Finite Group Databasebeta

← #303039

Subgroups — #303039

Order 512.

723 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 723 conjugacy classes of subgroups — too many to draw. Only its 172 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#59869C211yesmaximal, characteristic
2#29319C212yesmaximal
3#28515C211yesmaximal, characteristic
4#10577C212yesmaximal
5#10576C212yesmaximal
6#9868C212yesmaximal
7#8929C212yesmaximal
8#8928C212yesmaximal
9#8232C211yesmaximal, characteristic
10C2×#479—22no—
11#2859C2211yescharacteristic
12#2809—22no—
13#2571C2212yes—
14#2568—22no—
15#2546—22no—
16#2508—24no—
17#2369C2212yes—
18#2355—22no—
19C4×C4⋊Q8C2212yes—
20C4×C4⋊Q8C2211yescharacteristic
21#2097—22no—
22#2084—22no—
23#2057—22no—
24#1956C2211yescharacteristic
25#1956C2211yescharacteristic
26#1948C2214yes—
27#1947C2214yes—
28#1943—24no—
29#1845C2211yescharacteristic
30#1844C2211yescharacteristic
31#1767C2212yes—
32C4×C4⋊C8C2211yescharacteristic
33C2×#333C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8C2311yescharacteristic
40C2×#125—21no—
41C2×C4×Q8—22no—
42C2×C4×Q8D412yes—
43#479—48no—
44C2×#107—22no—
45C2×#107—24no—
46C2×#106—22no—
47C2×#106—24no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8C2314yes—
50C2×#103—22no—
51C2×#103—22no—
52C2×#97—22no—
53C2×C4×C8C2311yescharacteristic
54C2×C4×C8C2311yescharacteristic
55#400—24no—
56#400—24no—
57#398—28no—
58#391—24no—
59#391—24no—
60#391—22no—
61#391D412yes—
62#389—24no—
63#389—24no—
64#389—22no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389D414yes—
69#387—28no—
70#384D412yes—
71#384C2311yescharacteristic
72#384C2311yescharacteristic
73#384C2311yescharacteristic
74#382—24no—
75#382—24no—
76#382—21no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—24no—
79C4×C4⋊C4—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2314yes—
82C43C2311yescharacteristic
83#340—24no—
84#340—24no—
85#336—22no—
86#336—21no—
87#333Q818yescomplemented
88C22×Q8—22no—
89C22×Q8C2×D412yes—
90C22×C8—24no—
91C22×C8—21no—
92C22×C8—21no—
93C4⋊Q8—22no—
94C4⋊Q8—44no—
95C4⋊Q8—84no—
96C4⋊Q8C2×Q814yes—
97#125—44no—
98C4×Q8—48no—
99C4×Q8—44no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4C2×D414yes—
116C2×C4⋊C4C2×D414yes—
117C2×C4⋊C4C2×D412yes—
118C2×C42—22no—
119C2×C42—24no—
120C2×C42—24no—
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2411yescharacteristic
126#107—48no—
127#107—44no—
128#106—44no—
129#106—84no—
130C4⋊C8—24no—
131C4⋊C8—24no—
132C4⋊C8C2×Q818yes—
133#103—44no—
134#103—44no—
135#97—82no—
136C4×C8—22no—
137C4×C8—22no—
138#95—24no—
139#95—28no—
140#95—28no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—42no—
145C2×Q8—22no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×Q8—44no—
151C2×Q8C4⋊Q814yes—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—24no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4#12011yescharacteristic
159C22×C4#12012yes—
160C22×C4#12011yescharacteristic
161C22×C4C22×D411yescharacteristic
162C22×C4C22×D412yes—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—28no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—44no—
171C2×C8—44no—
172C4⋊C4—28no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—48no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—82no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4—82no—
194C42—24no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—41no—
199C42—48no—
200C42—44no—
201C42—84no—
202C42C22×Q812yes—
203C422−1+412yes—
204C23C2×#12011yescharacteristic
205Q8—44no—
206Q8—82no—
207Q8—88no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4#45312yes—
232C2×C4C2×C4⋊Q811yescharacteristic
233C2×C4#53611yescharacteristic
234C2×C4D4×Q814yes—
235C2×C4#57514yes—
236C2×C4#57712yes—
237C2×C4#57812yes—
238C8—42no—
239C8—42no—
240C8—48no—
241C22#243411yescharacteristic
242C22#300512yes—
243C22#301312yes—
244C22C2×#45311yescharacteristic
245C22#302211yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—41no—
250C4—42no—
251C4—44no—
252C4—82no—
253C4—84no—
254C4—84no—
255C4—161no—
256C4—161no—
257C4#338814yes—
258C4#339014yes—
259C2#1854711yescharacteristic
260C2#1868811yescharacteristic
261C2#1884912yes—
262C2#2582411yescharacteristic
263C2#2584612yes—
subgroupG/H
center, socle, prime radicalC23C2×#120
commutator, Frattini, cosocleC2×C42C24
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
87. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#598692
#293192
#285152
#105772
#105762
#98682
#89292
#89282
#82322