Finite Group Databasebeta

← #28097

Subgroups — #28097

Order 256.

603 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 139 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×C8×D4C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic, complemented
3#2487C211yesmaximal, characteristic, complemented
4#2408C211yesmaximal, characteristic
5#2065C212yesmaximal, complemented
6#2034C212yesmaximal, complemented
7#1934C212yesmaximal, complemented
8#1893C212yesmaximal, complemented
9#1824C211yesmaximal, characteristic, complemented
10#1807C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C23×C8C2211yescharacteristic, complemented
15C2×#121C2212yes—
16C2×#120—21no—
17C2×C4×D4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C8×D4—24no—
20C2×#106C2212yescomplemented
21C2×C4⋊C8C2211yescharacteristic, complemented
22C2×#102—24no—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic, complemented
29C2×C4×C8C2211yescharacteristic, complemented
30#400—21no—
31#399—21no—
32#394—21no—
33#392—21no—
34#390C2212yes—
35#389C2212yescomplemented
36#386—21no—
37#385—21no—
38#380C2211yescharacteristic
39#380C2211yescharacteristic
40#380C2212yescomplemented
41#379—21no—
42#340C2212yescomplemented
43#340C2212yescomplemented
44#340C2212yescomplemented
45C25—21no—
46C22×D4—24no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—22no—
51C22×D4—42no—
52C22×D4D411yescharacteristic
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4C2311yescharacteristic
56C23×C4C2311yescharacteristic
57C22×C8—22no—
58C22×C8—42no—
59C22×C8—42no—
60C22×C8D414yescomplemented
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65#121—24no—
66#121—42no—
67#120—44no—
68C4×D4—22no—
69C4×D4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4D411yescharacteristic
85C2×C22⋊C4D411yescharacteristic
86C2×C42C2311yescharacteristic
87#106—42no—
88C4⋊C8—22no—
89#102—24no—
90#102—24no—
91#102—24no—
92#102—42no—
93C22⋊C8—22no—
94C22⋊C8—22no—
95C4×C8—22no—
96#95—21no—
97#95—21no—
98#95—22no—
99#95—22no—
100C24—22no—
101C24—44no—
102C24—42no—
103C24—42no—
104C24—42no—
105C24—82no—
106C24C2×D411yescharacteristic
107C24C2×D411yescharacteristic
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—28no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—48no—
117C2×D4—44no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—42no—
122C2×D4—82no—
123C2×D4SD1614yes—
124C2×D4#4114yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—82no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D412yes—
140C22×C4C2×D411yescharacteristic
141C22×C4#4112yes—
142C22×C4C2411yescharacteristic
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—44no—
150C2×C8—44no—
151C2×C8C2×D412yes—
152C2×C8C2×D412yes—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—82no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—44no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—44no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C42—21no—
171C42—21no—
172C23—22no—
173C23—22no—
174C23—21no—
175C23—22no—
176C23—21no—
177C23—22no—
178C23—24no—
179C23—48no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—88no—
187C23—88no—
188C23—84no—
189C23—84no—
190C23#12112yes—
191C23C2×SD1612yes—
192C23C2×SD1612yes—
193C23#13512yes—
194C23C22×D411yescharacteristic
195D4—28no—
196D4—24no—
197D4—24no—
198D4—48no—
199D4—42no—
200D4—42no—
201D4—88no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—84no—
225C2×C4#12112yes—
226C2×C4C2×SD1612yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C2×#4112yes—
229C2×C4C2×#4111yescharacteristic
230C2×C4H(5,𝔽2)12yes—
231C8—22no—
232C8—22no—
233C8—41no—
234C8—41no—
235C22—24no—
236C22—24no—
237C22—22no—
238C22—22no—
239C22—24no—
240C22—22no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—44no—
245C22—44no—
246C22—44no—
247C22—816no—
248C22—82no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22#46514yes—
253C22C2×#12111yescharacteristic
254C22#54612yes—
255C22C22×SD1611yescharacteristic
256C22C2×#13511yescharacteristic
257C22#57612yes—
258C4—22no—
259C4—22no—
260C4—41no—
261C4—41no—
262C4—82no—
263C4—82no—
264C4—82no—
265C4#52511yescharacteristic
266C4#53411yescharacteristic
267C4#55112yes—
268C2—24no—
269C2—42no—
270C2—84no—
271C2#253411yescharacteristic
272C2C2×#46511yescharacteristic
273C2#305811yescharacteristic
274C2#329512yes—
275C2#329712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2408C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C8×D4C2semidirect
2. C22×#102C2semidirect
3. #2487C2semidirect
5. #2065C2semidirect
6. #2034C2semidirect
7. #1934C2semidirect
8. #1893C2semidirect
9. #1824C2semidirect
10. #1807C2semidirect
11. #1786C2semidirect
14. C23×C8C22semidirect
20. C2×#106C22semidirect
21. C2×C4⋊C8C22semidirect
28. C2×C22⋊C8C22semidirect
29. C2×C4×C8C22semidirect
35. #389C22semidirect
40. #380C22semidirect
42. #340C22semidirect
43. #340C22semidirect
44. #340C22semidirect
60. C22×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×C8×D42
C22×#1022
#24872
#24082
#20652
#20342
#19342
#18932
#18242
#18072
#17862