—24no—
22#480—24no—
23#450—24no—
24#447—24no—
25C2×#107C2212yescomplemented
26C2×#106C2212yescomplemented
27C2×#103C2212yescomplemented
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yescomplemented
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2212yescomplemented
34#391C2211yescharacteristic, complemented
35#390—22no—
36#390C2212yes—
37#390C2212yes—
38#390C2211yescharacteristic, complemented
39#388—22no—
40#386—22no—
41#382C2211yescharacteristic, complemented
42#336C2212yescomplemented
43C25—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—24no—
46C22×D4—21no—
47C22×D4—21no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—42no—
51C22×D4C2311yescharacteristic
52C22×D4C2312yes—
53C23×C4—21no—
54C23×C4D412yes—
55C23×C4C2311yescharacteristic
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—42no—
59C2×D8—24no—
60C2×D8—24no—
61C2×D8—42no—
62C22×C8C2312yes—
63C22×C8C2312yes—
64#124—42no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67#121—22no—
68#121—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—48no—
83C2×C42—42no—
84C2×C42D412yes—
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87#107—24no—
88#106—24no—
89#103—24no—
90#102—24no—
91#102—24no—
92#102—24no—
93C22⋊C8D414yes—
94C22⋊C8D414yes—
95#97—42no—
96#95—22no—
97#95—22no—
98C24—24no—
99C24—22no—
100C24—21no—
101C24—42no—
102C24—44no—
103C24—42no—
104C24—44no—
105C24C2×D411yescharacteristic
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—48no—
121C2×D4—48no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—44no—
125C2×D4—42no—
126C2×D4#4114yes—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—24no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—44no—
138C22×C4C2×D412yes—
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2411yescharacteristic
144SD16—48no—
145D8—48no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8C2×D414yes—
151C2×C8C2×D414yes—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C22⋊C4—44no—
161C22⋊C4—48no—
162C22⋊C4—48no—
163C22⋊C4—44no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C42—22no—
167C42—22no—
168C42—42no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—24no—
173C23—48no—
174C23—416no—
175C23—42no—
176C23—42no—
177C23—44no—
178C23—44no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—42no—
183C23—42no—
184C23—84no—
185C23—88no—
186C23—84no—
187C23#12712yes—
188C23AGL(1,ℤ/8)14yes—
189C23C22×D411yescharacteristic
190Q8—42no—
191Q8—42no—
192D4—24no—
193D4—24no—
194D4—48no—
195D4—42no—
196D4—42no—
197D4—42no—
198D4—42no—
199D4—84no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—48no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—84no—
221C2×C4#12712yes—
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C2×#4112yes—
225C2×C4H(5,𝔽2)12yes—
226C8—24no—
227C8—24no—
228C22—24no—
229C22—22no—
230C22—24no—
231C22—22no—
232C22—48no—
233C22—44no—
234C22—816no—
235C22—82no—
236C22—82no—
237C22—84no—
238C22—82no—
239C22—81no—
240C22—81no—
241C22—82no—
242C22—81no—
243C22#49614yes—
244C22#52511yescharacteristic
245C22C2×#12711yescharacteristic
246C22#53411yescharacteristic
247C22C2×AGL(1,ℤ/8)12yes—
248C22#57512yes—
249C4—22no—
250C4—42no—
251C4—42no—
252C4—82no—
253C4—81no—
254C4—81no—
255C4—81no—
256C4#54612yes—
257C4#54612yes—
258C2—24no—
259C2—84no—
260C2—161no—
261C2#253911yescharacteristic
262C2#311412yes—
263C2C2×#49611yescharacteristic
264C2#315311yescharacteristic
265C2#319412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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
1. C2×#483C2semidirect
2. C2×#480C2semidirect
3. C2×#450C2semidirect
5. #2850C2semidirect
6. C2×#390C2semidirect
7. #2092C2semidirect
8. #2012C2semidirect
9. #1973C2semidirect
10. #1972C2semidirect
11. #1749C2semidirect
15. C2×#124C22semidirect
25. C2×#107C22semidirect
26. C2×#106C22semidirect
27. C2×#103C22semidirect
30. C2×#102C22semidirect
33. C2×#97C22semidirect
34. #391C22semidirect
38. #390C22semidirect
41. #382C22semidirect
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×#4802
C2×#4502
C2×#4472
#28502
C2×#3902
#20922
#20122
#19732
#19722
#17492