Finite Group Databasebeta

← #513191

Subgroups — #513191

Order 512.

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

normal subgroups only

This group has 419 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#33983C212yesmaximal, complemented
2#33859C212yesmaximal, complemented
3#13728C211yesmaximal, characteristic, complemented
4#13702C212yesmaximal, complemented
5#13691C212yesmaximal, complemented
6#13672C212yesmaximal, complemented
7#13669C212yesmaximal, complemented
8#13655C211yesmaximal, characteristic, complemented
9#13647C211yesmaximal, characteristic, complemented
10#3417C2212yes—
11#3415C2212yescomplemented
12#3290—22no—
13#2264—22no—
14#2264C2212yescomplemented
15#2263—22no—
16#2263C2212yes—
17#2262—22no—
18#2262C2211yescharacteristic
19C2×Q64C2211yescharacteristic, complemented
20C2×SD64—22no—
21C2×SD64C2212yescomplemented
22C2×D32C2211yescharacteristic
23#2258C2212yescomplemented
24#2249C2212yescomplemented
25#2222C2212yescomplemented
26#2221C2212yes—
27#2212—22no—
28#2194—22no—
29#2193—22no—
30#2178C2212yescomplemented
31#2170C2211yescharacteristic
32#1425C2211yescharacteristic, complemented
33#1422C2211yescharacteristic, complemented
34#1420C2212yescomplemented
35#1419C2212yescomplemented
36#1417C2212yescomplemented
37#1401C2212yescomplemented
38C4×C32C2211yescharacteristic, complemented
39#578—22no—
40#577—42no—
41#576—22no—
42#510—22no—
43#510—42no—
44#509—22no—
45#509—42no—
46#508—22no—
47#508—22no—
48#508C2312yes—
49C2×Q32—22no—
50C2×Q32C2312yes—
51C2×SD32—22no—
52C2×D16—22no—
53C2×D16C2312yes—
54#504C2312yes—
55#495—22no—
56#473—22no—
57#472—22no—
58#471—42no—
59#455—42no—
60#454—42no—
61#443—22no—
62#443D411yescharacteristic
63Q64—22no—
64Q64—41no—
65Q64D411yescharacteristic
66SD64—22no—
67SD64—22no—
68SD64—22no—
69SD64—22no—
70SD64D412yes—
71D32—22no—
72D32—41no—
73D32D411yescharacteristic
74M64—22no—
75M64C2312yes—
76C2×C32—22no—
77C2×C32C2311yescharacteristic
78C2×C32C2311yescharacteristic
79#368C2312yes—
80#364D411yescharacteristic
81#362—22no—
82#361—22no—
83#359—22no—
84#350D412yes—
85C4×C16C2311yescharacteristic
862−1+4—42no—
87H(5,𝔽2)—42no—
88#137—42no—
89#137—42no—
90#137—82no—
91#137—82no—
92AGL(1,ℤ/8)—42no—
93AGL(1,ℤ/8)—42no—
94AGL(1,ℤ/8)—82no—
95AGL(1,ℤ/8)—82no—
96#135—22no—
97#135—42no—
98#135—42no—
99#135—82no—
100#135D811yescharacteristic
101C2×Q16—22no—
102C2×Q16—42no—
103C2×SD16—42no—
104C2×SD16—82no—
105C2×D8—22no—
106C2×D8—42no—
107#131—42no—
108#131C2×D412yes—
109#124—42no—
110Q32—22no—
111Q32—22no—
112Q32—42no—
113Q32C2×D412yes—
114SD32—22no—
115SD32—42no—
116SD32—42no—
117SD32—82no—
118D16—22no—
119D16—22no—
120D16—42no—
121D16C2×D412yes—
122M32—22no—
123M32C2×D412yes—
124C2×C16—22no—
125C2×C16C2×D411yescharacteristic
126C2×C16C2411yescharacteristic
127#108—22no—
128#108D811yescharacteristic
129#104—42no—
130#104D812yes—
131#101—42no—
132#100—42no—
133C4×C8C2×D411yescharacteristic
134C32—22no—
135C32—21no—
136C32C2×D411yescharacteristic
137C32C2×D411yescharacteristic
138#41—42no—
139#41—82no—
140#41—82no—
141#41—82no—
142#41—82no—
143#41—162no—
144#41C2×D812yes—
145C2×Q8—42no—
146C2×Q8—82no—
147C2×Q8—82no—
148C2×D4—42no—
149C2×D4—82no—
150C2×D4—82no—
151C2×D4—162no—
152Q16—22no—
153Q16—42no—
154Q16—42no—
155Q16—82no—
156Q16—82no—
157Q16D1611yescharacteristic
158SD16—42no—
159SD16—82no—
160SD16—82no—
161SD16—82no—
162SD16—82no—
163SD16—82no—
164SD16—82no—
165SD16D1612yes—
166D8—22no—
167D8—42no—
168D8—42no—
169D8—82no—
170D8—82no—
171D8D1611yescharacteristic
172M16—22no—
173M16—42no—
174M16—82no—
175M16C2×D812yes—
176C2×C8—22no—
177C2×C8—82no—
178C2×C8C2×D811yescharacteristic
179C2×C8C22×D411yescharacteristic
180C22⋊C4—162no—
181C42C2×D811yescharacteristic
182C16—22no—
183C16—21no—
184C16C22×D411yescharacteristic
185C16H(5,𝔽2)11yescharacteristic
186C23—82no—
187C23—162no—
188Q8—42no—
189Q8—82no—
190Q8—82no—
191Q8—162no—
192Q8C2×D1612yes—
193D4—22no—
194D4—42no—
195D4—82no—
196D4—82no—
197D4—82no—
198D4—162no—
199D4—162no—
200D4—162no—
201D4C2×D1612yes—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—162no—
207C2×C4—162no—
208C2×C4—162no—
209C2×C4C22×D811yescharacteristic
210C8—22no—
211C8—82no—
212C8C2×D1611yescharacteristic
213C8C2×D1611yescharacteristic
214C8D4211yescharacteristic
215C8#57611yescharacteristic
216C22—22no—
217C22—82no—
218C22—162no—
219C22—322no—
220C22—642no—
221C22#341511yescharacteristic
222C4—22no—
223C4—41no—
224C4—82no—
225C4—162no—
226C4D4×D811yescharacteristic
227C4C22×D1611yescharacteristic
228C2—21no—
229C2—42no—
230C2—162no—
231C2—322no—
232C2D4×D1611yescharacteristic
subgroupG/H
center, socleC2D4×D16
commutator, Frattini, cosocleC2×C16C24
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #2170 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33983C2semidirect
2. #33859C2semidirect
3. #13728C2semidirect
4. #13702C2semidirect
5. #13691C2semidirect
6. #13672C2semidirect
7. #13669C2semidirect
8. #13655C2semidirect
9. #13647C2semidirect
11. #3415C22semidirect
14. #2264C22semidirect
19. C2×Q64C22semidirect
21. C2×SD64C22semidirect
23. #2258C22semidirect
24. #2249C22semidirect
25. #2222C22semidirect
30. #2178C22semidirect
32. #1425C22semidirect
33. #1422C22semidirect
34. #1420C22semidirect
35. #1419C22semidirect
36. #1417C22semidirect
37. #1401C22semidirect
38. C4×C32C22semidirect

Maximal subgroups

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

subgroupindex
#339832
#338592
#137282
#137022
#136912
#136722
#136692
#136552
#136472