Finite Group Databasebeta

← #25776

Subgroups — #25776

Order 256.

828 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C23×SD16C211yesmaximal, characteristic, complemented
2C2×#495C212yesmaximal, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5C2×#447C211yesmaximal, characteristic
6#2436C211yesmaximal, characteristic, complemented
7#2435C211yesmaximal, characteristic, complemented
8#2015C212yesmaximal, complemented
9#2012C212yesmaximal, complemented
10#1968C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C23×D4C2211yescharacteristic
14C22×Q16C2212yescomplemented
15C22×SD16—28no—
16C22×SD16—24no—
17C22×SD16C2212yes—
18C22×SD16C2212yes—
19C22×SD16C2211yescharacteristic, complemented
20C22×D8C2212yes—
21C23×C8C2211yescharacteristic
22C2×#124C2212yes—
23C2×#124C2212yescomplemented
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26#495—28no—
27#465—24no—
28#451—24no—
29#447—24no—
30C2×#106C2211yescharacteristic, complemented
31C2×#103C2212yescomplemented
32C2×#103C2211yescharacteristic, complemented
33C2×#102C2212yes—
34C2×#102C2211yescharacteristic, complemented
35C2×C22⋊C8C2211yescharacteristic
36C2×C4×C8C2212yescomplemented
37#394—22no—
38#393—22no—
39#391C2212yescomplemented
40#390C2212yes—
41#386—22no—
42#379—22no—
43C4×C22⋊C4C2211yescharacteristic
44#336C2212yescomplemented
45C25—21no—
46C22×Q8—21no—
47C22×Q8—24no—
48C22×Q8—22no—
49C22×Q8—22no—
50C22×Q8—42no—
51C22×Q8C2311yescharacteristic
52C22×Q8C2312yes—
53C22×D4—24no—
54C22×D4—21no—
55C22×D4—22no—
56C22×D4—22no—
57C22×D4—21no—
58C22×D4—42no—
59C22×D4C2311yescharacteristic
60C22×D4C2312yes—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C2×Q16—24no—
64C2×Q16—24no—
65C2×Q16—42no—
66C2×SD16—232no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—24no—
71C2×SD16—24no—
72C2×SD16—22no—
73C2×SD16—24no—
74C2×SD16D418yescomplemented
75C2×D8—24no—
76C2×D8—24no—
77C2×D8—42no—
78C22×C8—24no—
79C22×C8D414yes—
80C22×C8C2311yescharacteristic
81C22×C8C2312yes—
82C22×C8C2311yescharacteristic
83#124—24no—
84#124—24no—
85#124—42no—
86#124—42no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89#121—22no—
90#121—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—24no—
97C2×C22⋊C4—24no—
98C2×C22⋊C4—24no—
99C2×C22⋊C4D412yes—
100C2×C42C2312yes—
101#106—22no—
102#103—24no—
103#103—22no—
104#102—24no—
105#102—22no—
106C22⋊C8D414yes—
107C4×C8—24no—
108#95—24no—
109#95D412yes—
110C24—22no—
111C24—21no—
112C24—44no—
113C24—42no—
114C24—44no—
115C24—82no—
116C24C2×D411yescharacteristic
117C2×Q8—22no—
118C2×Q8—28no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—44no—
128C2×Q8—48no—
129C2×Q8—44no—
130C2×Q8—42no—
131C2×Q8C2×D414yes—
132C2×D4—28no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—48no—
143C2×D4—44no—
144C2×D4—44no—
145C2×D4—42no—
146C2×D4—44no—
147C2×D4—42no—
148C2×D4C2×D414yes—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—44no—
156C22×C4—42no—
157C22×C4C2×D414yes—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D412yes—
160C22×C4C2×D412yes—
161C22×C4C2411yescharacteristic
162Q16—48no—
163SD16—216no—
164SD16—216no—
165SD16—44no—
166D8—48no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—28no—
171C2×C8C2×D414yes—
172C2×C8C2×D412yes—
173C2×C8C2×D414yes—
174C2×C8C2×D412yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—42no—
180C22⋊C4—24no—
181C22⋊C4—48no—
182C22⋊C4—48no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—84no—
186C42—22no—
187C42—22no—
188C23—22no—
189C23—22no—
190C23—24no—
191C23—48no—
192C23—42no—
193C23—42no—
194C23—44no—
195C23—44no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—816no—
201C23—84no—
202C23—84no—
203C23#12712yes—
204C23#13514yes—
205C23C22×D411yescharacteristic
206Q8—24no—
207Q8—24no—
208Q8—48no—
209Q8—42no—
210Q8—42no—
211Q8—41no—
212Q8—41no—
213Q8—42no—
214D4—24no—
215D4—24no—
216D4—48no—
217D4—41no—
218D4—42no—
219D4—41no—
220D4—42no—
221D4—42no—
222D4—84no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—48no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4#12712yes—
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D412yes—
243C2×C4C22×D411yescharacteristic
244C2×C4H(5,𝔽2)12yes—
245C8—22no—
246C8—24no—
247C8#12014yes—
248C22—22no—
249C22—22no—
250C22—28no—
251C22—44no—
252C22—48no—
253C22—816no—
254C22—82no—
255C22—82no—
256C22—84no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22—84no—
261C22#49514yescomplemented
262C22C2×#12711yescharacteristic
263C22D4212yes—
264C22C2×#13512yes—
265C22#57712yes—
266C4—22no—
267C4—44no—
268C4—44no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4C2×#12011yescharacteristic
273C4#53511yescharacteristic
274C4D4212yes—
275C2—24no—
276C2—84no—
277C2—161no—
278C2#243311yescharacteristic
279C2C2×#49511yescharacteristic
280C2#315411yescharacteristic
281C2#328214yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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. C23×SD16C2semidirect
2. C2×#495C2semidirect
3. C2×#465C2semidirect
4. C2×#451C2semidirect
6. #2436C2semidirect
7. #2435C2semidirect
8. #2015C2semidirect
9. #2012C2semidirect
10. #1968C2semidirect
11. C8×C22⋊C4C2semidirect
14. C22×Q16C22semidirect
19. C22×SD16C22semidirect
23. C2×#124C22semidirect
30. C2×#106C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
34. C2×#102C22semidirect
36. C2×C4×C8C22semidirect
39. #391C22semidirect
44. #336C22semidirect
74. C2×SD16D4semidirect
261. C22#495semidirect

Maximal subgroups

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

subgroupindex
C23×SD162
C2×#4952
C2×#4652
C2×#4512
C2×#4472
#24362
#24352
#20152
#20122
#19682
C8×C22⋊C42