Finite Group Databasebeta

← #302820

Subgroups — #302820

Order 512.

647 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 647 conjugacy classes of subgroups — too many to draw. Only its 156 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#46007C211yesmaximal, characteristic
2#30310C212yesmaximal
3#29227C211yesmaximal, characteristic
4#10586C212yesmaximal
5#10585C212yesmaximal
6#9895C212yesmaximal
7#8915C212yesmaximal
8#8913C212yesmaximal
9#8225C211yesmaximal, characteristic
10#2862C2212yes—
11#2859C2211yescharacteristic
12#2845C2212yes—
13#2806—22no—
14#2755—24no—
15#2546—22no—
16#2451—22no—
17#2446—22no—
18#2382C2212yes—
19C4×#125C2211yescharacteristic
20#2101—22no—
21#2085—24no—
22#2058—22no—
23#2057—22no—
24#1992C2211yescharacteristic
25#1992C2211yescharacteristic
26#1989—22no—
27#1847C2214yes—
28#1836C2211yescharacteristic
29#1836C2211yescharacteristic
30#1836C2214yes—
31#1767C2211yescharacteristic
32#1745C2212yes—
33C2×#330C2212yes—
34#1295C2212yes—
35#1295C2214yes—
36#1295C2214yes—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8D412yes—
39C2×C4⋊Q8D412yes—
40C2×#125—21no—
41C2×#125C2311yescharacteristic
42C2×C4×Q8—22no—
43C2×#107—24no—
44C2×#106—24no—
45C2×C4⋊C8—21no—
46C2×#97C2311yescharacteristic
47C2×#97C2311yescharacteristic
48C2×#97C2314yes—
49#400—24no—
50#398—24no—
51#398—24no—
52#398—24no—
53#398—24no—
54#395—24no—
55#391—22no—
56#391—24no—
57#391—22no—
58#391—22no—
59#391—22no—
60#391—22no—
61#389—22no—
62#389—24no—
63#389—24no—
64#389—22no—
65#389—22no—
66#389D412yes—
67#387—24no—
68#387—24no—
69#387—24no—
70#387—22no—
71#384C2311yescharacteristic
72#384C2311yescharacteristic
73#382—21no—
74#382D414yes—
75#382C2311yescharacteristic
76#382C2314yes—
77C4×C4⋊C4—22no—
78C4×C4⋊C4D412yes—
79C43C2311yescharacteristic
80#340—24no—
81#340—24no—
82#340—22no—
83#340—22no—
84#340—22no—
85#340—22no—
86#336—22no—
87#336—22no—
88#336—21no—
89#330Q818yescomplemented
90C22×Q8—22no—
91C22×Q8—22no—
92C22×C8—24no—
93C22×C8—21no—
94C22×C8—21no—
95C4⋊Q8—82no—
96C4⋊Q8—82no—
97C4⋊Q8—84no—
98#125—22no—
99#125—44no—
100#125C2×Q814yes—
101C4×Q8—84no—
102C2×C4⋊C4—22no—
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—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2×D412yes—
117C2×C4⋊C4C2×D414yes—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D412yes—
120C2×C42—22no—
121C2×C42—24no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D412yes—
126C2×C42C2411yescharacteristic
127#107—84no—
128#106—84no—
129C4⋊C8—42no—
130#97—24no—
131#97—22no—
132#97—22no—
133#97C2×Q818yes—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95—24no—
140#95—24no—
141#95—22no—
142#95—24no—
143#95—24no—
144#95—42no—
145C2×Q8—44no—
146C2×Q8—44no—
147C2×Q8—44no—
148C2×Q8—44no—
149C2×Q8—82no—
150C2×Q8—82no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4#12011yescharacteristic
158C22×C4#12012yes—
159C22×C4#12011yescharacteristic
160C22×C4C22×D411yescharacteristic
161C22×C4C22×D412yes—
162C2×C8—28no—
163C2×C8—44no—
164C2×C8—44no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C4⋊C4—22no—
172C4⋊C4—28no—
173C4⋊C4—44no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C4⋊C4—44no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4—82no—
184C4⋊C4—84no—
185C4⋊C4—84no—
186C4⋊C4—84no—
187C4⋊C4—84no—
188C4⋊C4—82no—
189C4⋊C4C4⋊Q814yes—
190C42—22no—
191C42—42no—
192C42—41no—
193C42—42no—
194C42—44no—
195C42—84no—
196C42C22×Q812yes—
197C422−1+412yes—
198C23C2×#12011yescharacteristic
199Q8—88no—
200Q8—164no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—44no—
215C2×C4—42no—
216C2×C4—44no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4#45412yes—
228C2×C4C2×C4⋊Q811yescharacteristic
229C2×C4#53611yescharacteristic
230C2×C4D4×Q814yes—
231C8—48no—
232C8—81no—
233C8—81no—
234C22#243411yescharacteristic
235C22C2×#45411yescharacteristic
236C22#301912yes—
237C22#302012yes—
238C22#302211yescharacteristic
239C4—22no—
240C4—21no—
241C4—21no—
242C4—24no—
243C4—41no—
244C4—42no—
245C4—44no—
246C4—84no—
247C4—82no—
248C4—84no—
249C4—161no—
250C4—161no—
251C2#1900111yescharacteristic
252C2#1906311yescharacteristic
253C2#1909112yes—
254C2#2582611yescharacteristic
255C2#2583212yes—
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
89. #330Q8semidirect

Maximal subgroups

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

subgroupindex
#460072
#303102
#292272
#105862
#105852
#98952
#89152
#89132
#82252