Finite Group Databasebeta

← #464499

Subgroups — #464499

Order 512.

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

normal subgroups only

This group has 934 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
1C22×#499C211yesmaximal, characteristic
2C2×#2090C212yesmaximal
3#23553C212yesmaximal
4#21020C211yesmaximal, characteristic
5#11383C212yesmaximal
6#11366C212yesmaximal
7#11331C212yesmaximal
8#11270C212yesmaximal
9#10822C211yesmaximal, characteristic
10C22×#125C2212yes—
11C2×#499—28no—
12C2×#499—24no—
13C2×#428C2214yes—
14C22×#107C2212yes—
15C22×#106C2212yes—
16C2×#424C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2296C2211yescharacteristic
19#2090—28no—
20#2059—28no—
21#1916—24no—
22C2×#343C2214yes—
23C2×#343C2214yes—
24C2×#341C2212yes—
25C2×#341C2212yes—
26#1736C2214yes—
27#1729C2211yescharacteristic
28#1729C2211yescharacteristic
29#1729C2211yescharacteristic
30#1728C2212yes—
31#1728C2211yescharacteristic
32C22×M16C2311yescharacteristic
33C22×M16C2311yescharacteristic
34C22×M16C2314yes—
35C23×C8C2311yescharacteristic
36C23×C8C2311yescharacteristic
37C2×#125—28no—
38C2×#125—24no—
39C2×#125—24no—
40C2×#125—44no—
41C2×#117C2311yescharacteristic
42C2×#117C2311yescharacteristic
43C22×C4⋊C4—24no—
44C22×C4⋊C4C2314yes—
45C22×C42C2311yescharacteristic
46#499—232no—
47#428—28no—
48#428—44no—
49C2×#107—24no—
50C2×#107—28no—
51C2×#107—24no—
52C2×#106—24no—
53C2×#106—24no—
54C2×#106—24no—
55C2×#106D418yes—
56#424—24no—
57C2×C4⋊C8—22no—
58C2×C4×C8—24no—
59C2×C4×C8D414yes—
60#389D414yes—
61#385—22no—
62C4×C4⋊C4—22no—
63#343—28no—
64#343—44no—
65#341—24no—
66#341Q818yes—
67#340—28no—
68#336—22no—
69#336—22no—
70#336—22no—
71#336—24no—
72#336—22no—
73C23×C4—24no—
74C23×C4#4112yes—
75C23×C4C2411yescharacteristic
76C2×M16—24no—
77C2×M16—24no—
78C2×M16—24no—
79C2×M16—24no—
80C2×M16—24no—
81C2×M16—22no—
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16C2×Q818yes—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—24no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8C2×D412yes—
96C22×C8C2×D414yes—
97C22×C8C2×D412yes—
98C22×C8C2×Q814yes—
99#125—216no—
100#125—416no—
101#125—48no—
102#117—22no—
103#117—22no—
104#117—22no—
105#117—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—416no—
113C2×C4⋊C4—48no—
114C2×C4⋊C4C2×D418yes—
115C2×C4⋊C4C2×D412yes—
116C2×C4⋊C4C2×D412yes—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C42—22no—
120C2×C42—22no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2×D412yes—
124C2×C42C2×D412yes—
125#107—216no—
126#107—48no—
127#106—216no—
128#106—48no—
129C4⋊C8—44no—
130C4×C8—28no—
131#95—24no—
132C24C2×#4111yescharacteristic
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—28no—
137C22×C4—44no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—88no—
141C22×C4#12114yes—
142C22×C4C22×D411yescharacteristic
143C22×C4C22×D412yes—
144C22×C4C22×D411yescharacteristic
145C22×C4C22×Q812yes—
146C22×C4H(5,𝔽2)12yes—
147C22×C42−1+412yes—
148C22×C42−1+412yes—
149C22×C42−1+412yes—
150M16—28no—
151M16—28no—
152M16—44no—
153M16—44no—
154M16—44no—
155M16—44no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—28no—
160C2×C8—22no—
161C2×C8—28no—
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—
171C2×C8#12014yes—
172C2×C8C4⋊Q814yes—
173C4⋊C4—28no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—28no—
177C4⋊C4—48no—
178C4⋊C4—432no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C42—22no—
184C42—22no—
185C42—24no—
186C42—41no—
187C42—41no—
188C42—41no—
189C42—41no—
190C23—24no—
191C23C2×#12111yescharacteristic
192C23#54112yes—
193C23#54612yes—
194C23#55712yes—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—48no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—88no—
205C2×C4—88no—
206C2×C4—88no—
207C2×C4C2×#12011yescharacteristic
208C2×C4C2×C4⋊Q811yescharacteristic
209C2×C4#53511yescharacteristic
210C2×C4#53611yescharacteristic
211C2×C4#53611yescharacteristic
212C2×C4#53611yescharacteristic
213C2×C4#53614yes—
214C2×C4#53612yes—
215C2×C4#54414yes—
216C2×C4D4212yes—
217C2×C4D4×Q812yes—
218C2×C4D4×Q814yes—
219C2×C4#56314yes—
220C8—24no—
221C8—24no—
222C8—48no—
223C8—81no—
224C8—81no—
225C22—24no—
226C22—24no—
227C22—24no—
228C22#240811yescharacteristic
229C22#254611yescharacteristic
230C22#262111yescharacteristic
231C22#271314yes—
232C22#306714yes—
233C4—44no—
234C4—82no—
235C4—82no—
236C4—88no—
237C4#243411yescharacteristic
238C4#261411yescharacteristic
239C4#266011yescharacteristic
240C4#267714yes—
241C4#282111yescharacteristic
242C2—24no—
243C2#1650711yescharacteristic
244C2#2427912yes—
245C2#2870212yes—
246C2#2988812yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#4992
C2×#20902
#235532
#210202
#113832
#113662
#113312
#112702
#108222