Finite Group Databasebeta

← #512985

Subgroups — #512985

Order 512.

363 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 130 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#33608C211yesmaximal, characteristic
2#33410C211yesmaximal, characteristic
3#13716C212yesmaximal, complemented
4#13716C212yesmaximal, complemented
5#13715C212yesmaximal, complemented
6#13714C212yesmaximal, complemented
7#13634C211yesmaximal, characteristic, complemented
8#13633C211yesmaximal, characteristic, complemented
9#13618C211yesmaximal, characteristic, complemented
10C2×#1416C211yesmaximal, characteristic, complemented
11#13554C211yesmaximal, characteristic
12C22×Q32C2211yescharacteristic
13C2×#504C2211yescharacteristic
14#3059—21no—
15#2913—21no—
16C2×M64C2211yescharacteristic
17C22×C32C2211yescharacteristic
18#2219—22no—
19#2216C2212yes—
20#2216C2212yes—
21#2159—21no—
22#2159—22no—
23#2158C2211yescharacteristic
24#2158C2211yescharacteristic
25#2158C2212yes—
26C2×#366C2211yescharacteristic
27C2×#366C2212yes—
28#2141—21no—
29#1424C2211yescharacteristic
30#1424C2211yescharacteristic
31#1424C2212yes—
32#1423C2211yescharacteristic
33#1423C2211yescharacteristic
34#1423C2212yes—
35#1416C2212yes—
36#1416C2212yes—
37#1416C2212yes—
38#1416C2212yes—
39C22⋊C32C2212yes—
40C22⋊C32C2212yes—
41C22×Q16—21no—
42C2×#131D411yescharacteristic
43#536—41no—
44#520—41no—
45C2×Q32—21no—
46C2×Q32—22no—
47C2×Q32—22no—
48C2×Q32C2312yes—
49#504—22no—
50#504—22no—
51C2×M32D411yescharacteristic
52C2×M32D412yes—
53C22×C16C2312yes—
54C22×C16C2311yescharacteristic
55#470—42no—
56#467—22no—
57#467—22no—
58#428—41no—
59#428—42no—
60#427—21no—
61#427—22no—
62#427—21no—
63C2×#107—22no—
64C2×#107—21no—
65#419—41no—
66M64—22no—
67C2×C32—22no—
68C2×C32C2311yescharacteristic
69C2×C32C2312yes—
70C2×C32C2311yescharacteristic
71#367—22no—
72#367—21no—
73#367—21no—
74#366—22no—
75#366—21no—
76#366C2312yes—
77#366C2311yescharacteristic
78#366C2311yescharacteristic
79#366C2311yescharacteristic
80#366C2311yescharacteristic
81#358—22no—
82#358—22no—
83C2×#41D811yescharacteristic
84C22×Q8—41no—
85C2×Q16—22no—
86C2×Q16—42no—
87C2×Q16—41no—
88C2×Q16—42no—
89#131—22no—
90#131—22no—
91C2×M16D811yescharacteristic
92C2×M16D812yes—
93C22×C8C2×D412yes—
94C22×C8C2×D411yescharacteristic
95C4⋊Q8—81no—
96#124—81no—
97#123—82no—
98C22⋊Q8—42no—
99C22⋊Q8—42no—
100C4×Q8—81no—
101C4×D4—82no—
102C4×D4—81no—
103#117—42no—
104#117—41no—
105#117—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108Q32—22no—
109Q32—22no—
110Q32—42no—
111M32—22no—
112M32—24no—
113C2×C16—22no—
114C2×C16—22no—
115C2×C16C2×D412yes—
116C2×C16C2×D411yescharacteristic
117C2×C16#4112yes—
118C2×C16#4112yes—
119C2×C16C2411yescharacteristic
120#107—21no—
121#107—21no—
122#107—21no—
123#107—22no—
124#107—21no—
125#107—42no—
126#107—41no—
127#106—42no—
128#106—41no—
129#106—41no—
130#103—42no—
131#103—42no—
132C32—21no—
133C32—21no—
134C32—22no—
135#41—22no—
136#41—22no—
137C2×Q8—42no—
138C2×Q8—81no—
139C2×Q8—82no—
140C2×Q8—82no—
141C2×Q8D1611yescharacteristic
142C2×D4D1612yes—
143C2×D4D1611yescharacteristic
144C22×C4—81no—
145C22×C4—82no—
146C22×C4—81no—
147C22×C4C2×D812yes—
148C22×C4C2×D811yescharacteristic
149Q16—42no—
150Q16—42no—
151Q16—82no—
152M16—22no—
153M16—24no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8#12312yes—
157C2×C8C2×D812yes—
158C2×C8C2×D811yescharacteristic
159C2×C8#13712yes—
160C2×C8C22×D411yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—82no—
167C4⋊C4—81no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C22⋊C4—82no—
171C22⋊C4—81no—
172C22⋊C4—81no—
173C22⋊C4—82no—
174C22⋊C4—82no—
175C42—82no—
176C42—81no—
177C42—81no—
178C16—22no—
179C16#12312yes—
180C16C2×#4111yescharacteristic
181C16C2×#4111yescharacteristic
182C23C2×D1612yes—
183C23C2×D1611yescharacteristic
184Q8—22no—
185Q8—82no—
186Q8—82no—
187Q8—162no—
188D4—24no—
189D4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—162no—
199C2×C4—321no—
200C2×C4—322no—
201C2×C4#48012yes—
202C2×C4C2×D1612yes—
203C2×C4C2×D1611yescharacteristic
204C2×C4#51012yes—
205C2×C4C22×D811yescharacteristic
206C8—22no—
207C8#48012yes—
208C8C2×#12311yescharacteristic
209C8C2×#13711yescharacteristic
210C22—21no—
211C22—21no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22#223212yes—
216C22C22×D1611yescharacteristic
217C4—22no—
218C4—162no—
219C4—161no—
220C4—162no—
221C4—161no—
222C4—161no—
223C4—161no—
224C4#223212yes—
225C4C2×#48011yescharacteristic
226C4C2×#51011yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#223211yescharacteristic
230C2#3398311yescharacteristic
231C2#3398411yescharacteristic
subgroupG/H
center, socleC22C22×D16
commutator, Frattini, cosocleC2×C16C24
prime radicalC2×#41D8
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G
LECSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13716C2semidirect
4. #13716C2semidirect
5. #13715C2semidirect
6. #13714C2semidirect
7. #13634C2semidirect
8. #13633C2semidirect
9. #13618C2semidirect
10. C2×#1416C2semidirect

Maximal subgroups

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

subgroupindex
#336082
#334102
#137162
#137152
#137142
#136342
#136332
#136182
C2×#14162
#135542