Finite Group Databasebeta

← #25864

Subgroups — #25864

Order 256.

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

normal subgroups only

This group has 727 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
1C22×#124C211yesmaximal, characteristic
2C2×#489C212yesmaximal, complemented
3#2857C211yesmaximal, characteristic, complemented
4#2042C214yesmaximal, complemented
5#1959C212yesmaximal, complemented
6#1787C212yesmaximal, complemented
7#1786C212yesmaximal
8#1749C211yesmaximal, characteristic, complemented
9C23×Q8C2211yescharacteristic
10C23×D4C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic, complemented
12C2×#124—28no—
13C2×#124—21no—
14C2×#124—21no—
15C2×#124C2212yes—
16C2×#124C2212yes—
17C2×C22⋊Q8—21no—
18C22×C22⋊C4—21no—
19C22×C22⋊C4—21no—
20C22×C42C2211yescharacteristic
21#489—28no—
22C2×#103C2214yescomplemented
23C2×#103C2214yescomplemented
24C2×#102C2214yes—
25C2×#102C2214yes—
26C2×C22⋊C8C2212yes—
27C2×#97C2212yescomplemented
28#391—21no—
29#389—21no—
30#389C2212yescomplemented
31#384C2211yescharacteristic, complemented
32#380—21no—
33#380C2212yes—
34#336C2212yescomplemented
35C25—21no—
36C22×Q8—24no—
37C22×Q8—21no—
38C22×Q8—21no—
39C22×Q8—24no—
40C22×Q8—42no—
41C22×Q8C2312yes—
42C22×D4—24no—
43C22×D4—21no—
44C22×D4—24no—
45C22×D4—42no—
46C22×D4C2312yes—
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2314yes—
52C4⋊Q8—24no—
53#124—28no—
54#124—28no—
55#124—24no—
56#124—24no—
57#124—24no—
58#124D418yes—
59C22⋊Q8—44no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—28no—
67C2×C22⋊C4—28no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C42—24no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78#103—28no—
79#103—44no—
80#102—28no—
81#102—44no—
82C22⋊C8—24no—
83#97—24no—
84#95—22no—
85#95—22no—
86C24—24no—
87C24—22no—
88C24—42no—
89C24—42no—
90C24—42no—
91C24—42no—
92C24—42no—
93C24—42no—
94C24C2×D411yescharacteristic
95C2×Q8—24no—
96C2×Q8—22no—
97C2×Q8—24no—
98C2×Q8—24no—
99C2×Q8—416no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8C2×D414yes—
106C2×D4—24no—
107C2×D4—22no—
108C2×D4—24no—
109C2×D4—24no—
110C2×D4—416no—
111C2×D4—42no—
112C2×D4—42no—
113C2×D4C2×D414yes—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4#4114yes—
134C22×C4C2411yescharacteristic
135C2×C8—24no—
136C2×C8—24no—
137C2×C8—24no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C22⋊C4—28no—
145C22⋊C4—28no—
146C22⋊C4—28no—
147C22⋊C4—28no—
148C22⋊C4—44no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42C2×D414yes—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—24no—
158C23—48no—
159C23—48no—
160C23—48no—
161C23—42no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—84no—
168C23—84no—
169C23—84no—
170C23—84no—
171C23#12412yes—
172C23AGL(1,ℤ/8)12yes—
173C23#13712yes—
174C23C22×D411yescharacteristic
175Q8—28no—
176Q8—48no—
177Q8—44no—
178Q8—84no—
179D4—28no—
180D4—48no—
181D4—44no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—28no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4#12012yes—
208C2×C4#12012yes—
209C2×C4#12412yes—
210C2×C4C22×D412yes—
211C2×C4C2×#4112yes—
212C2×C4H(5,𝔽2)12yes—
213C8—44no—
214C22—22no—
215C22—22no—
216C22—22no—
217C22—22no—
218C22—22no—
219C22—22no—
220C22—44no—
221C22—44no—
222C22—44no—
223C22—816no—
224C22—82no—
225C22—82no—
226C22—82no—
227C22—82no—
228C22#48914yescomplemented
229C22C2×#12011yescharacteristic
230C22C2×#12411yescharacteristic
231C22#53511yescharacteristic
232C22C2×AGL(1,ℤ/8)11yescharacteristic
233C22C2×#13711yescharacteristic
234C22#57511yescharacteristic
235C22#57511yescharacteristic
236C4—22no—
237C4—22no—
238C4—22no—
239C4—44no—
240C4—41no—
241C4—41no—
242C4—82no—
243C4—82no—
244C4#54614yes—
245C2—24no—
246C2—84no—
247C2#243511yescharacteristic
248C2#300611yescharacteristic
249C2#300711yescharacteristic
250C2C2×#48911yescharacteristic
251C2#314011yescharacteristic
252C2#321311yescharacteristic
253C2#321511yescharacteristic
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×#489C2semidirect
3. #2857C2semidirect
4. #2042C2semidirect
5. #1959C2semidirect
6. #1787C2semidirect
8. #1749C2semidirect
11. C2×C4⋊Q8C22semidirect
22. C2×#103C22semidirect
23. C2×#103C22semidirect
27. C2×#97C22semidirect
30. #389C22semidirect
31. #384C22semidirect
34. #336C22semidirect
228. C22#489semidirect

Maximal subgroups

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

subgroupindex
C22×#1242
C2×#4892
#28572
#20422
#19592
#17872
#17862
#17492