Finite Group Databasebeta

← #492296

Subgroups — #492296

Order 512.

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

normal subgroups only

This group has 552 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#33958C211yesmaximal, characteristic, complemented
2D4×D16C211yesmaximal, characteristic, complemented
3#19885C212yesmaximal, complemented
4#12515C211yesmaximal, characteristic, complemented
5#12491C212yesmaximal, complemented
6#12439C211yesmaximal, characteristic, complemented
7#12430C211yesmaximal, characteristic, complemented
8#12328C212yesmaximal, complemented
9#12316C212yesmaximal, complemented
10#12081C211yesmaximal, characteristic, complemented
11C16×SD16C211yesmaximal, characteristic, complemented
12C22×D16—21no—
13#3391C2212yescomplemented
14D4×D8C2212yes—
15#3274—22no—
16#2246—21no—
17#2246C2211yescharacteristic
18#2246C2211yescharacteristic, complemented
19#2246C2212yescomplemented
20#2215—21no—
21#2206—22no—
22#2206C2212yes—
23#2206C2212yescomplemented
24#2184—22no—
25Q8×C16C2211yescharacteristic, complemented
26C4×D16—21no—
27C4×D16C2211yescharacteristic
28D4×C16C2211yescharacteristic
29#1712C2212yescomplemented
30#1687—22no—
31#1668C2212yescomplemented
32#1667C2212yes—
33#1621—22no—
34#1619—22no—
35#1589C2212yescomplemented
36C8×SD16C2211yescharacteristic
37#1371C2211yescharacteristic, complemented
38#1367C2211yescharacteristic, complemented
39#1361C2212yescomplemented
40#1361C2212yescomplemented
41Q8⋊C16C2211yescharacteristic, complemented
42#1331C2212yescomplemented
43D4⋊C16C2211yescharacteristic
44C8×C16C2211yescharacteristic, complemented
45C2×AGL(1,ℤ/8)—42no—
46C22×D8—22no—
47#550—22no—
48D42—22no—
49C2×D16—22no—
50C2×D16—22no—
51C2×D16—21no—
52C2×D16—21no—
53C2×D16—21no—
54C2×D16—21no—
55C2×D16—42no—
56C2×D16—44no—
57C2×D16D411yescharacteristic
58C22×C16—21no—
59#493—22no—
60#493—22no—
61#493—22no—
62#493C2312yes—
63#493C2312yes—
64#469—42no—
65#466—22no—
66#460—22no—
67#459—22no—
68#459—22no—
69#459—22no—
70#459—42no—
71#449—42no—
72#447—42no—
73#447—42no—
74C8×Q8C2311yescharacteristic
75C4×SD16D411yescharacteristic
76C4×D8—22no—
77C4×D8C2312yes—
78#436—22no—
79C8×D4C2311yescharacteristic
80#366D411yescharacteristic
81C4⋊C16—21no—
82C4⋊C16C2311yescharacteristic
83C4⋊C16C2311yescharacteristic
84#357—22no—
85#357—22no—
86#357D412yes—
87C22⋊C16—21no—
88C4×C16—21no—
89C4×C16C2311yescharacteristic
90C4×C16C2311yescharacteristic
91#335C2312yes—
92#334D411yescharacteristic
93#331—22no—
94#331—22no—
95Q8⋊C8D411yescharacteristic
96D4⋊C8—22no—
97D4⋊C8D411yescharacteristic
98C82C2311yescharacteristic
99C2×#41—42no—
100C22×D4—42no—
101C22×D4—42no—
102AGL(1,ℤ/8)—88no—
103C2×SD16—42no—
104C2×SD16D811yescharacteristic
105C2×D8—22no—
106C2×D8—22no—
107C2×D8—22no—
108C2×D8—22no—
109C2×D8—42no—
110C2×D8—42no—
111C2×D8—44no—
112C2×D8—42no—
113C2×D8—42no—
114C2×D8—42no—
115C2×D8—82no—
116C2×D8—82no—
117C2×D8C2×D412yes—
118C2×M16—42no—
119C22×C8—21no—
120#127—22no—
121#127—22no—
122#127—42no—
123#121—42no—
124#121—42no—
125#121—42no—
126#121—82no—
127#120—82no—
128C4×Q8C2×D411yescharacteristic
129C4×D4—22no—
130C4×D4—42no—
131C4×D4C2×D411yescharacteristic
132D16—22no—
133D16—44no—
134D16—42no—
135D16—42no—
136D16—81no—
137D16—81no—
138D16—84no—
139D16—82no—
140C2×C16—21no—
141C2×C16—21no—
142C2×C16—21no—
143C2×C16—42no—
144C2×C16C2×D411yescharacteristic
145C2×C16C2×D411yescharacteristic
146#107C2×D412yes—
147#106D811yescharacteristic
148C4⋊C8—21no—
149C4⋊C8—22no—
150C4⋊C8C2×D411yescharacteristic
151C4⋊C8C2×D411yescharacteristic
