Finite Group Databasebeta

← #454534

Subgroups — #454534

Order 512.

1356 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1356 conjugacy classes of subgroups — too many to draw. Only its 218 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#35090C212yesmaximal
2#23565C214yesmaximal
3#11205C214yesmaximal
4#11070C214yesmaximal
5#10726C211yesmaximal, characteristic
6C2×#543C2212yes—
7C22×#117C2211yescharacteristic
8#2855—24no—
9#2627—24no—
10#2438—216no—
11#2315C2212yes—
12#2315C2214yes—
13#2315C2214yes—
14C2×#381C2214yes—
15C2×#381C2212yes—
16#2062—216no—
17C2×#342C2218yes—
18C2×#342C2214yes—
19C2×#342C2214yes—
20C24×C4C2311yescharacteristic
21#543—28no—
22#543—44no—
23C2×C22⋊Q8—24no—
24C2×C22⋊Q8—28no—
25C2×#117—28no—
26C2×#117—42no—
27C22×C4⋊C4D412yes—
28C22×C22⋊C4C2314yes—
29C22×C22⋊C4C2314yes—
30C22×C22⋊C4C2314yes—
31C22×C22⋊C4C2312yes—
32C22×C42D412yes—
33#400—28no—
34#398—216no—
35#395—28no—
36#391—28no—
37#389—28no—
38#387—216no—
39#385—28no—
40#385—28no—
41#385—216no—
42#385D4116yes—
43#381—24no—
44#381—24no—
45#381—28no—
46#381—42no—
47#381Q818yescomplemented
48#380—24no—
49#380—28no—
50#380—44no—
51C2×#95—28no—
52C2×#95—42no—
53#342—28no—
54#342—216no—
55#342—44no—
56C25C2411yescharacteristic
57C22×Q8—24no—
58C23×C4—24no—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4—24no—
62C23×C4—21no—
63C23×C4—24no—
64C23×C4—24no—
65C23×C4C2×D411yescharacteristic
66C23×C4C2×D411yescharacteristic
67C23×C4C2×D418yes—
68C23×C4C2×D414yes—
69C23×C4C2×Q814yes—
70C22⋊Q8—432no—
71C22⋊Q8—48no—
72C22⋊Q8—48no—
73#117—416no—
74C2×C4⋊C4—28no—
75C2×C4⋊C4—24no—
76C2×C4⋊C4—28no—
77C2×C4⋊C4—28no—
78C2×C4⋊C4—216no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—48no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4—28no—
84C2×C22⋊C4—24no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—24no—
87C2×C22⋊C4—24no—
88C2×C22⋊C4—28no—
89C2×C22⋊C4—24no—
90C2×C22⋊C4—24no—
91C2×C22⋊C4—28no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—28no—
94C2×C22⋊C4—24no—
95C2×C22⋊C4C2×D418yes—
96C2×C22⋊C4C2×D418yes—
97C2×C22⋊C4C2×Q818yes—
98C2×C42—28no—
99C2×C42—42no—
100#95—216no—
101#95—216no—
102#95—28no—
103#95—216no—
104#95—48no—
105#95—48no—
106C24—82no—
107C24C22×D411yescharacteristic
108C24C22×D414yes—
109C24C22×Q812yes—
110C24H(5,𝔽2)12yes—
111C24H(5,𝔽2)11yescharacteristic
112C242−1+411yescharacteristic
113C242−1+414yes—
114C2×Q8—48no—
115C2×Q8—88no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—24no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—24no—
126C22×C4—24no—
127C22×C4—216no—
128C22×C4—24no—
129C22×C4—28no—
130C22×C4—28no—
131C22×C4—28no—
132C22×C4—216no—
133C22×C4—44no—
134C22×C4—44no—
135C22×C4—44no—
136C22×C4—44no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—84no—
142C22×C4—88no—
143C22×C4—88no—
144C22×C4—88no—
145C22×C4#12018yes—
146C22×C4C4⋊Q812yes—
147C22×C4C4⋊Q812yes—
148C4⋊C4—48no—
149C4⋊C4—416no—
150C4⋊C4—48no—
151C4⋊C4—88no—
152C4⋊C4—816no—
153C4⋊C4—168no—
154C22⋊C4—216no—
155C22⋊C4—28no—
156C22⋊C4—28no—
157C22⋊C4—216no—
158C22⋊C4—44no—
159C22⋊C4—44no—
160C22⋊C4—44no—
161C22⋊C4—44no—
162C22⋊C4—48no—
163C22⋊C4—48no—
164C42—48no—
165C23—42no—
166C23—48no—
167C23—44no—
168C23—82no—
169C23—81no—
170C23—81no—
171C23—84no—
172C23C2×#12012yes—
173C23C2×C4⋊Q811yescharacteristic
174C23#53411yescharacteristic
175C23#53612yes—
176C23#53614yes—
177C23#53711yescharacteristic
178C23#53714yes—
179C23#54312yes—
180C23#54412yes—
181C23D4214yes—
182C23D4×Q818yes—
183C23#56314yes—
184Q8—168no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—48no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—48no—
198C2×C4—48no—
199C2×C4—48no—
200C2×C4—416no—
201C2×C4—416no—
202C2×C4—48no—
203C2×C4—48no—
204C2×C4—48no—
205C2×C4—84no—
206C2×C4—88no—
207C2×C4—816no—
208C2×C4—88no—
209C2×C4—164no—
210C2×C4—164no—
211C2×C4—164no—
212C22—24no—
213C22—28no—
214C22—22no—
215C22—42no—
216C22—44no—
217C22—44no—
218C22—44no—
219C22—41no—
220C22—42no—
221C22—41no—
222C22—42no—
223C22—48no—
224C22#240311yescharacteristic
225C22#243412yes—
226C22#261612yes—
227C22#265911yescharacteristic
228C22#267714yes—
229C22#268314yes—
230C22#281711yescharacteristic
231C4—48no—
232C4—42no—
233C4—42no—
234C4—88no—
235C4—88no—
236C4—88no—
237C2—22no—
238C2—24no—
239C2—21no—
240C2—21no—
241C2—24no—
242C2—22no—
243C2#1625411yescharacteristic
244C2#2389612yes—
subgroupG/H
center, socleC22#2403
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
47. #381Q8semidirect

Maximal subgroups

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

subgroupindex
#350902
#235652
#112052
#110702
#107262