Finite Group Databasebeta

← #491664

Subgroups — #491664

Order 512.

565 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 565 conjugacy classes of subgroups — too many to draw. Only its 146 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#33794C212yesmaximal, complemented
2#33771C212yesmaximal, complemented
3#19667C211yesmaximal, characteristic, complemented
4#19652C211yesmaximal, characteristic, complemented
5#12536C212yesmaximal, complemented
6#12525C212yesmaximal, complemented
7#12506C212yesmaximal, complemented
8#12503C212yesmaximal, complemented
9#11412C211yesmaximal, characteristic, complemented
10#3208—22no—
11#3207—22no—
12#3169C2212yes—
13#3160C2211yescharacteristic
14C2×#493C2211yescharacteristic
15#2254C2212yescomplemented
16#2252C2212yescomplemented
17#2246C2212yes—
18#2246C2212yes—
19#2240C2212yes—
20#2240C2212yes—
21#2240C2212yescomplemented
22#2240C2212yescomplemented
23#2234—24no—
24#2228—24no—
25#2206—24no—
26#2184—24no—
27#2162C2212yes—
28#1717—22no—
29#1715—22no—
30#1712C2211yescharacteristic, complemented
31#1710C2211yescharacteristic, complemented
32#1706C2212yes—
33#1706C2212yes—
34#1706C2212yescomplemented
35#1450C2211yescharacteristic
36#1312C2212yescomplemented
37#1311C2211yescharacteristic, complemented
38C8×C16C2211yescharacteristic, complemented
39C22×D8—22no—
40#533—21no—
41#532—22no—
42C2×#127—21no—
43C2×D16—24no—
44C2×D16—24no—
45#500C2312yes—
46#500C2312yes—
47#499—22no—
48#498—21no—
49#493—21no—
50#493—22no—
51#493—22no—
52#493C2312yes—
53#493C2312yes—
54#492—22no—
55#492—22no—
56#492—22no—
57#487—22no—
58#486—22no—
59#486—22no—
60#486—22no—
61#486—22no—
62#466—22no—
63#466—22no—
64#463—44no—
65#460—44no—
66#450—44no—
67#449—44no—
68C4×D8—22no—
69C4×D8—22no—
70#431D412yes—
71#427—22no—
72C2×C4×C8C2311yescharacteristic
73#367—24no—
74#366—24no—
75C4⋊C16D414yes—
76#357—24no—
77#357—24no—
78#357—24no—
79#357—24no—
80C22⋊C16D414yes—
81C4×C16C2312yes—
82C4×C16C2312yes—
83#322D412yes—
84C82C2311yescharacteristic
85C82C2311yescharacteristic
86C2×#41—42no—
87C22×D4—41no—
88C22×D4—42no—
89C2×SD16—44no—
90C2×SD16—44no—
91C2×SD16—44no—
92C2×D8—22no—
93C2×D8—22no—
94C2×D8—22no—
95C2×D8—22no—
96C2×D8—42no—
97C2×D8—44no—
98C2×D8—84no—
99C22×C8C2×D412yes—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102#127—22no—
103#127—22no—
104#127—41no—
105#127—42no—
106#125—41no—
107#124—42no—
108C22⋊Q8—42no—
109#121—42no—
110#121—42no—
111C4×Q8—42no—
112C4×D4—42no—
113C4×D4—42no—
114#117—41no—
115C2×C42C2×D411yescharacteristic
116D16—84no—
117D16—84no—
118C2×C16C2×D414yes—
119C2×C16C2×D414yes—
120#107—22no—
121#107—22no—
122#107—22no—
123#107—22no—
124#106—42no—
125C4⋊C8D814yes—
126#103—44no—
127#102—42no—
128#102—42no—
129#102—44no—
130#102—44no—
131#102—44no—
132C22⋊C8D814yes—
133C4×C8C2×D412yes—
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D412yes—
136C4×C8C2×D412yes—
137C4×C8C2411yescharacteristic
138C24—81no—
139#41—164no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—84no—
149C2×D4—82no—
150C2×D4—81no—
151C2×D4—82no—
152C2×D4—164no—
153C22×C4—82no—
154C22×C4#12711yescharacteristic
155C22×C4C2×D812yes—
156SD16—164no—
157SD16—164no—
158SD16—164no—
159D8—44no—
160D8—44no—
161D8—44no—
162D8—44no—
163D8—84no—
164D8—164no—
165C2×C8—24no—
166C2×C8#12712yes—
167C2×C8C2×D814yes—
168C2×C8C2×D814yes—
169C2×C8C22×D412yes—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—82no—
175C22⋊C4—81no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C42—81no—
179C42#12711yescharacteristic
180C42C2×D812yes—
181C42C22×D411yescharacteristic
182C16—24no—
183C16—24no—
184C23—82no—
185C23—82no—
186C23—162no—
187C23—164no—
188C23#49311yescharacteristic
189Q8—84no—
190Q8—84no—
191D4—82no—
192D4—82no—
193D4—82no—
194D4—82no—
195D4—84no—
196D4—84no—
197D4—84no—
198D4—84no—
199D4—162no—
200D4—164no—
201D4—164no—
202C2×C4—22no—
203C2×C4—24no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—164no—
207C2×C4#49311yescharacteristic
208C2×C4#49312yes—
209C2×C4C2×#12711yescharacteristic
210C2×C4C22×D812yes—
211C8—22no—
212C8—24no—
213C8—24no—
214C8#57714yes—
215C22—22no—
216C22—21no—
217C22—162no—
218C22—162no—
219C22—162no—
220C22—162no—
221C22—162no—
222C22—162no—
223C22—324no—
224C22C2×#49311yescharacteristic
225C4—22no—
226C4—21no—
227C4—162no—
228C4—162no—
229C4#315412yes—
230C4#341514yes—
231C2—41no—
232C2—322no—
233C2—322no—
234C2#1966011yescharacteristic
235C2#3372012yes—
subgroupG/H
center, socleC22C2×#493
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33794C2semidirect
2. #33771C2semidirect
3. #19667C2semidirect
4. #19652C2semidirect
5. #12536C2semidirect
6. #12525C2semidirect
7. #12506C2semidirect
8. #12503C2semidirect
9. #11412C2semidirect
15. #2254C22semidirect
16. #2252C22semidirect
21. #2240C22semidirect
22. #2240C22semidirect
30. #1712C22semidirect
31. #1710C22semidirect
34. #1706C22semidirect
36. #1312C22semidirect
37. #1311C22semidirect
38. C8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#337942
#337712
#196672
#196522
#125362
#125252
#125062
#125032
#114122