Finite Group Databasebeta

← #492226

Subgroups — #492226

Order 512.

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

normal subgroups only

This group has 488 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#33908C211yesmaximal, characteristic, complemented
2D4×D16C211yesmaximal, characteristic, complemented
3#19898C212yesmaximal, complemented
4#12503C211yesmaximal, characteristic, complemented
5#12485C212yesmaximal, complemented
6#12452C211yesmaximal, characteristic, complemented
7#12430C211yesmaximal, characteristic, complemented
8#12372C212yesmaximal, complemented
9#12343C212yesmaximal, complemented
10#12081C211yesmaximal, characteristic, complemented
11C16×D8C211yesmaximal, characteristic, complemented
12C22×D16—21no—
13#3341C2212yescomplemented
14#3325—22no—
15D4×D8C2212yes—
16#2252C2211yescharacteristic, complemented
17#2246C2211yescharacteristic
18#2240C2212yescomplemented
19#2234—22no—
20#2226C2212yescomplemented
21#2215—21no—
22#2215—21no—
23#2206C2212yes—
24#2184—22no—
25C4×D16C2211yescharacteristic
26D4×C16C2211yescharacteristic
27D4×C16C2211yescharacteristic, complemented
28#2158—21no—
29#1706C2212yescomplemented
30#1697—22no—
31#1675C2212yes—
32#1667C2212yescomplemented
33#1629—22no—
34#1624—22no—
35#1589C2212yescomplemented
36C8×D8C2211yescharacteristic
37#1371C2211yescharacteristic, complemented
38#1370C2211yescharacteristic, complemented
39#1365C2212yescomplemented
40#1361C2212yescomplemented
41#1331C2212yescomplemented
42D4⋊C16C2211yescharacteristic
43D4⋊C16C2211yescharacteristic, complemented
44C8×C16C2211yescharacteristic, complemented
45C22×D8—22no—
46#555—22no—
47D42—22no—
48C2×D16—22no—
49C2×D16—21no—
50C2×D16—21no—
51C2×D16—44no—
52C2×D16D411yescharacteristic
53C22×C16—21no—
54C22×C16—21no—
55#500C2312yes—
56#493C2312yes—
57#492—22no—
58#486—22no—
59#482—42no—
60#482—42no—
61#476—22no—
62#476—22no—
63#469—42no—
64#466—22no—
65#466—22no—
66#459—22no—
67#459—22no—
68#447—42no—
69#447—42no—
70C4×D8D411yescharacteristic
71C4×D8C2312yes—
72#436—22no—
73C8×D4C2311yescharacteristic
74C8×D4C2311yescharacteristic
75#427—22no—
76#418—42no—
77#367—22no—
78#366—21no—
79#366—21no—
80#366D411yescharacteristic
81C4⋊C16C2311yescharacteristic
82C4⋊C16C2311yescharacteristic
83#357—22no—
84#357—22no—
85#357—42no—
86#357D412yes—
87C22⋊C16—21no—
88C22⋊C16—21no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335D411yescharacteristic
92#335C2312yes—
93#333—22no—
94#331—22no—
95D4⋊C8—22no—
96D4⋊C8D411yescharacteristic
97D4⋊C8D411yescharacteristic
98C82C2311yescharacteristic
99C22×D4—42no—
100C22×D4—42no—
101C2×SD16—82no—
102C2×D8—22no—
103C2×D8—22no—
104C2×D8—42no—
105C2×D8—44no—
106C2×D8—42no—
107C2×D8—82no—
108C2×D8D811yescharacteristic
109C2×D8C2×D412yes—
110C2×M16—42no—
111C22×C8—21no—
112C22×C8—21no—
113C4⋊Q8—22no—
114#127—22no—
115#124—82no—
116#121—42no—
117#121—42no—
118#121—42no—
119#121—42no—
120#120—82no—
121C4×D4—22no—
122C4×D4C2×D411yescharacteristic
123C4×D4C2×D411yescharacteristic
124#117—42no—
125#117—42no—
126D16—22no—
127D16—44no—
128D16—42no—
129D16—81no—
130C2×C16—21no—
131C2×C16—21no—
132C2×C16—21no—
133C2×C16—42no—
134C2×C16—42no—
135C2×C16C2×D411yescharacteristic
136C2×C16C2×D411yescharacteristic
