Finite Group Databasebeta

← #32917

Subgroups — #32917

Order 256.

794 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 155 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, complemented
2C2×#463C212yesmaximal, complemented
3C2×#450C211yesmaximal, characteristic, complemented
4C2×#447C211yesmaximal, characteristic
5#2620C211yesmaximal, characteristic, complemented
6#1879C212yesmaximal, complemented
7#1877C212yesmaximal, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1873C211yesmaximal, characteristic, complemented
10#1802C212yesmaximal, complemented
11#1798C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C22×D8C2212yes—
15C2×#124C2211yescharacteristic, complemented
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18#518—28no—
19C2×C4×D4—22no—
20C2×C4×D4C2212yes—
21C2×C4×D4C2212yes—
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C22⋊C4—22no—
25#463—28no—
26#450—24no—
27#447—24no—
28C2×C4⋊C8C2212yescomplemented
29C2×#103C2212yescomplemented
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yescomplemented
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#396—22no—
36#394—22no—
37#391C2211yescharacteristic, complemented
38#390C2211yescharacteristic, complemented
39#384C2211yescharacteristic, complemented
40#340C2212yescomplemented
41#340C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×Q8C2311yescharacteristic
45C22×D4—24no—
46C22×D4—24no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4D414yes—
51C22×D4C2311yescharacteristic
52C22×D4C2312yes—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4C2311yescharacteristic
56C2×SD16—24no—
57C2×SD16—24no—
58C2×SD16—24no—
59C2×D8—24no—
60C2×D8—24no—
61C2×D8—24no—
62C22×C8C2312yes—
63C22×C8C2312yes—
64#124—24no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67#121—22no—
68#121—22no—
69C4×D4—216no—
70C4×D4—24no—
71C4×D4—24no—
72C4×D4—24no—
73C4×D4D418yescomplemented
74#117—24no—
75#117—24no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4D412yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
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⋊C4—22no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4D412yes—
91C2×C42—21no—
92C2×C42C2311yescharacteristic
93C2×C42C2311yescharacteristic
94C4⋊C8—24no—
95#103—24no—
96#102—24no—
97#102—24no—
98#102—24no—
99C22⋊C8—22no—
100C22⋊C8—22no—
101#95—22no—
102C24—24no—
103C24—24no—
104C24—22no—
105C24—24no—
106C24—21no—
107C24—82no—
108C24—82no—
109C24C2×D411yescharacteristic
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×D4—24no—
115C2×D4—24no—
116C2×D4—216no—
117C2×D4—216no—
118C2×D4—24no—
119C2×D4—22no—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—42no—
127C2×D4—82no—
128C2×D4C2×D414yes—
129C2×D4C2×D414yes—
130C2×D4#4114yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—28no—
135C22×C4—28no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—82no—
140C22×C4—82no—
141C22×C4—82no—
142C22×C4—82no—
143C22×C4C2×D412yes—
144C22×C4C2×D412yes—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D412yes—
147C22×C4#4112yes—
148C22×C4C2411yescharacteristic
149SD16—48no—
150D8—48no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—42no—
164C4⋊C4—82no—
165C4⋊C4C2×D414yes—
166C22⋊C4—24no—
167C22⋊C4—28no—
168C22⋊C4—24no—
169C22⋊C4—28no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—42no—
173C22⋊C4—82no—
174C42—24no—
175C42—22no—
176C42C2×D414yes—
177C23—216no—
178C23—21no—
179C23—24no—
180C23—24no—
181C23—21no—
182C23—24no—
183C23—22no—
184C23—22no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—84no—
195C23—84no—
196C23—88no—
197C23—88no—
198C23#12112yes—
199C23C22×D411yescharacteristic
200Q8—42no—
201Q8—42no—
202D4—216no—
203D4—24no—
204D4—24no—
205D4—24no—
206D4—42no—
207D4—42no—
208D4—84no—
209D4#12118yes—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—88no—
230C2×C4—88no—
231C2×C4#12014yes—
232C2×C4#12112yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D412yes—
235C2×C4C2×#4112yes—
236C2×C4C2×#4111yescharacteristic
237C8—42no—
238C8—42no—
239C22—22no—
240C22—28no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—44no—
245C22—44no—
246C22—44no—
247C22—44no—
248C22—42no—
249C22—44no—
250C22—44no—
251C22—816no—
252C22—81no—
253C22—82no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22C2×#12011yescharacteristic
258C22C2×#12111yescharacteristic
259C22#52511yescharacteristic
260C22#57612yes—
261C22#57712yes—
262C4—22no—
263C4—24no—
264C4—42no—
265C4—44no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4C2×#12112yes—
270C4#52512yes—
271C2—42no—
272C2—48no—
273C2—161no—
274C2#284711yescharacteristic
275C2#301212yes—
276C2#304111yescharacteristic
277C2#304211yescharacteristic
278C2#311812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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
1. C2×#518C2semidirect
2. C2×#463C2semidirect
3. C2×#450C2semidirect
5. #2620C2semidirect
6. #1879C2semidirect
7. #1877C2semidirect
8. #1875C2semidirect
9. #1873C2semidirect
10. #1802C2semidirect
11. #1798C2semidirect
15. C2×#124C22semidirect
28. C2×C4⋊C8C22semidirect
29. C2×#103C22semidirect
32. C2×#102C22semidirect
37. #391C22semidirect
38. #390C22semidirect
39. #384C22semidirect
40. #340C22semidirect
41. #340C22semidirect
42. #336C22semidirect
73. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4632
C2×#4502
C2×#4472
#26202
#18792
#18772
#18752
#18732
#18022
#17982