Finite Group Databasebeta

← #23456

Subgroups — #23456

Order 256.

681 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 175 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×#544C211yesmaximal, characteristic, complemented
2C2×#418C211yesmaximal, characteristic
3C22×#102C211yesmaximal, characteristic
4#2309C211yesmaximal, characteristic
5#2025C214yesmaximal, complemented
6#2023C214yesmaximal, complemented
7#1736C211yesmaximal, characteristic, complemented
8#1735C211yesmaximal, characteristic, complemented
9#1729C211yesmaximal, characteristic, complemented
10C23×D4C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C23×C8C2211yescharacteristic
13#544—24no—
14#544—24no—
15C2×C4⋊Q8—21no—
16C2×C4⋊Q8C2212yescomplemented
17C2×#125—21no—
18C2×#125C2212yescomplemented
19C2×C22⋊Q8—22no—
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#418—24no—
25C2×#102—24no—
26C2×#102C2214yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29#390—21no—
30#390—21no—
31#390C2214yes—
32#379—22no—
33#340C2212yescomplemented
34#340C2212yescomplemented
35#340C2214yescomplemented
36#336C2214yescomplemented
37C25—21no—
38C22×Q8—22no—
39C22×D4—22no—
40C22×D4—22no—
41C22×D4—21no—
42C22×D4—22no—
43C22×D4—22no—
44C22×D4—22no—
45C22×D4Q814yescomplemented
46C22×D4C2312yes—
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C2×M16—22no—
50C2×M16—22no—
51C2×M16—22no—
52C22×C8—22no—
53C22×C8—22no—
54C22×C8—22no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—28no—
58C4⋊Q8—24no—
59C4⋊Q8—24no—
60#125—28no—
61#125—24no—
62#125—24no—
63C22⋊Q8—216no—
64#117—22no—
65#117—22no—
66#117—22no—
67#117—22no—
68#117—22no—
69#117D414yes—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D414yes—
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2312yes—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4D412yes—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42C2312yes—
90#102—28no—
91#102—24no—
92#102—24no—
93C24—22no—
94C24—24no—
95C24—42no—
96C24—42no—
97C24—82no—
98C24—82no—
99C24C2×D411yescharacteristic
100C2×Q8—28no—
101C2×Q8—84no—
102C2×D4—28no—
103C2×D4—28no—
104C2×D4—24no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—42no—
111C2×D4—44no—
112C2×D4—44no—
113C2×D4—42no—
114C2×D4C2×Q814yes—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—22no—
120C22×C4—82no—
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2×Q812yes—
127C22×C4#4112yes—
128C22×C4#4112yes—
129C22×C4#4112yes—
130C22×C4C2411yescharacteristic
131M16—44no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—44no—
139C4⋊C4—28no—
140C4⋊C4—28no—
141C4⋊C4—28no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4C2×D414yes—
151C4⋊C4C2×D412yes—
152C4⋊C4C2×D412yes—
153C22⋊C4—24no—
154C22⋊C4—24no—
155C22⋊C4—42no—
156C22⋊C4—84no—
157C22⋊C4—84no—
158C42—24no—
159C42—22no—
160C42—22no—
161C42—42no—
162C23—22no—
163C23—22no—
164C23—24no—
165C23—42no—
166C23—42no—
167C23—44no—
168C23—44no—
169C23—48no—
170C23—88no—
171C23—88no—
172C23—84no—
173C23—84no—
174C23#12012yes—
175C23C22⋊Q812yes—
176C23#12412yes—
177C23C22×D411yescharacteristic
178Q8—88no—
179D4—24no—
180D4—24no—
181D4—42no—
182D4—42no—
183D4—48no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—44no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—84no—
198C2×C4#12012yes—
199C2×C4C22⋊Q814yes—
200C2×C4C22⋊Q814yes—
201C2×C4C22⋊Q812yes—
202C2×C4#12414yes—
203C2×C4#12414yes—
204C2×C4#12412yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C2×C4C22×Q811yescharacteristic
208C2×C4C2×#4111yescharacteristic
209C2×C4C2×#4111yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C8—42no—
212C8—42no—
213C22—24no—
214C22—24no—
215C22—22no—
216C22—22no—
217C22—48no—
218C22—44no—
219C22—816no—
220C22—84no—
221C22—82no—
222C22—82no—
223C22#39314yes—
224C22C2×#12011yescharacteristic
225C22C2×C22⋊Q811yescharacteristic
226C22C2×#12411yescharacteristic
227C22#57612yes—
228C22#57712yes—
229C4—42no—
230C4—44no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—82no—
235C4#39314yes—
236C4C2×C22⋊Q811yescharacteristic
237C4C2×C22⋊Q811yescharacteristic
238C4C2×#12411yescharacteristic
239C4C2×#12411yescharacteristic
240C2—24no—
241C2—84no—
242C2C2×#39311yescharacteristic
243C2#300811yescharacteristic
244C2#301011yescharacteristic
245C2#308112yes—
246C2#313612yes—
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
1. C2×#544C2semidirect
5. #2025C2semidirect
6. #2023C2semidirect
7. #1736C2semidirect
8. #1735C2semidirect
9. #1729C2semidirect
16. C2×C4⋊Q8C22semidirect
18. C2×#125C22semidirect
33. #340C22semidirect
34. #340C22semidirect
35. #340C22semidirect
36. #336C22semidirect
45. C22×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4182
C22×#1022
#23092
#20252
#20232
#17362
#17352
#17292