Finite Group Databasebeta

← #497149

Subgroups — #497149

Order 512.

540 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 160 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#33721C211yesmaximal, characteristic
2#22285C212yesmaximal, complemented
3#13216C212yesmaximal, complemented
4#13035C212yesmaximal, complemented
5#13034C212yesmaximal, complemented
6#13020C212yesmaximal, complemented
7#13016C212yesmaximal
8#12893C211yesmaximal, characteristic, complemented
9#12597C211yesmaximal, characteristic, complemented
10C2×#510C2212yes—
11C2×#508C2211yescharacteristic
12C22×Q32C2211yescharacteristic
13#3156C2212yes—
14#2957C2212yes—
15#2950C2212yes—
16#2251C414yescomplemented
17#2249C412yescomplemented
18#2247C412yescomplemented
19C2×#362C2212yes—
20#2145C2212yes—
21#2143C2212yes—
22#2141C2212yes—
23#2110C2211yescharacteristic
24#2108C2211yescharacteristic
25#1979C2212yes—
26#1976C2212yes—
27#1912—22no—
28#1909—22no—
29#1905—22no—
30#1902—22no—
31#1833C2212yes—
32#1747C2211yescharacteristic
33#1385C2211yescharacteristic
34#1384C2211yescharacteristic
35#1382C2212yes—
36#1379C2212yes—
37#1378C2212yes—
38C2×#137—22no—
39C2×#135C2312yes—
40C22×Q16C2312yes—
41#568—22no—
42#536—22no—
43#519—22no—
44#510—42no—
45#510—42no—
46#508—42no—
47C2×Q32—42no—
48C2×Q32C2×C412yes—
49C2×Q32C2×C412yes—
50C2×Q32C2×C411yescharacteristic
51C2×Q32C2×C414yes—
52C2×SD32C2×C412yes—
53C2×SD32C2×C414yes—
54C2×D16C2×C411yescharacteristic
55C2×M32C2312yes—
56C2×M32C2311yescharacteristic
57C22×C16C2311yescharacteristic
58#497—24no—
59#495C2×C414yes—
60#494C2×C414yes—
61#444—42no—
62#443—42no—
63#441—42no—
64C4×Q16—42no—
65C2×#108C2312yes—
66#427C2312yes—
67#421—22no—
68#421—22no—
69#419—22no—
70#419—22no—
71C2×#101—22no—
72#410—22no—
73#405C2312yes—
74#405C2311yescharacteristic
75#362—42no—
76#360—42no—
77#359—42no—
78#358—42no—
79#347—22no—
80#346C2×C412yes—
81C4×C16C2×C412yes—
82#344—22no—
83#338D411yescharacteristic
84H(3,ℤ/4)—22no—
85H(3,ℤ/4)D411yescharacteristic
86H(3,ℤ/4)D412yes—
87C2×#41—22no—
88C22×Q8—22no—
89#137—82no—
90#137—82no—
91#135—22no—
92#135—22no—
93#135—22no—
94#135—22no—
95C2×Q16—24no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—22no—
101C2×Q16C22×C414yes—
102C2×Q16C22×C412yes—
103C2×SD16—24no—
104C2×SD16—22no—
105C2×D8C22×C412yes—
106#131—44no—
107#131—82no—
108C2×M16—22no—
109C2×M16—82no—
110C2×M16C2×D412yes—
111C2×M16C2×D411yescharacteristic
112C22×C8C2411yescharacteristic
113C4⋊Q8—24no—
114#124—24no—
115#123—82no—
116C22⋊Q8—82no—
117C4×Q8—44no—
118C4×Q8—82no—
119#117—22no—
120#117—82no—
121#117C2×D412yes—
122#117C2×D411yescharacteristic
123Q32—42no—
124Q32—42no—
125Q32—41no—
126Q32—44no—
127SD32—42no—
128SD32—44no—
129D16—41no—
130M32—22no—
131M32—22no—
132M32—22no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16C22×C411yescharacteristic
136C2×C16C22×C412yes—
137C2×C16C22×C411yescharacteristic
138#108—22no—
139#108—22no—
140#107—22no—
141#106—22no—
142#104—44no—
143#104—82no—
144#103—44no—
145#103—82no—
146#101—82no—
147#99—82no—
148#97—22no—
149#97#4112yes—
150C4×C8—22no—
151C4×C8C22×C412yes—
152#41—42no—
153#41—42no—
154#41—42no—
155#41—42no—
156C2×Q8—24no—
157C2×Q8—22no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×D4—22no—
164C2×D4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4C22×D411yescharacteristic
168Q16—24no—
169Q16—22no—
170Q16—24no—
171Q16—22no—
172Q16—42no—
173Q16—42no—
174Q16—84no—
175SD16—42no—
176SD16—42no—
177SD16—84no—
178D8—22no—
179D8—22no—
180M16—22no—
181M16—22no—
182M16—22no—
183M16—42no—
184M16—42no—
185M16—84no—
186C2×C8—22no—
187C2×C8—42no—
188C2×C8—84no—
189C2×C8C4×D412yes—
190C2×C8C23×C411yescharacteristic
191C2×C8C2×#4111yescharacteristic
192C2×C8H(5,𝔽2)12yes—
193C2×C8H(5,𝔽2)12yes—
194C4⋊C4—21no—
195C4⋊C4—22no—
196C4⋊C4—42no—
197C4⋊C4—84no—
198C4⋊C4—84no—
199C22⋊C4—21no—
200C22⋊C4—22no—
201C22⋊C4—42no—
202C22⋊C4—84no—
203C42—22no—
204C42—42no—
205C42—84no—
206C42C4×D412yes—
207C16—21no—
208C16—21no—
209C16—22no—
210C16—42no—
211C23—42no—
212C23#57611yescharacteristic
213Q8—42no—
214Q8—44no—
215Q8—44no—
216Q8—42no—
217Q8—82no—
218Q8—82no—
219D4—42no—
220D4—42no—
221D4—82no—
222D4—82no—
223C2×C4—22no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4C2×C4×D411yescharacteristic
233C2×C4#53512yes—
234C2×C4#57612yes—
235C2×C4#57611yescharacteristic
236C8—22no—
237C8—42no—
238C8—82no—
239C8#51811yescharacteristic
240C8#51811yescharacteristic
241C8#54012yes—
242C22—41no—
243C22—82no—
244C22—162no—
245C22#294811yescharacteristic
246C22#322012yes—
247C4—42no—
248C4—82no—
249C4—82no—
250C4—84no—
251C4—162no—
252C4#232811yescharacteristic
253C4#294811yescharacteristic
254C4#322012yes—
255C2—21no—
256C2—22no—
257C2—162no—
258C2#2227611yescharacteristic
subgroupG/H
center, socleC2#22276
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#508C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22285C2semidirect
3. #13216C2semidirect
4. #13035C2semidirect
5. #13034C2semidirect
6. #13020C2semidirect
8. #12893C2semidirect
9. #12597C2semidirect
16. #2251C4semidirect
17. #2249C4semidirect
18. #2247C4semidirect

Maximal subgroups

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

subgroupindex
#337212
#222852
#132162
#130352
#130342
#130202
#130162
#128932
#125972