Finite Group Databasebeta

← #513196

Subgroups — #513196

Order 512.

391 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 122 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#33905C211yesmaximal, characteristic, complemented
2D4×D16C211yesmaximal, characteristic
3#13719C211yesmaximal, characteristic, complemented
4#13712C212yesmaximal, complemented
5#13705C211yesmaximal, characteristic, complemented
6#13696C212yesmaximal, complemented
7#13686C211yesmaximal, characteristic, complemented
8#13664C212yesmaximal, complemented
9C4×SD64C211yesmaximal, characteristic, complemented
10D4×C32C211yesmaximal, characteristic, complemented
11C2×#1415C212yesmaximal, complemented
12C22×D16C2212yes—
13#3338—21no—
14D4×D8—21no—
15C2×SD64C2211yescharacteristic
16C2×SD64C2212yescomplemented
17C22×C32C2212yescomplemented
18#2252C2211yescharacteristic, complemented
19#2246C2211yescharacteristic
20#2236—22no—
21#2227—21no—
22#2216C2212yescomplemented
23#2215C2212yes—
24#2206—21no—
25#2184—22no—
26C4×Q32C2211yescharacteristic, complemented
27C4×D16C2211yescharacteristic
28D4×C16C2211yescharacteristic
29C2×#366C2212yescomplemented
30#1424C2211yescharacteristic, complemented
31#1424C2212yescomplemented
32C4⋊C32C2211yescharacteristic, complemented
33#1416C2211yescharacteristic, complemented
34#1415C2211yescharacteristic
35#1415C2212yes—
36#1415C2212yescomplemented
37#1415C2212yescomplemented
38#1415C2212yescomplemented
39C22⋊C32C2212yescomplemented
40C4×C32C2211yescharacteristic, complemented
41C22×D8—22no—
42#554—41no—
43D42—41no—
44C2×Q32C2311yescharacteristic
45C2×D16—24no—
46C2×D16—22no—
47C2×D16—21no—
48C2×D16—42no—
49C2×D16C2312yes—
50C2×D16C2311yescharacteristic
51C22×C16C2312yes—
52#500—21no—
53#493—21no—
54#484—42no—
55#477—41no—
56#467—22no—
57#466—22no—
58#459—41no—
59#447—42no—
60C4×Q16—21no—
61C4×D8—21no—
62C8×D4D411yescharacteristic
63C2×#107—22no—
64SD64—22no—
65SD64—42no—
66C2×C32—42no—
67C2×C32C2311yescharacteristic
68C2×C32C2311yescharacteristic
69C2×C32C2312yes—
70#366—22no—
71#366—21no—
72#366C2311yescharacteristic
73#366C2312yes—
74#366C2311yescharacteristic
75C4⋊C16D411yescharacteristic
76#358—21no—
77#358—22no—
78#357—21no—
79#357—22no—
80C22⋊C16D412yes—
81C4×C16C2311yescharacteristic
82C22×D4—42no—
83C22×D4—81no—
84C2×Q16—21no—
85C2×D8—21no—
86C2×D8—22no—
87C2×D8—44no—
88C2×D8—42no—
89C2×D8—41no—
90C2×D8—82no—
91C22×C8C2×D412yes—
92C4⋊Q8—41no—
93#127—41no—
94#125—81no—
95C22⋊Q8—42no—
96C22⋊Q8—82no—
97#121—42no—
98#121—81no—
99#120—82no—
100C4×Q8—41no—
101C4×D4—41no—
102C4×D4—81no—
103C4×D4D811yescharacteristic
104C2×C4⋊C4—42no—
105Q32—21no—
106Q32—21no—
107D16—22no—
108D16—21no—
109D16—22no—
110D16—44no—
111D16—81no—
112D16#4112yes—
113C2×C16—22no—
114C2×C16—22no—
115C2×C16C2×D411yescharacteristic
116C2×C16C2×D412yes—
117C2×C16C2411yescharacteristic
118#107—21no—
119#107—21no—
120#107—22no—
121#107—42no—
122#107—41no—
