Finite Group Databasebeta

← #492535

Subgroups — #492535

Order 512.

490 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 126 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
1#33862C212yesmaximal, complemented
2#20006C211yesmaximal, characteristic, complemented
3D82C211yesmaximal, characteristic
4#12509C211yesmaximal, characteristic, complemented
5#12468C212yesmaximal, complemented
6#12447C211yesmaximal, characteristic, complemented
7#12425C211yesmaximal, characteristic, complemented
8#12372C212yesmaximal, complemented
9#12342C212yesmaximal, complemented
10C8×SD32C211yesmaximal, characteristic, complemented
11C16×D8C211yesmaximal, characteristic, complemented
12#3380—21no—
13#3338C2212yescomplemented
14D4×D8—21no—
15D4×D8C2212yes—
16#2240C2212yescomplemented
17#2240C2212yescomplemented
18#2233—22no—
19#2226C2212yes—
20#2217—22no—
21#2207C2212yescomplemented
22#2187—22no—
23C4×SD32C2211yescharacteristic
24D4×C16C2212yescomplemented
25C2×#357—22no—
26#1713C2211yescharacteristic, complemented
27#1712C2211yescharacteristic
28#1676—21no—
29#1675C2212yescomplemented
30#1667—21no—
31#1667C2212yes—
32#1632—22no—
33#1619—22no—
34C8×Q16C2211yescharacteristic, complemented
35C8×D8C2211yescharacteristic
36C8×D8C2211yescharacteristic
37#1372C2211yescharacteristic, complemented
38#1370C2211yescharacteristic, complemented
39#1365C2212yescomplemented
40#1361C2212yescomplemented
41#1332C2211yescharacteristic, complemented
42#1331C2211yescharacteristic
43D4⋊C16C2212yescomplemented
44C8×C16C2211yescharacteristic, complemented
45C22×D8—22no—
46C22×D8—41no—
47#554—22no—
48D42—22no—
49C2×SD32—22no—
50C2×SD32D411yescharacteristic
51C22×C16—22no—
52#500—21no—
53#500C2312yes—
54#499—41no—
55#494—21no—
56#493—21no—
57#493—21no—
58#493C2312yes—
59#484—42no—
60#478—42no—
61#477—22no—
62#476—22no—
63#467—22no—
64#466—22no—
65#466—41no—
66#459—22no—
67#459—22no—
68#447—42no—
69#447—42no—
70C8×Q8—21no—
71C4×Q16C2311yescharacteristic
72C4×D8—41no—
73C4×D8D411yescharacteristic
74C4×D8C2311yescharacteristic
75C8×D4—21no—
76C8×D4C2312yes—
77C2×#107—22no—
78#367—22no—
79#367D411yescharacteristic
80C4⋊C16C2312yes—
81#358D411yescharacteristic
82#357—22no—
83#357—22no—
84#357—22no—
85#357—22no—
86#357—42no—
87#357D411yescharacteristic
88C22⋊C16—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#333—22no—
95#331—22no—
96Q8⋊C8—21no—
97D4⋊C8—21no—
98D4⋊C8D412yes—
99C82C2311yescharacteristic
100C22×D4—42no—
101C22×D4—42no—
102C2×Q16—81no—
103C2×Q16C2×D411yescharacteristic
104C2×D8—22no—
105C2×D8—22no—
106C2×D8—42no—
107C2×D8—44no—
108C2×D8—42no—
109C2×D8—41no—
110C2×D8—41no—
111C2×D8—81no—
112C2×D8—82no—
113C2×D8—84no—
114C2×D8D811yescharacteristic
115C2×D8C2×D411yescharacteristic
116C22×C8—22no—
117C22×C8—41no—
118C4⋊Q8—22no—
119#127—22no—
120#125—42no—
121C22⋊Q8—42no—
122C22⋊Q8—82no—
123#121—42no—
124#121—42no—
125#120—82no—
126C4×Q8—21no—
127C4×D4—21no—
128C4×D4—42no—
129C4×D4C2×D412yes—
130C2×C4⋊C4—42no—
131SD32—22no—
132SD32—82no—
133C2×C16—22no—
134C2×C16—21no—
135C2×C16—82no—
136C2×C16C2×D411yescharacteristic
137C2×C16C2×D411yescharacteristic
138#107—22no—
139#107—22no—
140#107—42no—
141#107—41no—
142#107—41no—
143#107D811yescharacteristic
144#107C2×D411yescharacteristic
145#107C2×D411yescharacteristic