152#103D811yescharacteristic
153#102—22no—
154#102—42no—
155#102—42no—
156#102—42no—
157#102—42no—
158#102D811yescharacteristic
159C22⋊C8—21no—
160C22⋊C8—42no—
161C4×C8—21no—
162C4×C8C2×D411yescharacteristic
163C4×C8C2×D411yescharacteristic
164C4×C8C2411yescharacteristic
165C24—82no—
166#41—84no—
167#41—162no—
168C2×Q8C2×D811yescharacteristic
169C2×D4—22no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4—82no—
179C2×D4—84no—
180C2×D4—82no—
181C2×D4—84no—
182C2×D4—162no—
183C2×D4—162no—
184C2×D4C2×D811yescharacteristic
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—82no—
188SD16—84no—
189SD16D1614yescomplemented
190D8—22no—
191D8—22no—
192D8—42no—
193D8—42no—
194D8—42no—
195D8—42no—
196D8—82no—
197D8—82no—
198D8—84no—
199D8—84no—
200D8—84no—
201D8—162no—
202D8—164no—
203D8—162no—
204M16—84no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8C2×D811yescharacteristic
212C2×C8C2×D811yescharacteristic
213C2×C8C22×D411yescharacteristic
214C2×C8H(5,𝔽2)11yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—22no—
217C4⋊C4C2×D811yescharacteristic
218C4⋊C4C2×D811yescharacteristic
219C22⋊C4—21no—
220C22⋊C4—42no—
221C22⋊C4—82no—
222C22⋊C4—82no—
223C42—21no—
224C42C22×D411yescharacteristic
225C16—22no—
226C16—21no—
227C16—41no—
228C16—41no—
229C16AGL(1,ℤ/8)12yescomplemented
230C23—21no—
231C23—42no—
232C23—82no—
233C23—82no—
234C23—164no—
235C23—164no—
236C23—322no—
237Q8—21no—
238Q8C2×D1612yescomplemented
239D4—21no—
240D4—42no—
241D4—42no—
242D4—82no—
243D4—82no—
244D4—82no—
245D4—82no—
246D4—82no—
247D4—82no—
248D4—82no—
249D4—82no—
250D4—164no—
251D4—162no—
252D4—162no—
253D4—164no—
254D4—164no—
255D4—322no—
256D4C2×D1612yes—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—164no—
266C2×C4D4211yescharacteristic
267C2×C4C22×D811yescharacteristic
268C2×C4#57611yescharacteristic
269C8—22no—
270C8—21no—
271C8—21no—
272C8—41no—
273C8—41no—
274C8—82no—
275C8C2×D1612yescomplemented
276C8C2×AGL(1,ℤ/8)11yescharacteristic
277C8#57611yescharacteristic
278C22—22no—
279C22—41no—
280C22—41no—
281C22—82no—
282C22—82no—
283C22—82no—
284C22—162no—
285C22—162no—
286C22—162no—
287C22—162no—
288C22—162no—
289C22—162no—
290C22—328no—
291C22D4×D811yescharacteristic
292C4—22no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—82no—
297C4#327411yescharacteristic
298C4#329211yescharacteristic
299C4C22×D1611yescharacteristic
300C4#341511yescharacteristic
301C2—42no—
302C2—162no—
303C2—322no—
304C2—322no—
305C2#1988511yescharacteristic
306C2D4×D1611yescharacteristic
307C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×SD16 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33958C2semidirect
2. D4×D16C2semidirect
3. #19885C2semidirect
4. #12515C2semidirect
5. #12491C2semidirect
6. #12439C2semidirect
7. #12430C2semidirect
8. #12328C2semidirect
9. #12316C2semidirect
10. #12081C2semidirect
11. C16×SD16C2semidirect
13. #3391C22semidirect
18. #2246C22semidirect
19. #2246C22semidirect
23. #2206C22semidirect
25. Q8×C16C22semidirect
29. #1712C22semidirect
31. #1668C22semidirect
35. #1589C22semidirect
37. #1371C22semidirect
38. #1367C22semidirect
39. #1361C22semidirect
40. #1361C22semidirect
41. Q8⋊C16C22semidirect
42. #1331C22semidirect
44. C8×C16C22semidirect
189. SD16D16semidirect
229. C16AGL(1,ℤ/8)semidirect
238. Q8C2×D16semidirect
275. C8C2×D16semidirect

Maximal subgroups

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

subgroupindex
#339582
D4×D162
#198852
#125152
#124912
#124392
#124302
#123282
#123162
#120812
C16×SD162