123C4⋊C8D811yescharacteristic
124#103—41no—
125#103—42no—
126#102—41no—
127#102—42no—
128C22⋊C8D812yes—
129C4×C8C2×D411yescharacteristic
130C32—21no—
131C32—21no—
132C32—22no—
133C24—82no—
134C2×Q8—41no—
135C2×Q8—81no—
136C2×D4—41no—
137C2×D4—42no—
138C2×D4—84no—
139C2×D4—82no—
140C2×D4—82no—
141C2×D4—81no—
142C2×D4—81no—
143C2×D4—162no—
144C2×D4—164no—
145C2×D4D1611yescharacteristic
146C22×C4—82no—
147C22×C4—81no—
148C22×C4C2×D812yes—
149Q16—41no—
150Q16—41no—
151D8—22no—
152D8—42no—
153D8—41no—
154D8—42no—
155D8—84no—
156D8—161no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8C2×D811yescharacteristic
160C2×C8C2×D812yes—
161C2×C8C22×D411yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—82no—
166C4⋊C4—81no—
167C4⋊C4—81no—
168C4⋊C4—82no—
169C4⋊C4D1611yescharacteristic
170C22⋊C4—82no—
171C22⋊C4—81no—
172C22⋊C4—82no—
173C22⋊C4D1612yes—
174C42—81no—
175C42C2×D811yescharacteristic
176C16—22no—
177C16—21no—
178C16C2×#4111yescharacteristic
179C16H(5,𝔽2)11yescharacteristic
180C23—81no—
181C23—82no—
182C23—164no—
183C23—164no—
184C23—322no—
185C23C2×D1612yes—
186Q8—81no—
187Q8—81no—
188Q8—321no—
189D4—42no—
190D4—81no—
191D4—82no—
192D4—82no—
193D4—164no—
194D4—164no—
195D4—162no—
196D4—321no—
197D4SD6414yescomplemented
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—162no—
205C2×C4—322no—
206C2×C4C2×D1611yescharacteristic
207C2×C4C2×D1612yes—
208C2×C4C2×D1611yescharacteristic
209C2×C4C22×D811yescharacteristic
210C8—21no—
211C8—22no—
212C8#54611yescharacteristic
213C8#57711yescharacteristic
214C22—22no—
215C22—22no—
216C22—82no—
217C22—161no—
218C22—161no—
219C22—162no—
220C22—162no—
221C22—162no—
222C22—328no—
223C22C2×SD6414yes—
224C22C22×D1611yescharacteristic
225C4—22no—
226C4—21no—
227C4—161no—
228C4—161no—
229C4—161no—
230C4—162no—
231C4C2×SD6412yes—
232C4#329411yescharacteristic
233C4#341611yescharacteristic
234C2—24no—
235C2—162no—
236C2—322no—
237C2#3386111yescharacteristic
238C2C22×SD6411yescharacteristic
239C2#3398211yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radicalD4×D16C2
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33905C2semidirect
3. #13719C2semidirect
4. #13712C2semidirect
5. #13705C2semidirect
6. #13696C2semidirect
7. #13686C2semidirect
8. #13664C2semidirect
9. C4×SD64C2semidirect
10. D4×C32C2semidirect
11. C2×#1415C2semidirect
16. C2×SD64C22semidirect
17. C22×C32C22semidirect
18. #2252C22semidirect
22. #2216C22semidirect
26. C4×Q32C22semidirect
29. C2×#366C22semidirect
30. #1424C22semidirect
31. #1424C22semidirect
32. C4⋊C32C22semidirect
33. #1416C22semidirect
36. #1415C22semidirect
37. #1415C22semidirect
38. #1415C22semidirect
39. C22⋊C32C22semidirect
40. C4×C32C22semidirect
197. D4SD64semidirect

Maximal subgroups

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

subgroupindex
#339052
D4×D162
#137192
#137122
#137052
#136962
#136862
#136642
C4×SD642
D4×C322
C2×#14152