Finite Group Databasebeta

← #513165

Subgroups — #513165

Order 512.

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

normal subgroups only

This group has 455 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
1C2×#2262C211yesmaximal, characteristic, complemented
2C22×D32C211yesmaximal, characteristic, complemented
3#33846C212yesmaximal, complemented
4#13728C211yesmaximal, characteristic, complemented
5#13695C211yesmaximal, characteristic, complemented
6#13694C211yesmaximal, characteristic, complemented
7#13689C212yesmaximal, complemented
8#13664C212yesmaximal, complemented
9#13663C212yesmaximal, complemented
10C4×D32C211yesmaximal, characteristic, complemented
11D4×C32C211yesmaximal, characteristic, complemented
12C2×#508C2212yescomplemented
13C22×D16C2212yes—
14#3280—22no—
15#2262—24no—
16C2×Q64C2211yescharacteristic, complemented
17C2×SD64C2212yescomplemented
18C2×SD64C2212yescomplemented
19C2×D32—24no—
20C2×D32C2212yes—
21C2×D32C2211yescharacteristic
22C2×D32C2211yescharacteristic
23C22×C32C2211yescharacteristic
24C22×C32C2211yescharacteristic, complemented
25#2249C2212yescomplemented
26#2216C2212yes—
27#2215C2212yescomplemented
28#2210—22no—
29#2187—22no—
30#2186—22no—
31C4×D16C2212yescomplemented
32D4×C16C2211yescharacteristic
33#1423C2211yescharacteristic, complemented
34C4⋊C32C2211yescharacteristic, complemented
35#1416C2212yescomplemented
36#1415C2212yescomplemented
37#1415C2212yescomplemented
38C22⋊C32C2211yescharacteristic
39C22⋊C32C2211yescharacteristic, complemented
40C4×C32C2211yescharacteristic, complemented
41C2×#135—22no—
42C22×D8—22no—
43#546—42no—
44#508—24no—
45#508—42no—
46C2×Q32C2312yes—
47C2×SD32—22no—
48C2×SD32—22no—
49C2×D16—24no—
50C2×D16—22no—
51C2×D16—42no—
52C2×D16C2312yes—
53C2×D16C2312yes—
54C22×C16C2311yescharacteristic
55C22×C16C2311yescharacteristic
56#495—22no—
57#467—22no—
58#466—22no—
59#463—42no—
60#450—42no—
61#449—42no—
62C4×D8—22no—
63C8×D4D411yescharacteristic
64Q64—22no—
65SD64—24no—
66SD64—42no—
67D32—24no—
68D32—22no—
69D32D414yescomplemented
70C2×C32—22no—
71C2×C32—22no—
72C2×C32C2311yescharacteristic
73C2×C32C2311yescharacteristic
74C2×C32C2311yescharacteristic
75C2×C32C2311yescharacteristic
76#366C2312yes—
77C4⋊C16D411yescharacteristic
78#358—22no—
79#357—22no—
80#357—22no—
81C22⋊C16D411yescharacteristic
82C22⋊C16D411yescharacteristic
83C4×C16C2311yescharacteristic
84C2×#41—42no—
85C22×D4—42no—
86#135—44no—
87#135—82no—
88C2×Q16—22no—
89C2×SD16—42no—
90C2×SD16—42no—
91C2×D8—22no—
92C2×D8—22no—
93C2×D8—44no—
94C2×D8—42no—
95C2×D8—82no—
96C22×C8C2×D411yescharacteristic
97C22×C8C2×D411yescharacteristic
98#124—42no—
99#123—82no—
100C22⋊Q8—42no—
101#121—42no—
102#121—82no—
103#120—82no—
104C4×D4—42no—
105C4×D4D811yescharacteristic
106C2×C22⋊C4—82no—
107Q32—22no—
108Q32—22no—
109SD32—44no—
110SD32—82no—
111D16—22no—
112D16—22no—
113D16—22no—
114D16—44no—
115D16C2×D414yes—
116C2×C16—21no—
117C2×C16—21no—
118C2×C16—21no—
119C2×C16—21no—
120C2×C16C2×D411yescharacteristic
121C2×C16C2×D411yescharacteristic
122C2×C16C2×D411yescharacteristic
123C2×C16C2411yescharacteristic
124#107—22no—
125C4⋊C8D811yescharacteristic