137#107—22no—
138#107—22no—
139#107D811yescharacteristic
140#107C2×D412yes—
141#106—42no—
142#106—42no—
143C4⋊C8—22no—
144C4⋊C8C2×D411yescharacteristic
145C4⋊C8C2×D411yescharacteristic
146#102—22no—
147#102—42no—
148#102—42no—
149#102—42no—
150#102—42no—
151#102—42no—
152#102—42no—
153#102D811yescharacteristic
154#102D811yescharacteristic
155C22⋊C8—21no—
156C22⋊C8—21no—
157C22⋊C8—42no—
158C4×C8C2×D411yescharacteristic
159C4×C8C2×D411yescharacteristic
160C4×C8C2411yescharacteristic
161C24—82no—
162C2×Q8—42no—
163C2×D4—22no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—82no—
168C2×D4—84no—
169C2×D4—82no—
170C2×D4—84no—
171C2×D4—82no—
172C2×D4—162no—
173C2×D4—162no—
174C2×D4C2×D811yescharacteristic
175C2×D4C2×D811yescharacteristic
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179SD16—164no—
180D8—22no—
181D8—22no—
182D8—22no—
183D8—42no—
184D8—42no—
185D8—82no—
186D8—84no—
187D8—162no—
188M16—162no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—42no—
197C2×C8C2×D811yescharacteristic
198C2×C8C2×D811yescharacteristic
199C2×C8C22×D411yescharacteristic
200C2×C8H(5,𝔽2)11yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4C2×D811yescharacteristic
205C4⋊C4C2×D811yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—42no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—82no—
213C42C22×D411yescharacteristic
214C16—21no—
215C16—41no—
216C16—41no—
217C16—41no—
218C16AGL(1,ℤ/8)12yes—
219C23—21no—
220C23—21no—
221C23—42no—
222C23—82no—
223C23—164no—
224C23—164no—
225C23—322no—
226Q8—162no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—42no—
231D4—42no—
232D4—82no—
233D4—82no—
234D4—82no—
235D4—82no—
236D4—162no—
237D4—164no—
238D4—164no—
239D4—322no—
240D4#50812yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4D4211yescharacteristic
252C2×C4C22×D811yescharacteristic
253C2×C4#57611yescharacteristic
254C8—22no—
255C8—21no—
256C8—21no—
257C8—21no—
258C8—41no—
259C8—41no—
260C8—82no—
261C8C2×AGL(1,ℤ/8)11yescharacteristic
262C8#57711yescharacteristic
263C22—22no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22—162no—
273C22—162no—
274C22—162no—
275C22—328no—
276C22D4×D811yescharacteristic
277C4—21no—
278C4—41no—
279C4—41no—
280C4—82no—
281C4—162no—
282C4#327211yescharacteristic
283C4#329011yescharacteristic
284C4C2×#50811yescharacteristic
285C4#341511yescharacteristic
286C2—42no—
287C2—81no—
288C2—162no—
289C2—322no—
290C2#1988311yescharacteristic
291C2#3384611yescharacteristic
292C2#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. #33908C2semidirect
2. D4×D16C2semidirect
3. #19898C2semidirect
4. #12503C2semidirect
5. #12485C2semidirect
6. #12452C2semidirect
7. #12430C2semidirect
8. #12372C2semidirect
9. #12343C2semidirect
10. #12081C2semidirect
11. C16×D8C2semidirect
13. #3341C22semidirect
16. #2252C22semidirect
18. #2240C22semidirect
20. #2226C22semidirect
27. D4×C16C22semidirect
29. #1706C22semidirect
32. #1667C22semidirect
35. #1589C22semidirect
37. #1371C22semidirect
38. #1370C22semidirect
39. #1365C22semidirect
40. #1361C22semidirect
41. #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
#339082
D4×D162
#198982
#125032
#124852
#124522
#124302
#123722
#123432
#120812
C16×D82