Finite Group Databasebeta

← #32905

Subgroups — #32905

Order 256.

934 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C22×C4×D4C211yesmaximal, characteristic
2C2×#461C212yesmaximal, complemented
3C2×#450C212yesmaximal
4#2857C211yesmaximal, characteristic, complemented
5#1877C214yesmaximal, complemented
6#1872C212yesmaximal, complemented
7#1802C212yesmaximal, complemented
8#1797C211yesmaximal, characteristic, complemented
9C23×D4C2211yescharacteristic
10C24×C4—21no—
11C22×SD16C2214yes—
12C2×C4⋊Q8C2211yescharacteristic, complemented
13C2×C22⋊Q8C2212yes—
14C2×C4×D4—28no—
15C2×C4×D4—21no—
16C2×C4×D4—21no—
17C2×C4×D4C2212yes—
18C2×C4×D4C2212yes—
19C22×C4⋊C4C2211yescharacteristic
20C22×C22⋊C4—21no—
21C22×C42C2211yescharacteristic
22#461—28no—
23#450—28no—
24C2×C4⋊C8C2212yescomplemented
25C2×#103C2214yescomplemented
26C2×#102C2214yes—
27C2×C22⋊C8C2212yes—
28#391C2212yescomplemented
29#389—21no—
30#389—21no—
31#384C2211yescharacteristic, complemented
32#380—21no—
33#380—21no—
34#340C2214yescomplemented
35#336C2212yescomplemented
36C25—21no—
37C22×Q8C2312yes—
38C22×D4—24no—
39C22×D4—24no—
40C22×D4—21no—
41C22×D4—22no—
42C22×D4D414yes—
43C22×D4C2312yes—
44C23×C4—28no—
45C23×C4—21no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4—44no—
49C23×C4—44no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C2×SD16—28no—
54C2×SD16—28no—
55C2×SD16—28no—
56C22×C8C2314yes—
57C4⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C4×D4—28no—
61C4×D4—28no—
62C4×D4—24no—
63C4×D4—24no—
64C4×D4—24no—
65C4×D4D418yes—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—24no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2312yes—
75C2×C22⋊C4—28no—
76C2×C22⋊C4—22no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C42—24no—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84C4⋊C8—24no—
85#103—28no—
86#102—28no—
87C22⋊C8—24no—
88#95—22no—
89#95—22no—
90C24—24no—
91C24—24no—
92C24—24no—
93C24—22no—
94C24—42no—
95C24—42no—
96C24—42no—
97C24—42no—
98C24C2×D411yescharacteristic
99C2×Q8—24no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—24no—
103C2×D4—24no—
104C2×D4—24no—
105C2×D4—216no—
106C2×D4—216no—
107C2×D4—24no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4C2×D414yes—
112C2×D4C2×D414yes—
113C2×D4#4114yes—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—28no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—28no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—44no—
128C22×C4—432no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—44no—
132C22×C4—44no—
133C22×C4—44no—
134C22×C4SD1614yescomplemented
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4#4112yes—
142C22×C4C2411yescharacteristic
143SD16—416no—
144C2×C8—24no—
145C2×C8—24no—
146C2×C8—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—42no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—44no—
155C4⋊C4—42no—
156C4⋊C4C2×D414yes—
157C22⋊C4—28no—
158C22⋊C4—28no—
159C22⋊C4—44no—
160C42—22no—
161C42—22no—
162C42—22no—
163C42C2×D414yes—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—216no—
169C23—24no—
170C23—24no—
171C23—24no—
172C23—48no—
173C23—48no—
174C23—48no—
175C23—48no—
176C23—48no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—48no—
182C23#12112yes—
183C23C2×SD1612yes—
184C23#13712yes—
185C23C22×D411yescharacteristic
186Q8—44no—
187Q8—44no—
188D4—216no—
189D4—24no—
190D4—24no—
191D4—24no—
192D4#12118yes—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—28no—
202C2×C4—21no—
203C2×C4—24no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—44no—
207C2×C4—416no—
208C2×C4—48no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—416no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4#12012yes—
219C2×C4#12012yes—
220C2×C4#12112yes—
221C2×C4C2×SD1612yes—
222C2×C4C2×SD1612yes—
223C2×C4AGL(1,ℤ/8)12yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D412yes—
226C2×C4C2×#4112yes—
227C2×C4C2×#4111yescharacteristic
228C8—44no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—28no—
236C22—432no—
237C22—44no—
238C22—44no—
239C22—44no—
240C22—44no—
241C22—44no—
242C22—44no—
243C22#46114yescomplemented
244C22C2×#12011yescharacteristic
245C22C2×#12111yescharacteristic
246C22#52511yescharacteristic
247C22C22×SD1611yescharacteristic
248C22C2×AGL(1,ℤ/8)11yescharacteristic
249C22C2×#13711yescharacteristic
250C22#57511yescharacteristic
251C4—22no—
252C4—22no—
253C4—22no—
254C4—44no—
255C4—41no—
256C4—82no—
257C4—82no—
258C4#45014yescomplemented
259C4C2×#12112yes—
260C4#52512yes—
261C2—24no—
262C2—48no—
263C2#284711yescharacteristic
264C2C2×#45011yescharacteristic
265C2#300711yescharacteristic
266C2C2×#46111yescharacteristic
267C2#303611yescharacteristic
268C2#310111yescharacteristic
269C2#311011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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
2. C2×#461C2semidirect
4. #2857C2semidirect
5. #1877C2semidirect
6. #1872C2semidirect
7. #1802C2semidirect
8. #1797C2semidirect
12. C2×C4⋊Q8C22semidirect
24. C2×C4⋊C8C22semidirect
25. C2×#103C22semidirect
28. #391C22semidirect
31. #384C22semidirect
34. #340C22semidirect
35. #336C22semidirect
134. C22×C4SD16semidirect
243. C22#461semidirect
258. C4#450semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D42
C2×#4612
C2×#4502
#28572
#18772
#18722
#18022
#17972