Finite Group Databasebeta

← #454526

Subgroups — #454526

Order 512.

1270 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1270 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#35151C212yesmaximal
2#23561C214yesmaximal, complemented
3#11224C214yesmaximal, complemented
4#11100C214yesmaximal
5#10728C211yesmaximal, characteristic
6C2×#544C2212yes—
7C22×#117C2211yescharacteristic
8#2911C2214yes—
9C2×#407C2212yes—
10#2855—24no—
11#2626—24no—
12#2441—28no—
13#2439—28no—
14#2317C2214yes—
15#2316C2212yes—
16#2280C2214yes—
17#2061—28no—
18#2060—28no—
19C2×#343C2214yes—
20#1733C2218yes—
21C2×H(3,ℤ/4)C2214yes—
22C24×C4C2311yescharacteristic
23C22×M16C2314yes—
24#544—28no—
25#544—44no—
26C2×C4⋊Q8—24no—
27C2×#125—24no—
28C2×C22⋊Q8—24no—
29C2×#117—28no—
30C2×#117—42no—
31C2×#117C2314yes—
32C2×#117C2314yes—
33C22×C4⋊C4C2312yes—
34C22×C22⋊C4D412yes—
35C22×C42D412yes—
36#428—28no—
37#428—28no—
38#427—44no—
39#407—42no—
40#407Q818yescomplemented
41C2×C22⋊C8—42no—
42#399—28no—
43#396—216no—
44#393—28no—
45#391—28no—
46#389—28no—
47#389—28no—
48#389—28no—
49#387—216no—
50#385—28no—
51#384D418yes—
52#382D418yes—
53#381—24no—
54#380—24no—
55#380—24no—
56#380—24no—
57#379—24no—
58C4×C22⋊C4—28no—
59#343—28no—
60#339—216no—
61H(3,ℤ/4)—44no—
62C25C2×D411yescharacteristic
63C22×Q8—24no—
64C23×C4—24no—
65C23×C4—22no—
66C23×C4—21no—
67C23×C4—24no—
68C23×C4—21no—
69C23×C4C2×D411yescharacteristic
70C23×C4C2×D414yes—
71C23×C4C2×Q814yes—
72C23×C4C2411yescharacteristic
73C2×M16—24no—
74C2×M16—28no—
75C2×M16—28no—
76C2×M16C2×Q818yes—
77C22×C8—24no—
78C4⋊Q8—416no—
79#125—416no—
80C22⋊Q8—48no—
81C22⋊Q8—48no—
82#117—24no—
83#117—24no—
84#117—24no—
85#117—24no—
86#117—28no—
87#117—28no—
88#117—416no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—28no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—216no—
96C2×C4⋊C4C2×D414yes—
97C2×C4⋊C4C2×D414yes—
98C2×C4⋊C4C2×D418yes—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—28no—
102C2×C22⋊C4—28no—
103C2×C22⋊C4—28no—
104C2×C22⋊C4—42no—
105C2×C42—28no—
106C2×C42—24no—
107C2×C42—42no—
108C2×C42C2×D418yes—
109#107—48no—
110#106—44no—
111#106—44no—
112C22⋊C8—48no—
113#95—28no—
114#95—28no—
115#95—28no—
116#95—216no—
117C24—22no—
118C24—21no—
119C24—22no—
120C24—21no—
121C24—82no—
122C24C4⋊Q812yes—
123C24C22×D411yescharacteristic
124C2×Q8—48no—
125C2×Q8—88no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—216no—
135C22×C4—24no—
136C22×C4—28no—
137C22×C4—24no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4—44no—
142C22×C4—84no—
143C22×C4#12018yes—
144C22×C4C4⋊Q812yes—
145C22×C4C22×D414yes—
146C22×C4C22×Q812yes—
147C22×C4H(5,𝔽2)11yescharacteristic
148C22×C4H(5,𝔽2)12yes—
149C22×C42−1+411yescharacteristic
150C22×C42−1+414yes—
151M16—216no—
152M16—48no—
153C2×C8—28no—
154C2×C8—44no—
155C2×C8—44no—
156C4⋊C4—216no—
157C4⋊C4—28no—
158C4⋊C4—28no—
159C4⋊C4—48no—
160C4⋊C4—416no—
161C4⋊C4—44no—
162C4⋊C4—44no—
163C4⋊C4—88no—
164C4⋊C4—816no—
165C22⋊C4—48no—
166C22⋊C4—44no—
167C22⋊C4—44no—
168C22⋊C4—44no—
169C22⋊C4—44no—
170C22⋊C4—88no—
171C22⋊C4—88no—
172C42—28no—
173C42—28no—
174C42—48no—
175C42—48no—
176C23—22no—
177C23—24no—
178C23—42no—
179C23—44no—
180C23—44no—
181C23—44no—
182C23—41no—
183C23—42no—
184C23—41no—
185C23—82no—
186C23—84no—
187C23—82no—
188C23C2×#12012yes—
189C23C2×C4⋊Q811yescharacteristic
190C23#53411yescharacteristic
191C23#53612yes—
192C23#53711yescharacteristic
193Q8—168no—
194C2×C4—28no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—48no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—48no—
202C2×C4—48no—
203C2×C4—48no—
204C2×C4—48no—
205C2×C4—48no—
206C2×C4—48no—
207C2×C4—48no—
208C2×C4—48no—
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—84no—
212C2×C4#53614yes—
213C2×C4#53714yes—
214C2×C4#54312yes—
215C2×C4#54412yes—
216C2×C4D4214yes—
217C2×C4D4×Q818yes—
218C2×C4#56314yes—
219C8—48no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—41no—
224C22—42no—
225C22—41no—
226C22—44no—
227C22—48no—
228C22—42no—
229C22—84no—
230C22—82no—
231C22—82no—
232C22#240311yescharacteristic
233C22#243412yes—
234C22#261612yes—
235C22#265911yescharacteristic
236C22#281711yescharacteristic
237C4—48no—
238C4—42no—
239C4—88no—
240C4—84no—
241C4—84no—
242C4#267714yes—
243C4#268314yes—
244C2—22no—
245C2—21no—
246C2—22no—
247C2—21no—
248C2—82no—
249C2#1625411yescharacteristic
250C2#2389812yes—
subgroupG/H
center, socleC22#2403
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23561C2semidirect
3. #11224C2semidirect
40. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#351512
#235612
#112242
#111002
#107282