Finite Group Databasebeta

← #464498

Subgroups — #464498

Order 512.

1042 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 226 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×#498C211yesmaximal, characteristic
2C2×#2090C212yesmaximal
3#23552C212yesmaximal
4#21020C211yesmaximal, characteristic
5#11382C212yesmaximal
6#11366C212yesmaximal
7#11330C212yesmaximal
8#11271C212yesmaximal
9#10822C211yesmaximal, characteristic
10C22×C4⋊Q8C2212yes—
11C2×#498—28no—
12C2×#498—24no—
13C2×#428C2214yes—
14C22×#106C2212yes—
15C22×#106C2212yes—
16C2×#424C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2296C2211yescharacteristic
19#2090—28no—
20#2058—24no—
21#2057—24no—
22#1916—24no—
23C2×#343C2214yes—
24C2×#343C2214yes—
25C2×#341C2212yes—
26C2×#341C2212yes—
27#1736C2214yes—
28#1729C2211yescharacteristic
29#1729C2211yescharacteristic
30#1729C2211yescharacteristic
31#1728C2212yes—
32#1728C2211yescharacteristic
33C23×Q8—22no—
34C22×M16C2311yescharacteristic
35C22×M16C2311yescharacteristic
36C22×M16C2314yes—
37C23×C8C2311yescharacteristic
38C23×C8C2311yescharacteristic
39C2×C4⋊Q8—28no—
40C2×C4⋊Q8—24no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—44no—
43C2×#117C2311yescharacteristic
44C2×#117C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2314yes—
47C22×C42C2311yescharacteristic
48#498—232no—
49#428—28no—
50#428—44no—
51C2×#107—24no—
52C2×#106—24no—
53C2×#106—24no—
54C2×#106—24no—
55C2×#106—28no—
56C2×#106—24no—
57C2×#106D418yes—
58#424—24no—
59C2×C4⋊C8—22no—
60C2×C4×C8—24no—
61C2×C4×C8D414yes—
62#389D414yes—
63#385—22no—
64C4×C4⋊C4—22no—
65#343—28no—
66#343—44no—
67#341—24no—
68#341Q818yescomplemented
69#340—28no—
70#336—22no—
71#336—22no—
72#336—22no—
73#336—24no—
74#336—22no—
75C22×Q8—48no—
76C22×Q8—44no—
77C22×Q8—88no—
78C23×C4—24no—
79C23×C4#4112yes—
80C23×C4C2411yescharacteristic
81C2×M16—24no—
82C2×M16—24no—
83C2×M16—24no—
84C2×M16—24no—
85C2×M16—24no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16C2×Q818yes—
93C22×C8—22no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8—24no—
98C22×C8—21no—
99C22×C8—21no—
100C22×C8C2×D412yes—
101C22×C8C2×D414yes—
102C22×C8C2×D412yes—
103C22×C8C2×Q814yes—
104C4⋊Q8—216no—
105C4⋊Q8—416no—
106C4⋊Q8—48no—
107#117—22no—
108#117—22no—
109#117—22no—
110#117—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4C2×D418yes—
120C2×C4⋊C4C2×D412yes—
121C2×C4⋊C4C2×D412yes—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42C2×D412yes—
129C2×C42C2×D412yes—
130#107—48no—
131#106—216no—
132#106—216no—
133#106—48no—
134C4⋊C8—44no—
135C4×C8—28no—
136#95—24no—
137C24C2×#4111yescharacteristic
138C2×Q8—416no—
139C2×Q8—832no—
140C2×Q8—88no—
141C2×Q8—88no—
142C22×C4—24no—
143C22×C4—28no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—28no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—88no—
150C22×C4#12114yes—
151C22×C4C22×D411yescharacteristic
152C22×C4C22×D412yes—
153C22×C4C22×D411yescharacteristic
154C22×C4C22×Q812yes—
155C22×C4H(5,𝔽2)12yes—
156C22×C42−1+412yes—
157C22×C42−1+412yes—
158C22×C42−1+412yes—
159M16—28no—
160M16—28no—
161M16—44no—
162M16—44no—
163M16—44no—
164M16—44no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—28no—
169C2×C8—22no—
170C2×C8—28no—
171C2×C8—28no—
172C2×C8—44no—
173C2×C8—44no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8#12014yes—
181C2×C8C4⋊Q814yes—
182C4⋊C4—28no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—28no—
186C4⋊C4—416no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—48no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C42—22no—
193C42—22no—
194C42—24no—
195C42—41no—
196C42—41no—
197C42—41no—
198C42—41no—
199C23—24no—
200C23C2×#12111yescharacteristic
201C23#54112yes—
202C23#54612yes—
203C23#55712yes—
204Q8—832no—
205C2×C4—216no—
206C2×C4—24no—
207C2×C4—24no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—48no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—88no—
214C2×C4—88no—
215C2×C4—88no—
216C2×C4#47118yes—
217C2×C4C2×#12011yescharacteristic
218C2×C4C2×C4⋊Q811yescharacteristic
219C2×C4#53511yescharacteristic
220C2×C4#53611yescharacteristic
221C2×C4#53611yescharacteristic
222C2×C4#53611yescharacteristic
223C2×C4#53614yes—
224C2×C4#53612yes—
225C2×C4#54414yes—
226C2×C4D4212yes—
227C2×C4D4×Q812yes—
228C2×C4D4×Q814yes—
229C2×C4#56314yes—
230C8—24no—
231C8—24no—
232C8—48no—
233C8—81no—
234C8—81no—
235C22—24no—
236C22—24no—
237C22—24no—
238C22#240811yescharacteristic
239C22#254611yescharacteristic
240C22#262111yescharacteristic
241C22#271314yes—
242C22C2×#47114yes—
243C4—28no—
244C4—82no—
245C4—82no—
246C4—88no—
247C4#243411yescharacteristic
248C4#261411yescharacteristic
249C4#266011yescharacteristic
250C4#267714yes—
251C4#282111yescharacteristic
252C2—24no—
253C2#1650711yescharacteristic
254C2#2427812yes—
255C2#2870012yes—
256C2#2988712yes—
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

Complements

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

NH ≅ G/Ntype
68. #341Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
C2×#20902
#235522
#210202
#113822
#113662
#113302
#112712
#108222