Finite Group Databasebeta

← #303024

Subgroups — #303024

Order 512.

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

normal subgroups only

This group has 827 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#59864C211yesmaximal, characteristic
2#29320C212yesmaximal, complemented
3#28514C211yesmaximal, characteristic
4#10577C212yesmaximal, complemented
5#10576C212yesmaximal, complemented
6#9869C212yesmaximal, complemented
7#8924C212yesmaximal
8#8920C212yesmaximal
9#8231C211yesmaximal, characteristic
10C2×#478—22no—
11#2859C2211yescharacteristic
12#2804—22no—
13#2580—24no—
14#2571C2212yes—
15#2563—22no—
16#2487—22no—
17#2366C2212yes—
18D4×C4⋊C4—22no—
19C4×C4⋊Q8C2212yes—
20C4×#127C2211yescharacteristic
21#2092—22no—
22#2083—22no—
23#2058—22no—
24#1957C2211yescharacteristic
25#1957C2211yescharacteristic
26#1948C2214yes—
27#1947C2214yes—
28#1944—24no—
29#1845C2211yescharacteristic
30#1844C2211yescharacteristic
31#1767C2212yes—
32C4×C4⋊C8C2211yescharacteristic
33C2×#331C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C2×C4⋊Q8D412yes—
38C2×#127C2311yescharacteristic
39C2×#125—21no—
40C2×C4×Q8—22no—
41C2×C4×D4—24no—
42C2×C4×D4—22no—
43C2×C4×D4D412yes—
44C22×C4⋊C4—22no—
45#478—48no—
46C2×#107—22no—
47C2×#107—24no—
48C2×#106—22no—
49C2×#106—24no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8C2314yes—
52C2×#102—22no—
53C2×#102—22no—
54C2×#97—22no—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#400—24no—
58#399—24no—
59#393—28no—
60#391—22no—
61#390—24no—
62#390—22no—
63#390D412yes—
64#389—24no—
65#389—22no—
66#389—22no—
67#389—22no—
68#389D414yes—
69#386—24no—
70#385—24no—
71#385—28no—
72#384D412yes—
73#384C2311yescharacteristic
74#384C2311yescharacteristic
75#384C2311yescharacteristic
76#382—24no—
77#382—21no—
78#380—22no—
79#380—24no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4C2314yes—
82C43C2311yescharacteristic
83#340—24no—
84#340—24no—
85#336—22no—
86#336—21no—
87#331Q818yescomplemented
88C22×Q8—22no—
89C22×D4—22no—
90C22×D4C2×D412yes—
91C23×C4—24no—
92C23×C4—22no—
93C22×C8—24no—
94C22×C8—21no—
95C22×C8—21no—
96C4⋊Q8—44no—
97#127—22no—
98#127C2×Q814yes—
99#125—44no—
100C4×Q8—82no—
101C4×Q8—82no—
102C4×D4—48no—
103C4×D4—44no—
104C4×D4—84no—
105C4×D4—84no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—84no—
119C2×C4⋊C4C2×D414yes—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D412yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C42—24no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2×D411yescharacteristic
131C2×C42C2411yescharacteristic
132#107—44no—
133#107—84no—
134#106—48no—
135#106—44no—
136C4⋊C8—24no—
137C4⋊C8—24no—
138C4⋊C8C2×Q818yes—
139#102—44no—
140#102—44no—
141#97—82no—
142C4×C8—22no—
143C4×C8—22no—
144#95—24no—
145#95—28no—
146#95—24no—
147#95—24no—
148#95—42no—
149C24—22no—
150C2×Q8—44no—
151C2×Q8—82no—
152C2×Q8—82no—
153C2×D4—22no—
154C2×D4—28no—
155C2×D4—42no—
156C2×D4—42no—
157C2×D4—44no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4C4⋊Q814yes—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—48no—
165C22×C4—44no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—84no—
169C22×C4—88no—
170C22×C4#12012yes—
171C22×C4#12011yescharacteristic
172C22×C4#12011yescharacteristic
173C22×C4C22×D411yescharacteristic
174C22×C4C22×D412yes—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—28no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—44no—
183C2×C8—44no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—84no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—84no—
201C4⋊C4—82no—
202C22⋊C4—48no—
203C22⋊C4—82no—
204C22⋊C4—84no—
205C22⋊C4—88no—
206C42—24no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—41no—
211C42—44no—
212C42—44no—
213C42C22×Q812yes—
214C422−1+412yes—
215C23—24no—
216C23—44no—
217C23—44no—
218C23—42no—
219C23—42no—
220C23C2×#12011yescharacteristic
221Q8—88no—
222D4—44no—
223D4—48no—
224D4—48no—
225D4—82no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—88no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—168no—
248C2×C4#45312yes—
249C2×C4C2×C4⋊Q811yescharacteristic
250C2×C4#53611yescharacteristic
251C2×C4D4×Q814yes—
252C2×C4#57514yes—
253C2×C4#57612yes—
254C2×C4#57712yes—
255C8—42no—
256C8—42no—
257C8—48no—
258C22—44no—
259C22—48no—
260C22—82no—
261C22—84no—
262C22—82no—
263C22#243411yescharacteristic
264C22#300512yes—
265C22#301212yes—
266C22C2×#45311yescharacteristic
267C22#302211yescharacteristic
268C4—22no—
269C4—21no—
270C4—21no—
271C4—42no—
272C4—41no—
273C4—82no—
274C4—84no—
275C4—84no—
276C4—161no—
277C4—161no—
278C4#338814yes—
279C4#338914yes—
280C2—84no—
281C2#1852611yescharacteristic
282C2#1868611yescharacteristic
283C2#1886412yes—
284C2#2582411yescharacteristic
285C2#2584512yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #29320C2semidirect
4. #10577C2semidirect
5. #10576C2semidirect
6. #9869C2semidirect
87. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#293202
#285142
#105772
#105762
#98692
#89242
#89202
#82312