Finite Group Databasebeta

← #492549

Subgroups — #492549

Order 512.

489 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 118 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#33892C212yesmaximal, complemented
2#20019C211yesmaximal, characteristic, complemented
3D82C211yesmaximal, characteristic, complemented
4#12509C211yesmaximal, characteristic, complemented
5#12472C212yesmaximal, complemented
6#12447C211yesmaximal, characteristic, complemented
7#12424C211yesmaximal, characteristic, complemented
8#12372C212yesmaximal, complemented
9#12343C212yesmaximal, complemented
10C8×D16C211yesmaximal, characteristic, complemented
11#12039C211yesmaximal, characteristic, complemented
12#3341—21no—
13#3341C2212yescomplemented
14D4×D8—21no—
15D4×D8C2212yes—
16#2240C2212yescomplemented
17#2240C2212yescomplemented
18#2234—22no—
19#2226C2212yes—
20#2218—22no—
21#2206C2212yescomplemented
22#2184—22no—
23C4×D16C2211yescharacteristic
24#2169C2212yescomplemented
25#2140—22no—
26#1713C2211yescharacteristic, complemented
27#1712C2211yescharacteristic
28#1675—21no—
29#1675C2212yescomplemented
30#1667—21no—
31#1667C2212yes—
32#1634—22no—
33#1619—22no—
34C8×D8C2211yescharacteristic
35C8×D8C2211yescharacteristic
36C8×D8C2211yescharacteristic, complemented
37#1371C2211yescharacteristic, complemented
38#1370C2211yescharacteristic, complemented
39#1365C2212yescomplemented
40#1361C2212yescomplemented
41#1331C2211yescharacteristic
42#1331C2211yescharacteristic, complemented
43D4⋊C16C2212yescomplemented
44C8×C16C2211yescharacteristic, complemented
45C22×D8—22no—
46C22×D8—41no—
47#555—22no—
48D42—22no—
49C2×D16—22no—
50C2×D16D411yescharacteristic
51C2×M32—22no—
52#500—21no—
53#500C2312yes—
54#494—21no—
55#493—21no—
56#493—21no—
57#493C2312yes—
58#482—42no—
59#482—42no—
60#476—22no—
61#476—22no—
62#466—22no—
63#466—22no—
64#466—41no—
65#466—41no—
66#459—22no—
67#459—22no—
68#447—42no—
69#447—42no—
70C4×D8D411yescharacteristic
71C4×D8C2311yescharacteristic
72C4×D8C2311yescharacteristic
73C8×D4—21no—
74C8×D4—21no—
75C8×D4C2312yes—
76#427—22no—
77#427—41no—
78#367—22no—
79#366D411yescharacteristic
80C4⋊C16C2312yes—
81#357—22no—
82#357—22no—
83#357—22no—
84#357—22no—
85#357—42no—
86#357D411yescharacteristic
87#357D411yescharacteristic
88C22⋊C16—22no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2311yescharacteristic
93#335C2311yescharacteristic
94#333—22no—
95#331—22no—
96D4⋊C8—21no—
97D4⋊C8—21no—
98D4⋊C8D412yes—
99C82C2311yescharacteristic
100C22×D4—42no—
101C22×D4—42no—
102C2×Q16—81no—
103C2×D8—22no—
104C2×D8—22no—
105C2×D8—42no—
106C2×D8—44no—
107C2×D8—42no—
108C2×D8—41no—
109C2×D8—41no—
110C2×D8—81no—
111C2×D8—82no—
112C2×D8—84no—
113C2×D8D811yescharacteristic
114C2×D8C2×D411yescharacteristic
115C2×D8C2×D411yescharacteristic
116C22×C8—22no—
117C22×C8—41no—
118C22×C8—41no—
119C4⋊Q8—22no—
120#127—22no—
121#124—82no—
122#121—42no—
123#121—42no—
124#121—42no—
125#121—42no—
126#120—82no—
127C4×D4—21no—
128C4×D4—21no—
129C4×D4C2×D412yes—
130#117—42no—
131#117—42no—
132D16—22no—
133D16—82no—
134M32—82no—
135C2×C16—22no—
136C2×C16—21no—
137C2×C16C2×D411yescharacteristic
138C2×C16C2×D411yescharacteristic
139#107—22no—
140#107—22no—
