Finite Group Databasebeta

← #492278

Subgroups — #492278

Order 512.

474 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 114 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#33892C212yesmaximal, complemented
2#19898C212yesmaximal, complemented
3#12503C211yesmaximal, characteristic, complemented
4#12485C212yesmaximal, complemented
5#12472C212yesmaximal, complemented
6#12399C212yesmaximal, complemented
7#12316C212yesmaximal, complemented
8#12081C211yesmaximal, characteristic, complemented
9#12039C211yesmaximal, characteristic, complemented
10#3341C2212yes—
11#3325—22no—
12D4×D8C2212yescomplemented
13#2252C2211yescharacteristic, complemented
14#2246C2211yescharacteristic
15#2240C2212yescomplemented
16#2234—22no—
17#2226C2212yescomplemented
18#2218—22no—
19#2206C2212yes—
20#2184—22no—
21C4×D16C2211yescharacteristic
22#2169C2212yescomplemented
23#2140—22no—
24#1706C2212yescomplemented
25#1697—22no—
26#1675C2212yescomplemented
27#1667C2212yes—
28#1629—22no—
29#1624—22no—
30#1589C2212yescomplemented
31C8×D8C2211yescharacteristic
32#1371C2211yescharacteristic, complemented
33#1370C2211yescharacteristic, complemented
34#1365C2212yescomplemented
35#1361C2212yescomplemented
36#1331C2212yescomplemented
37D4⋊C16C2212yescomplemented
38C8×C16C2211yescharacteristic, complemented
39C22×D8—22no—
40#555—22no—
41D42—22no—
42C2×D16—22no—
43C2×D16—41no—
44C2×D16D411yescharacteristic
45C2×M32—22no—
46#500C2312yes—
47#493C2312yes—
48#492—22no—
49#486—22no—
50#482—42no—
51#482—42no—
52#476—22no—
53#476—22no—
54#469—42no—
55#466—22no—
56#466—22no—
57#459—22no—
58#459—22no—
59#447—42no—
60#447—42no—
61C4×D8D411yescharacteristic
62C4×D8C2312yes—
63#436—22no—
64C8×D4C2312yes—
65#427—22no—
66#418—42no—
67#367—22no—
68#366—41no—
69#366D411yescharacteristic
70C4⋊C16C2312yes—
71#357—22no—
72#357—22no—
73#357—22no—
74#357—22no—
75#357D412yes—
76C22⋊C16—22no—
77C4×C16C2311yescharacteristic
78C4×C16C2311yescharacteristic
79#335D411yescharacteristic
80#335C2312yes—
81#333—22no—
82#331—22no—
83D4⋊C8—22no—
84D4⋊C8D412yes—
85C82C2311yescharacteristic
86C22×D4—42no—
87C22×D4—42no—
88C2×SD16—82no—
89C2×D8—22no—
90C2×D8—22no—
91C2×D8—42no—
92C2×D8—44no—
93C2×D8—42no—
94C2×D8—82no—
95C2×D8D811yescharacteristic
96C2×D8C2×D412yes—
97C2×M16—42no—
98C22×C8—22no—
99C4⋊Q8—22no—
100#127—22no—
101#124—82no—
102#121—42no—
103#121—42no—
104#121—42no—
105#121—42no—
106#120—82no—
107C4×D4—22no—
108C4×D4C2×D412yes—
109#117—42no—
110#117—42no—
111D16—41no—
112D16—82no—
113D16—82no—
114M32—82no—
115C2×C16—21no—
116C2×C16—22no—
117C2×C16C2×D411yescharacteristic
118C2×C16C2×D411yescharacteristic
119#107—22no—
120#107—22no—
121#107D811yescharacteristic
122#107C2×D412yes—
123#106—42no—
124#106—42no—
125C4⋊C8—22no—
126C4⋊C8C2×D412yes—
127#102—22no—
128#102—42no—
129#102—42no—
130#102—42no—
131#102—42no—
132#102—42no—
133#102—42no—
134#102D812yes—
135C22⋊C8—22no—
136C22⋊C8—42no—
137C4×C8C2×D411yescharacteristic
138C4×C8C2×D411yescharacteristic
139C4×C8C2411yescharacteristic
140C24—82no—
141C2×Q8—42no—
142C2×D4—22no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—82no—
147C2×D4—84no—
148C2×D4—82no—
149C2×D4—84no—
150C2×D4—82no—
151C2×D4—162no—
152C2×D4—162no—
153C2×D4C2×D812yes—
154C22×C4—22no—
155C22×C4—42no—
156SD16—164no—
157D8—22no—
158D8—22no—
159D8—41no—
160D8—42no—
161D8—42no—
162D8—82no—
163D8—84no—
164D8—162no—
165M16—162no—
166C2×C8—21no—
167C2×C8—22no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8C2×D811yescharacteristic
172C2×C8C2×D811yescharacteristic
173C2×C8C22×D411yescharacteristic
174C2×C8H(5,𝔽2)11yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4C2×D812yes—
179C22⋊C4—22no—
180C22⋊C4—42no—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C42—82no—
185C42C22×D411yescharacteristic
186C16—22no—
187C16—21no—
188C16—21no—
189C16—42no—
190C23—22no—
191C23—42no—
192C23—82no—
193C23—164no—
194C23—164no—
195C23—322no—
196Q8—162no—
197D4—22no—
198D4—22no—
199D4—42no—
200D4—42no—
201D4—82no—
202D4—82no—
203D4—82no—
204D4—82no—
205D4—162no—
206D4—164no—
207D4—164no—
208D4—322no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4D4211yescharacteristic
217C2×C4C22×D811yescharacteristic
218C2×C4#57611yescharacteristic
219C8—22no—
220C8—21no—
221C8—21no—
222C8—21no—
223C8—42no—
224C8—82no—
225C8#57511yescharacteristic
226C8#57611yescharacteristic
227C22—42no—
228C22—42no—
229C22—42no—
230C22—82no—
231C22—82no—
232C22—82no—
233C22—162no—
234C22—162no—
235C22—162no—
236C22—328no—
237C22D4×D811yescharacteristic
238C4—21no—
239C4—42no—
240C4—82no—
241C4—162no—
242C4#327311yescharacteristic
243C4#329111yescharacteristic
244C4#341411yescharacteristic
245C4#341511yescharacteristic
246C2—82no—
247C2—162no—
248C2—322no—
249C2#1988411yescharacteristic
250C2#3383911yescharacteristic
251C2#3385811yescharacteristic
subgroupG/H
center, socleC22D4×D8
commutator, Frattini, cosocleC4×C8C24
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33892C2semidirect
2. #19898C2semidirect
3. #12503C2semidirect
4. #12485C2semidirect
5. #12472C2semidirect
6. #12399C2semidirect
7. #12316C2semidirect
8. #12081C2semidirect
9. #12039C2semidirect
12. D4×D8C22semidirect
13. #2252C22semidirect
15. #2240C22semidirect
17. #2226C22semidirect
22. #2169C22semidirect
24. #1706C22semidirect
26. #1675C22semidirect
30. #1589C22semidirect
32. #1371C22semidirect
33. #1370C22semidirect
34. #1365C22semidirect
35. #1361C22semidirect
36. #1331C22semidirect
37. D4⋊C16C22semidirect
38. C8×C16C22semidirect

Maximal subgroups

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

subgroupindex
#338922
#198982
#125032
#124852
#124722
#123992
#123162
#120812
#120392