Finite Group Databasebeta

← #23933

Subgroups — #23933

Order 256.

744 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 149 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×#533C211yesmaximal, characteristic, complemented
2C2×#450C212yesmaximal
3C2×#390C211yesmaximal, characteristic
4#2023C212yesmaximal, complemented
5#2013C212yesmaximal, complemented
6#1880C214yesmaximal, complemented
7#1872C212yesmaximal, complemented
8#1817C211yesmaximal, characteristic, complemented
9C23×D4C2211yescharacteristic
10C22×SD16C2214yes—
11#533—28no—
12C2×C4⋊Q8C2211yescharacteristic, complemented
13C2×C4⋊Q8C2211yescharacteristic, complemented
14C2×#125—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2212yes—
17C2×C4×Q8—21no—
18C2×C4×Q8C2212yescomplemented
19C2×#117—21no—
20C22×C4⋊C4C2211yescharacteristic
21C22×C22⋊C4—21no—
22C22×C22⋊C4—21no—
23C22×C42C2211yescharacteristic
24#450—28no—
25C2×#103C2214yescomplemented
26C2×#102C2214yes—
27C2×C22⋊C8C2212yes—
28#390—21no—
29#390—21no—
30#390C2212yes—
31#390C2212yes—
32#340C2214yescomplemented
33#336C2212yescomplemented
34#336C2212yescomplemented
35C25—21no—
36C22×Q8—21no—
37C22×Q8C2312yes—
38C22×D4—24no—
39C22×D4—21no—
40C22×D4—24no—
41C22×D4—42no—
42C22×D4C2312yes—
43C23×C4—21no—
44C23×C4D411yescharacteristic
45C23×C4D411yescharacteristic
46C23×C4C2311yescharacteristic
47C2×SD16—28no—
48C2×SD16—28no—
49C2×SD16—44no—
50C22×C8C2314yes—
51C4⋊Q8—24no—
52C4⋊Q8—24no—
53#125—28no—
54C22⋊Q8—28no—
55C22⋊Q8—24no—
56C22⋊Q8D418yes—
57C4×Q8—28no—
58C4×Q8—24no—
59C4×Q8—24no—
60#117—28no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—42no—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4—44no—
76C2×C22⋊C4—44no—
77C2×C42—22no—
78C2×C42—24no—
79C2×C42D411yescharacteristic
80C2×C42D411yescharacteristic
81C2×C42C2311yescharacteristic
82C2×C42C2311yescharacteristic
83#103—44no—
84#102—28no—
85C22⋊C8—24no—
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—42no—
100C2×Q8—42no—
101C2×Q8C2×D414yes—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—24no—
106C2×D4—416no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4#4114yes—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—41no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4SD1614yescomplemented
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D414yes—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2411yescharacteristic
131SD16—416no—
132C2×C8—24no—
133C2×C8—24no—
134C2×C8—24no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—28no—
138C4⋊C4—24no—
139C4⋊C4—28no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—42no—
143C4⋊C4C2×D414yes—
144C22⋊C4—28no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C22⋊C4—44no—
149C42—28no—
150C42—22no—
151C42—22no—
152C42—22no—
153C42—22no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—21no—
158C23—24no—
159C23—48no—
160C23—48no—
161C23—48no—
162C23—42no—
163C23—42no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23—84no—
169C23—84no—
170C23—84no—
171C23—84no—
172C23#12012yes—
173C23C2×SD1612yes—
174C23#13712yes—
175C23C22×D411yescharacteristic
176Q8—28no—
177Q8—48no—
178Q8—44no—
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—21no—
189C2×C4—21no—
190C2×C4—28no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—84no—
206C2×C4#12012yes—
207C2×C4C2×SD1612yes—
208C2×C4C2×SD1612yes—
209C2×C4AGL(1,ℤ/8)12yes—
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#44814yes—
229C22C2×#12011yescharacteristic
230C22#52511yescharacteristic
231C22#53511yescharacteristic
232C22C22×SD1611yescharacteristic
233C22C2×AGL(1,ℤ/8)11yescharacteristic
234C22C2×#13711yescharacteristic
235C22#57511yescharacteristic
236C4—22no—
237C4—22no—
238C4—22no—
239C4—22no—
240C4—44no—
241C4—41no—
242C4—44no—
243C4—82no—
244C4#54614yes—
245C2—24no—
246C2—84no—
247C2#240411yescharacteristic
248C2C2×#44811yescharacteristic
249C2#300611yescharacteristic
250C2#310111yescharacteristic
251C2#311011yescharacteristic
252C2#320811yescharacteristic
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
1. C2×#533C2semidirect
4. #2023C2semidirect
5. #2013C2semidirect
6. #1880C2semidirect
7. #1872C2semidirect
8. #1817C2semidirect
12. C2×C4⋊Q8C22semidirect
13. C2×C4⋊Q8C22semidirect
18. C2×C4×Q8C22semidirect
25. C2×#103C22semidirect
32. #340C22semidirect
33. #336C22semidirect
34. #336C22semidirect
123. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4502
C2×#3902
#20232
#20132
#18802
#18722
#18172