Finite Group Databasebeta

← #302869

Subgroups — #302869

Order 512.

817 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 208 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#28687C212yesmaximal
3#28512C211yesmaximal, characteristic
4#10574C214yesmaximal
5#9864C212yesmaximal
6#8928C214yesmaximal
7C4×#333C211yesmaximal, characteristic
8C2×#498—22no—
9C2×#475—22no—
10#2868C2211yescharacteristic
11#2811—22no—
12#2567—24no—
13#2567—22no—
14#2551—22no—
15#2551C2212yes—
16#2369—22no—
17#2369C2212yes—
18C4×C4⋊Q8C2212yes—
19C4×C4⋊Q8C2211yescharacteristic
20#2096—22no—
21#2084—24no—
22#1990—22no—
23#1956C2212yes—
24#1948C2218yes—
25#1844C2212yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#333C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8D412yes—
36C2×C4⋊Q8C2311yescharacteristic
37C2×C4×Q8—22no—
38C2×C4×Q8D412yes—
39#498—48no—
40#475—48no—
41C2×#106—24no—
42C2×#106—28no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8C2314yes—
45C2×#103—24no—
46C2×C4×C8—22no—
47C2×C4×C8C2312yes—
48#395—24no—
49#395—28no—
50#395—24no—
51#391—28no—
52#391—24no—
53#391—22no—
54#391D412yes—
55#389—24no—
56#389—24no—
57#389—24no—
58#389—22no—
59#389—22no—
60#389D414yes—
61#384—24no—
62#384—22no—
63#384—21no—
64#384D412yes—
65#384C2312yes—
66#384C2311yescharacteristic
67C4×C4⋊C4—22no—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4C2314yes—
72C43C2311yescharacteristic
73#340—28no—
74#336—22no—
75#336—21no—
76#333Q818yescomplemented
77C22×Q8—24no—
78C22×Q8—22no—
79C22×Q8—41no—
80C22×Q8C2×D412yes—
81C22×C8—24no—
82C22×C8—22no—
83C4⋊Q8—24no—
84C4⋊Q8—22no—
85C4⋊Q8—44no—
86C4⋊Q8—44no—
87C4⋊Q8—44no—
88C4⋊Q8—88no—
89C4⋊Q8—84no—
90C4⋊Q8SD1618yes—
91C4⋊Q8C2×Q814yes—
92C4×Q8—44no—
93C4×Q8—84no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—24no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4C2×D414yes—
108C2×C4⋊C4C2×D414yes—
109C2×C4⋊C4C2×D412yes—
110C2×C42—22no—
111C2×C42—24no—
112C2×C42—24no—
113C2×C42C2×D412yes—
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2411yescharacteristic
118#106—48no—
119#106—416no—
120C4⋊C8—24no—
121C4⋊C8—24no—
122C4⋊C8C2×Q818yes—
123#103—48no—
124C4×C8—24no—
125C4×C8—44no—
126#95—28no—
127#95—24no—
128#95—24no—
129#95—24no—
130#95—24no—
131C2×Q8—22no—
132C2×Q8—28no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—82no—
136C2×Q8—82no—
137C2×Q8—88no—
138C2×Q8—84no—
139C2×Q8—84no—
140C2×Q8—82no—
141C2×Q8C4⋊Q814yes—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—24no—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4#12011yescharacteristic
148C22×C4#12012yes—
149C22×C4#12011yescharacteristic
150C22×C4C22×D411yescharacteristic
151C22×C4C22×D412yes—
152C2×C8—24no—
153C2×C8—28no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—44no—
158C4⋊C4—28no—
159C4⋊C4—24no—
160C4⋊C4—28no—
161C4⋊C4—28no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—84no—
171C4⋊C4—82no—
172C4⋊C4—88no—
173C4⋊C4—84no—
174C4⋊C4—84no—
175C4⋊C4C2×SD1618yes—
176C42—24no—
177C42—22no—
178C42—41no—
179C42—48no—
180C42—44no—
181C42—84no—
182C42C2×SD1614yes—
183C42#13714yes—
184C42C22×Q812yes—
185C422−1+412yes—
186C23C2×#12011yescharacteristic
187Q8—44no—
188Q8—82no—
189Q8—88no—
190Q8—168no—
191Q8—164no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—28no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4#44814yes—
212C2×C4#44814yes—
213C2×C4#45312yes—
214C2×C4C2×C4⋊Q811yescharacteristic
215C2×C4#53611yescharacteristic
216C2×C4D4×Q814yes—
217C2×C4C22×SD1612yes—
218C2×C4C2×#13712yes—
219C2×C4#57714yes—
220C8—44no—
221C8—48no—
222C22#243411yescharacteristic
223C22C2×#44812yes—
224C22#300912yes—
225C22C2×#45311yescharacteristic
226C22#302211yescharacteristic
227C4—24no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—48no—
232C4—44no—
233C4—84no—
234C4—82no—
235C4—162no—
236C4#162314yes—
237C4Q8×SD1614yes—
238C4#338714yes—
239C2C2×#162311yescharacteristic
240C2#1871511yescharacteristic
241C2#1890412yes—
242C2#2582411yescharacteristic
243C2#2583712yes—
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
76. #333Q8semidirect

Maximal subgroups

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

subgroupindex
#599242
#286872
#285122
#105742
#98642
#89282
C4×#3332