Finite Group Databasebeta

← #23568

Subgroups — #23568

Order 256.

620 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 153 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×#533C211yesmaximal, characteristic, complemented
2C2×C4×SD16C212yesmaximal, complemented
3C4×#121C211yesmaximal, characteristic
4#2058C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2035C212yesmaximal, complemented
7#2023C211yesmaximal, characteristic, complemented
8#2013C211yesmaximal, characteristic, complemented
9#2000C212yesmaximal
10C4×#103C212yesmaximal, complemented
11C4×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13#533—28no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×#121C2211yescharacteristic
20C2×C4×Q8—21no—
21C2×C4×Q8C2212yes—
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25C22×C42C2211yescharacteristic
26C4×SD16—28no—
27C2×#106C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2212yes—
33C2×C4×C8C2212yes—
34#390C2211yescharacteristic
35#389C2211yescharacteristic
36#388—21no—
37#386—21no—
38#380C2211yescharacteristic
39C4×C4⋊C4C2211yescharacteristic
40C4×C22⋊C4—21no—
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C22×Q8—21no—
46C22×Q8C2312yes—
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C2×SD16—24no—
54C2×SD16—24no—
55C2×SD16—24no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—24no—
59C4⋊Q8—24no—
60#125—28no—
61C22⋊Q8—28no—
62C22⋊Q8—24no—
63C22⋊Q8—24no—
64#121—22no—
65#121—22no—
66C4×Q8—28no—
67C4×Q8—24no—
68C4×Q8D418yescomplemented
69C4×D4—24no—
70C4×D4—44no—
71#117—28no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C42—22no—
89C2×C42—24no—
90C2×C42—21no—
91C2×C42D411yescharacteristic
92C2×C42D411yescharacteristic
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95#106—24no—
96#103—24no—
97#103—24no—
98#103—24no—
99#102—24no—
100C22⋊C8—24no—
101C4×C8—24no—
102#95—21no—
103#95—21no—
104#95—21no—
105C24—21no—
106C24C2×D411yescharacteristic
107C2×Q8—24no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8C2×D414yes—
116C2×D4—24no—
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4#4112yes—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4SD1614yescomplemented
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147SD16—216no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8#4114yes—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—28no—
157C4⋊C4—24no—
158C4⋊C4—28no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4C2×D414yes—
165C4⋊C4#4112yes—
166C22⋊C4—28no—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—22no—
173C42—22no—
174C42—28no—
175C42—22no—
176C42—42no—
177C42C2×D414yes—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—22no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23C2×SD1612yes—
192C23C22×D411yescharacteristic
193Q8—24no—
194Q8—24no—
195Q8—48no—
196Q8—42no—
197Q8—42no—
198D4—24no—
199D4—48no—
200D4—42no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—28no—
208C2×C4—21no—
209C2×C4—24no—
210C2×C4—24no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—84no—
230C2×C4#12012yes—
231C2×C4#12012yes—
232C2×C4C2×SD1612yes—
233C2×C4C2×SD1612yes—
234C2×C4#13512yes—
235C2×C4#13712yes—
236C2×C4C22×D412yes—
237C2×C4C2×#4111yescharacteristic
238C2×C4C2×#4112yes—
239C2×C4C2×#4111yescharacteristic
240C8—24no—
241C8—42no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—41no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22C2×#12011yescharacteristic
257C22#52511yescharacteristic
258C22#53211yescharacteristic
259C22C22×SD1611yescharacteristic
260C22C2×#13511yescharacteristic
261C22C2×#13711yescharacteristic
262C22#57511yescharacteristic
263C4—22no—
264C4—22no—
265C4—22no—
266C4—42no—
267C4—41no—
268C4—44no—
269C4—44no—
270C4—81no—
271C4—81no—
272C4#44814yes—
273C4#52512yes—
274C4#53212yes—
275C2—42no—
276C2—82no—
277C2#238811yescharacteristic
278C2C2×#44811yescharacteristic
279C2#300211yescharacteristic
280C2#310111yescharacteristic
281C2#310711yescharacteristic
282C2#316711yescharacteristic
283C2#317211yescharacteristic
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#533C2semidirect
2. C2×C4×SD16C2semidirect
4. #2058C2semidirect
5. #2043C2semidirect
6. #2035C2semidirect
7. #2023C2semidirect
8. #2013C2semidirect
10. C4×#103C2semidirect
11. C4×C22⋊C8C2semidirect
68. C4×Q8D4semidirect
140. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×C4×SD162
C4×#1212
#20582
#20432
#20352
#20232
#20132
#20002
C4×#1032
C4×C22⋊C82