146#106—42no—
147C4⋊C8—21no—
148C4⋊C8—21no—
149C4⋊C8—41no—
150C4⋊C8C2×D412yes—
151#103—21no—
152#103—42no—
153#102—21no—
154#102—42no—
155#102—42no—
156#102—42no—
157#102D812yes—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C4×C8—41no—
161C4×C8C2×D411yescharacteristic
162C4×C8C2×D411yescharacteristic
163C4×C8C2411yescharacteristic
164C24—82no—
165C2×Q8—21no—
166C2×Q8—42no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—82no—
172C2×D4—84no—
173C2×D4—82no—
174C2×D4—84no—
175C2×D4—162no—
176C2×D4—162no—
177C2×D4C2×D812yes—
178C22×C4—22no—
179C22×C4—41no—
180C22×C4—82no—
181Q16—21no—
182Q16—21no—
183Q16—162no—
184D8—21no—
185D8—42no—
186D8—42no—
187D8—82no—
188D8—84no—
189D8—84no—
190D8—82no—
191D8—161no—
192D8—162no—
193D8SD3214yescomplemented
194D8#13512yes—
195C2×C8—21no—
196C2×C8—22no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—82no—
202C2×C8C2×D811yescharacteristic
203C2×C8C2×D811yescharacteristic
204C2×C8C22×D411yescharacteristic
205C2×C8H(5,𝔽2)11yescharacteristic
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—82no—
212C4⋊C4—82no—
213C4⋊C4C2×D812yes—
214C22⋊C4—22no—
215C22⋊C4—41no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C42—41no—
219C42C22×D411yescharacteristic
220C16—21no—
221C16—21no—
222C16—41no—
223C16—42no—
224C23—22no—
225C23—41no—
226C23—82no—
227C23—164no—
228C23—164no—
229C23—322no—
230Q8—41no—
231Q8—41no—
232Q8—162no—
233D4—22no—
234D4—22no—
235D4—41no—
236D4—82no—
237D4—82no—
238D4—82no—
239D4—82no—
240D4—162no—
241D4—164no—
242D4—164no—
243D4C2×SD3214yes—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—322no—
254C2×C4D4211yescharacteristic
255C2×C4C22×D811yescharacteristic
256C2×C4#57611yescharacteristic
257C8—22no—
258C8—21no—
259C8—21no—
260C8—42no—
261C8—81no—
262C8—81no—
263C8C2×SD3212yes—
264C8C2×#13511yescharacteristic
265C8#57611yescharacteristic
266C22—24no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—81no—
271C22—81no—
272C22—162no—
273C22—162no—
274C22—162no—
275C22—328no—
276C22D4×D811yescharacteristic
277C4—21no—
278C4—42no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4—162no—
283C4#328011yescharacteristic
284C4#329011yescharacteristic
285C4C22×SD3211yescharacteristic
286C4#341611yescharacteristic
287C2—44no—
288C2—82no—
289C2—322no—
290C2#1996811yescharacteristic
291C2D4×SD3211yescharacteristic
292C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
prime radicalD82C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33862C2semidirect
2. #20006C2semidirect
4. #12509C2semidirect
5. #12468C2semidirect
6. #12447C2semidirect
7. #12425C2semidirect
8. #12372C2semidirect
9. #12342C2semidirect
10. C8×SD32C2semidirect
11. C16×D8C2semidirect
13. #3338C22semidirect
16. #2240C22semidirect
17. #2240C22semidirect
21. #2207C22semidirect
24. D4×C16C22semidirect
26. #1713C22semidirect
29. #1675C22semidirect
34. C8×Q16C22semidirect
37. #1372C22semidirect
38. #1370C22semidirect
39. #1365C22semidirect
40. #1361C22semidirect
41. #1332C22semidirect
43. D4⋊C16C22semidirect
44. C8×C16C22semidirect
193. D8SD32semidirect

Maximal subgroups

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

subgroupindex
#338622
#200062
D822
#125092
#124682
#124472
#124252
#123722
#123422
C8×SD322
C16×D82