Finite Group Databasebeta

← #24476

Subgroups — #24476

Order 256.

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

normal subgroups only

This group has 630 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×#480C211yesmaximal, characteristic, complemented
2C22×#102C211yesmaximal, characteristic, complemented
3#2433C211yesmaximal, characteristic
4#2344C211yesmaximal, characteristic, complemented
5#2067C212yesmaximal, complemented
6#2041C212yesmaximal, complemented
7#1982C212yesmaximal, complemented
8#1925C212yesmaximal, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1850C211yesmaximal, characteristic, complemented
11#1786C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×#127C2212yes—
15C2×#121C2211yescharacteristic
16C2×#120—21no—
17C2×#120—21no—
18C2×#117—21no—
19C22×C4⋊C4C2211yescharacteristic
20#480—24no—
21C2×#107C2211yescharacteristic, complemented
22C2×#107C2211yescharacteristic, complemented
23C2×C4⋊C8C2212yescomplemented
24C2×#102—24no—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2211yescharacteristic, complemented
30C2×#102C2211yescharacteristic, complemented
31C2×#102C2212yescomplemented
32C2×C22⋊C8C2211yescharacteristic
33#392—22no—
34#390C2212yes—
35#389C2212yescomplemented
36#387—22no—
37#381—21no—
38#380C2211yescharacteristic
39C4×C4⋊C4C2212yescomplemented
40C4×C22⋊C4C2211yescharacteristic
41#340C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×D4—24no—
45C22×D4—21no—
46C22×D4—22no—
47C22×D4—21no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—42no—
52C22×D4C2311yescharacteristic
53C22×D4C2312yes—
54C23×C4—21no—
55C23×C4C2311yescharacteristic
56C22×C8—22no—
57C22×C8—42no—
58C22×C8C2312yes—
59C22×C8C2311yescharacteristic
60C22×C8C2311yescharacteristic
61#127—24no—
62#127—42no—
63#121—22no—
64#121—22no—
65#120—44no—
66#120—44no—
67#117—44no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4D414yescomplemented
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4D411yescharacteristic
81C2×C22⋊C4D411yescharacteristic
82C2×C42—22no—
83C2×C42C2312yes—
84#107—22no—
85#107—22no—
86C4⋊C8—42no—
87#102—24no—
88#102—24no—
89#102—24no—
90#102—22no—
91#102—22no—
92#102—42no—
93#102—42no—
94C22⋊C8—22no—
95#95—22no—
96#95—21no—
97#95—22no—
98#95D412yes—
99C24—22no—
100C24—21no—
101C24—44no—
102C24—42no—
103C24—42no—
104C24—42no—
105C24—82no—
106C24C2×D411yescharacteristic
107C2×D4—28no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—21no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4—48no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—44no—
121C2×D4—44no—
122C2×D4—44no—
123C2×D4—42no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—44no—
128C2×D4—82no—
129C2×D4—82no—
130C2×D4#4114yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D412yes—
143C22×C4C2×D412yes—
144C22×C4#4112yes—
145C22×C4C2411yescharacteristic
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—44no—
154C4⋊C4—24no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—44no—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C22⋊C4—22no—
165C22⋊C4—44no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—44no—
169C22⋊C4D814yes—
170C42—22no—
171C42—22no—
172C42—44no—
173C23—22no—
174C23—22no—
175C23—24no—
176C23—48no—
177C23—42no—
178C23—42no—
179C23—42no—
180C23—42no—
181C23—42no—
182C23—42no—
183C23—41no—
184C23—42no—
185C23—41no—
186C23—41no—
187C23—42no—
188C23—88no—
189C23—88no—
190C23—84no—
191C23—84no—
192C23#12112yes—
193C23C2×D812yes—
194C23AGL(1,ℤ/8)12yes—
195C23C22×D411yescharacteristic
196D4—24no—
197D4—24no—
198D4—48no—
199D4—41no—
200D4—42no—
201D4—41no—
202D4—42no—
203D4—42no—
204D4—88no—
205D4—88no—
206D4—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—48no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4#12112yes—
230C2×C4C2×D814yes—
231C2×C4C22×D411yescharacteristic
232C2×C4C2×#4112yes—
233C2×C4C2×#4111yescharacteristic
234C2×C4H(5,𝔽2)12yes—
235C8—41no—
236C8—41no—
237C8—42no—
238C22—22no—
239C22—24no—
240C22—22no—
241C22—24no—
242C22—44no—
243C22—44no—
244C22—44no—
245C22—816no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—82no—
250C22—81no—
251C22—81no—
252C22—82no—
253C22—81no—
254C22—82no—
255C22#45914yes—
256C22C2×#12111yescharacteristic
257C22#54612yes—
258C22C22×D811yescharacteristic
259C22C2×AGL(1,ℤ/8)11yescharacteristic
260C22#57712yes—
261C4—22no—
262C4—24no—
263C4—42no—
264C4—42no—
265C4—42no—
266C4—81no—
267C4—82no—
268C4#52511yescharacteristic
269C4#53511yescharacteristic
270C4#55112yes—
271C2—24no—
272C2—84no—
273C2—161no—
274C2#241111yescharacteristic
275C2C2×#45911yescharacteristic
276C2#304311yescharacteristic
277C2#329412yes—
278C2#331012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2433C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#480C2semidirect
2. C22×#102C2semidirect
4. #2344C2semidirect
5. #2067C2semidirect
6. #2041C2semidirect
7. #1982C2semidirect
8. #1925C2semidirect
9. #1893C2semidirect
10. #1850C2semidirect
11. #1786C2semidirect
21. C2×#107C22semidirect
22. C2×#107C22semidirect
23. C2×C4⋊C8C22semidirect
29. C2×#102C22semidirect
30. C2×#102C22semidirect
31. C2×#102C22semidirect
35. #389C22semidirect
39. C4×C4⋊C4C22semidirect
41. #340C22semidirect
42. #336C22semidirect
71. C2×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C22×#1022
#24332
#23442
#20672
#20412
#19822
#19252
#18932
#18502
#17862