126#103—42no—
127#102—42no—
128#102—42no—
129C22⋊C8D811yescharacteristic
130C22⋊C8D811yescharacteristic
131C4×C8C2×D411yescharacteristic
132C32—21no—
133C32—21no—
134C32—21no—
135C32C2×D412yes—
136C24—82no—
137#41—84no—
138#41—162no—
139C2×Q8—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—84no—
143C2×D4—82no—
144C2×D4—82no—
145C2×D4—82no—
146C2×D4—162no—
147C2×D4D1611yescharacteristic
148C22×C4—82no—
149C22×C4—82no—
150C22×C4C2×D811yescharacteristic
151C22×C4C2×D811yescharacteristic
152Q16—42no—
153Q16—42no—
154SD16—84no—
155SD16—162no—
156D8—24no—
157D8—42no—
158D8—42no—
159D8—42no—
160D8—84no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8C2×D811yescharacteristic
166C2×C8C2×D811yescharacteristic
167C2×C8C2×D811yescharacteristic
168C2×C8C22×D411yescharacteristic
169C4⋊C4—42no—
170C4⋊C4—82no—
171C4⋊C4D1611yescharacteristic
172C22⋊C4—82no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C22⋊C4—82no—
176C22⋊C4—82no—
177C22⋊C4D1611yescharacteristic
178C22⋊C4D1611yescharacteristic
179C42C2×D811yescharacteristic
180C16—21no—
181C16—21no—
182C16—21no—
183C16C22×D411yescharacteristic
184C16H(5,𝔽2)11yescharacteristic
185C23—82no—
186C23—82no—
187C23—164no—
188C23—164no—
189C23—322no—
190C23C2×D1611yescharacteristic
191C23C2×D1611yescharacteristic
192Q8—82no—
193Q8—82no—
194D4—22no—
195D4—44no—
196D4—82no—
197D4—82no—
198D4—82no—
199D4—164no—
200D4—164no—
201D4—322no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4—164no—
209C2×C4—322no—
210C2×C4C2×D1611yescharacteristic
211C2×C4C2×D1611yescharacteristic
212C2×C4C2×D1611yescharacteristic
213C2×C4C2×D1611yescharacteristic
214C2×C4C22×D811yescharacteristic
215C8—21no—
216C8—21no—
217C8—21no—
218C8D4211yescharacteristic
219C8#57611yescharacteristic
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—84no—
226C22—162no—
227C22—162no—
228C22—162no—
229C22—162no—
230C22—162no—
231C22—328no—
232C22#226212yescomplemented
233C22C22×D1611yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—162no—
239C4—162no—
240C4D4×D811yescharacteristic
241C4#341511yescharacteristic
242C2—22no—
243C2—41no—
244C2—164no—
245C2—322no—
246C2D4×D1611yescharacteristic
247C2C2×#226211yescharacteristic
248C2#3398211yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
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. C2×#2262C2semidirect
2. C22×D32C2semidirect
3. #33846C2semidirect
4. #13728C2semidirect
5. #13695C2semidirect
6. #13694C2semidirect
7. #13689C2semidirect
8. #13664C2semidirect
9. #13663C2semidirect
10. C4×D32C2semidirect
11. D4×C32C2semidirect
12. C2×#508C22semidirect
16. C2×Q64C22semidirect
17. C2×SD64C22semidirect
18. C2×SD64C22semidirect
24. C22×C32C22semidirect
25. #2249C22semidirect
27. #2215C22semidirect
31. C4×D16C22semidirect
33. #1423C22semidirect
34. C4⋊C32C22semidirect
35. #1416C22semidirect
36. #1415C22semidirect
37. #1415C22semidirect
39. C22⋊C32C22semidirect
40. C4×C32C22semidirect
69. D32D4semidirect
232. C22#2262semidirect

Maximal subgroups

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

subgroupindex
C2×#22622
C22×D322
#338462
#137282
#136952
#136942
#136892
#136642
#136632
C4×D322
D4×C322