141#107—41no—
142#107—41no—
143#107D811yescharacteristic
144#107C2×D411yescharacteristic
145#107C2×D411yescharacteristic
146#106—42no—
147#106—42no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8C2×D412yes—
151#102—21no—
152#102—21no—
153#102—42no—
154#102—42no—
155#102—42no—
156#102—42no—
157#102D812yes—
158C22⋊C8—22no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C4×C8C2×D411yescharacteristic
162C4×C8C2×D411yescharacteristic
163C4×C8C2411yescharacteristic
164C24—82no—
165C2×Q8—42no—
166C2×D4—21no—
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—82no—
176C2×D4—162no—
177C2×D4—162no—
178C2×D4C2×D812yes—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—41no—
182Q16—162no—
183D8—21no—
184D8—21no—
185D8—21no—
186D8—22no—
187D8—42no—
188D8—42no—
189D8—82no—
190D8—84no—
191D8—84no—
192D8—82no—
193D8—161no—
194D8—162no—
195D8#13512yes—
196C2×C8—21no—
197C2×C8—22no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8C2×D811yescharacteristic
205C2×C8C2×D811yescharacteristic
206C2×C8C22×D411yescharacteristic
207C2×C8H(5,𝔽2)11yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4C2×D812yes—
213C22⋊C4—22no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—82no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C42—82no—
220C42C22×D411yescharacteristic
221C16—21no—
222C16—21no—
223C16—41no—
224C16—42no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—82no—
229C23—164no—
230C23—164no—
231C23—322no—
232Q8—162no—
233D4—22no—
234D4—22no—
235D4—22no—
236D4—41no—
237D4—41no—
238D4—41no—
239D4—82no—
240D4—82no—
241D4—82no—
242D4—82no—
243D4—162no—
244D4—164no—
245D4—164no—
246D4—322no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4D4211yescharacteristic
258C2×C4C22×D811yescharacteristic
259C2×C4#57611yescharacteristic
260C8—22no—
261C8—21no—
262C8—21no—
263C8—42no—
264C8—81no—
265C8—81no—
266C8#50912yes—
267C8C2×#13511yescharacteristic
268C8#57611yescharacteristic
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—162no—
279C22—162no—
280C22—162no—
281C22—328no—
282C22D4×D811yescharacteristic
283C4—21no—
284C4—42no—
285C4—81no—
286C4—81no—
287C4—162no—
288C4#328011yescharacteristic
289C4#329011yescharacteristic
290C4C2×#50911yescharacteristic
291C4#341611yescharacteristic
292C2—82no—
293C2—82no—
294C2—161no—
295C2—322no—
296C2#1996811yescharacteristic
297C2#3383811yescharacteristic
298C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
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. #33892C2semidirect
2. #20019C2semidirect
3. D82C2semidirect
4. #12509C2semidirect
5. #12472C2semidirect
6. #12447C2semidirect
7. #12424C2semidirect
8. #12372C2semidirect
9. #12343C2semidirect
10. C8×D16C2semidirect
11. #12039C2semidirect
13. #3341C22semidirect
16. #2240C22semidirect
17. #2240C22semidirect
21. #2206C22semidirect
24. #2169C22semidirect
26. #1713C22semidirect
29. #1675C22semidirect
36. C8×D8C22semidirect
37. #1371C22semidirect
38. #1370C22semidirect
39. #1365C22semidirect
40. #1361C22semidirect
42. #1331C22semidirect
43. D4⋊C16C22semidirect
44. C8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#338922
#200192
D822
#125092
#124722
#124472
#124242
#123722
#123432
C8×D162
#120392