Finite Group Databasebeta

← #303032

Subgroups — #303032

Order 512.

767 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 180 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#59924C211yesmaximal, characteristic
2#29320C212yesmaximal
3#28515C211yesmaximal, characteristic
4#10574C212yesmaximal
5#10573C212yesmaximal
6#9869C212yesmaximal
7#8929C212yesmaximal
8#8928C212yesmaximal
9#8232C211yesmaximal, characteristic
10C2×#479—22no—
11#2859C2211yescharacteristic
12#2811—22no—
13#2567—24no—
14#2567—22no—
15#2551—22no—
16#2551C2212yes—
17#2369—22no—
18#2369C2212yes—
19C4×C4⋊Q8C2212yes—
20C4×C4⋊Q8C2211yescharacteristic
21#2097—22no—
22#2083—22no—
23#2058—22no—
24#1956C2211yescharacteristic
25#1956C2211yescharacteristic
26#1948C2214yes—
27#1947C2214yes—
28#1944—24no—
29#1845C2211yescharacteristic
30#1844C2211yescharacteristic
31#1767C2212yes—
32C4×C4⋊C8C2211yescharacteristic
33C2×#333C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—24no—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8C2311yescharacteristic
41C2×#125—21no—
42C2×C4×Q8—22no—
43C2×C4×Q8D412yes—
44#479—48no—
45C2×#107—22no—
46C2×#107—24no—
47C2×#106—22no—
48C2×#106—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8C2314yes—
51C2×#103—22no—
52C2×#103—22no—
53C2×#97—22no—
54C2×C4×C8C2311yescharacteristic
55C2×C4×C8C2311yescharacteristic
56#395—24no—
57#395—28no—
58#395—24no—
59#391—28no—
60#391—24no—
61#391—22no—
62#391D412yes—
63#389—24no—
64#389—24no—
65#389—24no—
66#389—22no—
67#389—22no—
68#389—22no—
69#389—22no—
70#389D414yes—
71#384—24no—
72#384D412yes—
73#384C2311yescharacteristic
74#384C2311yescharacteristic
75#384C2311yescharacteristic
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—24no—
89C22×Q8—22no—
90C22×Q8C2×D412yes—
91C22×C8—24no—
92C22×C8—21no—
93C22×C8—21no—
94C4⋊Q8—28no—
95C4⋊Q8—22no—
96C4⋊Q8—84no—
97C4⋊Q8—84no—
98C4⋊Q8—84no—
99C4⋊Q8C2×Q814yes—
100#125—44no—
101C4×Q8—44no—
102C4×Q8—82no—
103C4×Q8—82no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D412yes—
122C2×C42—22no—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2411yescharacteristic
130#107—44no—
131#107—84no—
132#106—44no—
133#106—48no—
134C4⋊C8—24no—
135C4⋊C8—24no—
136C4⋊C8C2×Q818yes—
137#103—44no—
138#103—44no—
139#97—82no—
140C4×C8—22no—
141C4×C8—22no—
142#95—28no—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—42no—
148C2×Q8—28no—
149C2×Q8—22no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—82no—
153C2×Q8—82no—
154C2×Q8—88no—
155C2×Q8—84no—
156C2×Q8C4⋊Q814yes—
157C22×C4—22no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#12012yes—
164C22×C4#12011yescharacteristic
165C22×C4#12011yescharacteristic
166C22×C4C22×D411yescharacteristic
167C22×C4C22×D412yes—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—28no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—44no—
176C2×C8—44no—
177C4⋊C4—28no—
178C4⋊C4—28no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4—84no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—88no—
196C4⋊C4—84no—
197C4⋊C4—82no—
198C4⋊C4#13718yes—
199C42—24no—
200C42—22no—
201C42—22no—
202C42—22no—
203C42—41no—
204C42—44no—
205C42—44no—
206C42—44no—
207C42—84no—
208C42C22×Q812yes—
209C422−1+412yes—
210C23C2×#12011yescharacteristic
211Q8—44no—
212Q8—82no—
213Q8—88no—
214Q8—168no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—28no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4#45312yes—
239C2×C4C2×C4⋊Q811yescharacteristic
240C2×C4#53611yescharacteristic
241C2×C4D4×Q814yes—
242C2×C4C2×#13714yes—
243C2×C4#57612yes—
244C2×C4#57712yes—
245C8—42no—
246C8—42no—
247C8—48no—
248C22#243411yescharacteristic
249C22#300412yes—
250C22#301212yes—
251C22C2×#45311yescharacteristic
252C22#302211yescharacteristic
253C4—22no—
254C4—21no—
255C4—21no—
256C4—42no—
257C4—48no—
258C4—44no—
259C4—41no—
260C4—84no—
261C4—82no—
262C4—161no—
263C4—161no—
264C4#338514yes—
265C4#338714yes—
266C2#1854611yescharacteristic
267C2#1868611yescharacteristic
268C2#1884712yes—
269C2#2582411yescharacteristic
270C2#2584712yes—
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
#599242
#293202
#285152
#105742
#105732
#98692
#89292
#89282
#82322