Finite Group Databasebeta

← #244358

Subgroups — #244358

Order 512.

806 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 162 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#22269C211yesmaximal, characteristic
2#22268C212yesmaximal, complemented
3#22247C211yesmaximal, characteristic, complemented
4#21646C212yesmaximal, complemented
5#10682C212yesmaximal, complemented
6#10678C212yesmaximal, complemented
7#10308C212yesmaximal, complemented
8#10305C212yesmaximal
9#8134C211yesmaximal, characteristic, complemented
10C2×#500C2211yescharacteristic
11C2×#493C2211yescharacteristic
12C2×#492C2212yes—
13C2×#486C2212yes—
14C4×C4⋊Q8C2212yes—
15C4×#127C2212yes—
16#1990C2212yes—
17#1990C2212yes—
18#1957C2212yes—
19#1957C2212yes—
20#1957C2212yes—
21#1956C2212yes—
22#1936—24no—
23#1918—24no—
24#1880—24no—
25#1879—24no—
26#1877—24no—
27#1834C2212yes—
28#1834C2211yescharacteristic
29#1834C2212yes—
30#1834C2211yescharacteristic
31#1762—24no—
32#1744C2211yescharacteristic
33#1744C2211yescharacteristic
34C4×#97C2211yescharacteristic
35#1273C2212yes—
36#1273C2212yes—
37C22×SD16—24no—
38C22×D8—22no—
39C2×C4⋊Q8C2312yes—
40C2×#127C2312yes—
41C2×C4×Q8—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45#500—24no—
46#493—24no—
47#492—24no—
48#492—24no—
49#486—24no—
50#486—24no—
51C2×#107—22no—
52C2×#103—24no—
53C2×#102—24no—
54C2×#102—24no—
55C2×#102—24no—
56C2×#102—24no—
57C2×#97D414yes—
58C2×C4×C8C2312yes—
59C2×C4×C8C2312yes—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62#391—22no—
63#390—22no—
64#390—22no—
65#390—22no—
66#389—22no—
67#389—22no—
68#384C2312yes—
69#384C2312yes—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C43C2311yescharacteristic
73#340—24no—
74#340—22no—
75#340—22no—
76#340—24no—
77#340—24no—
78#340—24no—
79#336D412yes—
80#336D414yes—
81#336D412yes—
82C22×Q8—22no—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4—22no—
86C23×C4—42no—
87C2×SD16—216no—
88C2×SD16—84no—
89C2×SD16—84no—
90C2×D8—28no—
91C2×D8—84no—
92C22×C8C2×D412yes—
93C22×C8C2×D414yes—
94C22×C8C2×D412yes—
95C22×C8C2×D414yes—
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98C4⋊Q8—22no—
99C4⋊Q8#4114yes—
100#127—22no—
101#127—22no—
102#127—22no—
103#127#4114yes—
104C4×Q8—84no—
105C4×D4—48no—
106C4×D4—84no—
107C4×D4—84no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—42no—
117C2×C22⋊C4—42no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4—42no—
120C2×C42—42no—
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2411yescharacteristic
126#107—28no—
127#103—84no—
128#102—216no—
129#102—48no—
130#102—84no—
131#102—84no—
132#97—44no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—22no—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8—22no—
139#95—42no—
140#95—42no—
141C24—42no—
142C2×Q8—24no—
143C2×Q8—42no—
144C2×Q8—44no—
145C2×Q8—44no—
146C2×D4—24no—
147C2×D4—24no—
148C2×D4—24no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—44no—
157C2×D4—44no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—84no—
162C22×C4—84no—
163C22×C4—84no—
164C22×C4#12712yes—
165C22×C4#12712yes—
166C22×C4C22×D411yescharacteristic
167C22×C4C22×D411yescharacteristic
168C22×C4C22×D411yescharacteristic
169SD16—816no—
170D8—88no—
171C2×C8—24no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—24no—
175C2×C8—24no—
176C2×C8—24no—
177C2×C8—24no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—24no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—84no—
192C22⋊C4—84no—
193C22⋊C4—84no—
194C22⋊C4—84no—
195C42—24no—
196C42—22no—
197C42—22no—
198C42—24no—
199C42—84no—
200C42C2×#4112yes—
201C42C2×#4112yes—
202C23—44no—
203C23—82no—
204C23—82no—
205C23—82no—
206C23—82no—
207C23—82no—
208C23—82no—
209C23C2×#12711yescharacteristic
210Q8—48no—
211Q8—84no—
212D4—48no—
213D4—48no—
214D4—48no—
215D4—84no—
216D4—84no—
217D4—84no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—44no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—168no—
234C2×C4#52512yes—
235C2×C4#52512yes—
236C2×C4#52511yescharacteristic
237C2×C4#52511yescharacteristic
238C2×C4#57514yes—
239C2×C4#57612yes—
240C2×C4#57612yes—
241C2×C4#57714yes—
242C8—44no—
243C8—44no—
244C8—42no—
245C8—42no—
246C22—84no—
247C22—84no—
248C22—84no—
249C22—162no—
250C22—162no—
251C22—162no—
252C22—162no—
253C22#239711yescharacteristic
254C22#315312yes—
255C22#315412yes—
256C22#315511yescharacteristic
257C22#315511yescharacteristic
258C4—42no—
259C4—42no—
260C4—84no—
261C4—162no—
262C4—162no—
263C4#311114yes—
264C4#312214yes—
265C4#312312yes—
266C4#312612yes—
267C2—164no—
268C2#1971512yes—
269C2#1972212yes—
270C2#2380411yescharacteristic
271C2#2381011yescharacteristic
272C2#2381111yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22268C2semidirect
3. #22247C2semidirect
4. #21646C2semidirect
5. #10682C2semidirect
6. #10678C2semidirect
7. #10308C2semidirect
9. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#222692
#222682
#222472
#216462
#106822
#106782
#103082
#103052
#81342