Finite Group Databasebeta

← #33018

Subgroups — #33018

Order 256.

716 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 143 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×#518C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic
4#2625C211yesmaximal, characteristic, complemented
5#1877C212yesmaximal, complemented
6#1877C212yesmaximal, complemented
7#1864C212yesmaximal, complemented
8#1862C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C2×C22⋊Q8C2211yescharacteristic
15#518—24no—
16#518—24no—
17C2×C4×D4—21no—
18C2×C4×D4—21no—
19C2×C4×D4C2212yes—
20C2×C4×D4C2212yes—
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C22⋊C4—22no—
24#485—24no—
25#450—24no—
26C2×#107C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×#103C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#400—21no—
36#398—21no—
37#397—22no—
38#391C2212yescomplemented
39#389C2212yescomplemented
40#380C2211yescharacteristic
41#340C2212yescomplemented
42#340C2212yescomplemented
43#336C2212yescomplemented
44#336C2212yescomplemented
45C25—21no—
46C22×Q8C2311yescharacteristic
47C22×D4—24no—
48C22×D4—24no—
49C22×D4—21no—
50C22×D4—22no—
51C22×D4D414yes—
52C22×D4C2312yes—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4C2311yescharacteristic
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—24no—
59C22×C8C2312yes—
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C4×D4—28no—
65C4×D4—28no—
66C4×D4—24no—
67C4×D4—24no—
68C4×D4—24no—
69C4×D4—24no—
70#117—22no—
71#117—22no—
72#117—22no—
73#117D414yes—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—216no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4D412yes—
89C2×C42—21no—
90C2×C42C2312yes—
91#107—22no—
92#106—22no—
93#103—22no—
94#103—22no—
95#102—24no—
96#102—24no—
97#102—24no—
98C22⋊C8—22no—
99C22⋊C8—22no—
100#95—21no—
101#95—22no—
102#95—22no—
103#95—22no—
104C24—28no—
105C24—22no—
106C24—24no—
107C24—81no—
108C24—82no—
109C24—81no—
110C24C2×D411yescharacteristic
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×D4—28no—
116C2×D4—28no—
117C2×D4—28no—
118C2×D4—216no—
119C2×D4—24no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4C2×D414yes—
124C2×D4#4114yes—
125C2×D4#4114yes—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—28no—
131C22×C4—28no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4C2×D412yes—
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D412yes—
143C22×C4C2×D412yes—
144C22×C4C2411yescharacteristic
145SD16—48no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—82no—
163C4⋊C4C2×D412yes—
164C4⋊C4C2×D412yes—
165C22⋊C4—28no—
166C22⋊C4—28no—
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—42no—
170C22⋊C4—81no—
171C42—24no—
172C42—22no—
173C42—22no—
174C23—216no—
175C23—24no—
176C23—22no—
177C23—24no—
178C23—24no—
179C23—21no—
180C23—22no—
181C23—21no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—84no—
187C23—84no—
188C23—88no—
189C23—88no—
190C23#12012yes—
191C23C22×D411yescharacteristic
192Q8—42no—
193Q8—41no—
194Q8—41no—
195D4—216no—
196D4—24no—
197D4—24no—
198D4—24no—
199D4#12318yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—88no—
224C2×C4—88no—
225C2×C4#12012yes—
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C2×#4112yes—
229C2×C4C2×#4112yes—
230C8—42no—
231C8—41no—
232C8—41no—
233C22—22no—
234C22—28no—
235C22—22no—
236C22—22no—
237C22—42no—
238C22—44no—
239C22—44no—
240C22—42no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—44no—
245C22—816no—
246C22C2×#12011yescharacteristic
247C22#52512yes—
248C22#57611yescharacteristic
249C22#57712yes—
250C22#57711yescharacteristic
251C4—22no—
252C4—42no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—81no—
257C4—82no—
258C4—81no—
259C4C2×#12312yes—
260C4#52511yescharacteristic
261C4#52511yescharacteristic
262C2—42no—
263C2—48no—
264C2#284911yescharacteristic
265C2#300911yescharacteristic
266C2#301211yescharacteristic
267C2#311812yes—
268C2#312312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#485C2semidirect
4. #2625C2semidirect
5. #1877C2semidirect
6. #1877C2semidirect
7. #1864C2semidirect
8. #1862C2semidirect
9. #1825C2semidirect
10. #1818C2semidirect
26. C2×#107C22semidirect
27. C2×#106C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
38. #391C22semidirect
39. #389C22semidirect
41. #340C22semidirect
42. #340C22semidirect
43. #336C22semidirect
44. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4852
C2×#4502
#26252
#18772
#18642
#18622
#18252
#18182
#17862