Finite Group Databasebeta

← #325432

Subgroups — #325432

Order 512.

569 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 162 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#33258C211yesmaximal, characteristic, complemented
2#28721C212yesmaximal
3#21631C211yesmaximal, characteristic
4#10624C212yesmaximal, complemented
5#10622C212yesmaximal
6#9899C212yesmaximal, complemented
7#8653C212yesmaximal, complemented
8#8652C212yesmaximal
9C4×#334C211yesmaximal, characteristic, complemented
10C2×#488—22no—
11C2×#446C2211yescharacteristic
12C2×#436C2211yescharacteristic
13#2859C2212yes—
14C4×#124C2211yescharacteristic
15#2074—22no—
16#2073—22no—
17#1991—22no—
18#1984C2211yescharacteristic
19#1954C2211yescharacteristic
20#1942—22no—
21#1942—22no—
22#1923—22no—
23#1845C2212yes—
24#1834C2212yes—
25#1831C2211yescharacteristic
26#1803C2214yes—
27#1802C2214yes—
28C4×C4⋊C8C2212yes—
29#1753—22no—
30C42×C8C2211yescharacteristic
31C2×#334C2212yes—
32#1296C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C22×M16—21no—
36C2×#125—22no—
37C2×#124D411yescharacteristic
38C2×C4×Q8C2311yescharacteristic
39C2×C4×D4C2311yescharacteristic
40#488—48no—
41#446—24no—
42#436—24no—
43C2×#107—22no—
44C2×#107—22no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8C2314yes—
47C2×C4⋊C8C2311yescharacteristic
48C2×C4⋊C8C2311yescharacteristic
49C2×#103—24no—
50C2×#102—24no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—22no—
53C2×#97—21no—
54C2×C4×C8—22no—
55C2×C4×C8D412yes—
56C2×C4×C8D412yes—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2311yescharacteristic
59#391D411yescharacteristic
60#390D411yescharacteristic
61#389—24no—
62#389—24no—
63#388—22no—
64#384C2314yes—
65#382—22no—
66#382D411yescharacteristic
67C4×C22⋊C4—22no—
68C43C2311yescharacteristic
69#340—24no—
70#340—24no—
71#340—24no—
72#336—22no—
73#336—22no—
74#336—22no—
75#334—24no—
76C22×Q8C2×D411yescharacteristic
77C22×D4C2×D411yescharacteristic
78C23×C4—21no—
79C2×M16—42no—
80C2×M16—44no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—24no—
84C22×C8C2×D412yes—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87#125—44no—
88#125—44no—
89#124—24no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×D4—22no—
93C4×D4—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—44no—
99C2×C4⋊C4C2×D411yescharacteristic
100C2×C4⋊C4C2×D411yescharacteristic
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C42—21no—
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D412yes—
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C2×C42C2411yescharacteristic
110#107—44no—
111#107—44no—
112C4⋊C8—24no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115C4⋊C8—24no—
116C4⋊C8—42no—
117#103—48no—
118#102—48no—
119C22⋊C8—44no—
120C22⋊C8—42no—
121#97—42no—
122C4×C8—24no—
123C4×C8—22no—
124C4×C8—42no—
125C4×C8—44no—
126C4×C8#4114yes—
127#95—22no—
128#95—22no—
129#95—44no—
130C24—21no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4#12112yes—
146C22×C4#12112yes—
147C22×C4C22×D411yescharacteristic
148C22×C4C22×D411yescharacteristic
149C22×C4C2×#4111yescharacteristic
150M16—84no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—44no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—42no—
161C2×C8—44no—
162C2×C8#12412yes—
163C2×C8#12412yes—
164C2×C8#13514yes—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—44no—
172C4⋊C4—84no—
173C4⋊C4—88no—
174C22⋊C4—44no—
175C22⋊C4—42no—
176C22⋊C4—44no—
177C42—22no—
178C42—22no—
179C42—22no—
180C42—42no—
181C42—41no—
182C42—41no—
183C42AGL(1,ℤ/8)14yes—
184C42C2×#4112yes—
185C422−1+412yes—
186C23—42no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—42no—
191C23C2×#12111yescharacteristic
192Q8—44no—
193D4—44no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—84no—
211C2×C4—84no—
212C2×C4—84no—
213C2×C4#46914yes—
214C2×C4C2×#12411yescharacteristic
215C2×C4#53611yescharacteristic
216C2×C4#54712yes—
217C2×C4#55612yes—
218C2×C4C2×#13512yes—
219C2×C4C2×AGL(1,ℤ/8)12yes—
220C2×C4#57512yes—
221C2×C4#57812yes—
222C8—22no—
223C8—42no—
224C8—84no—
225C8—81no—
226C8—81no—
227C8#48814yescomplemented
228C22—82no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22—82no—
233C22#254711yescharacteristic
234C22#303512yes—
235C22C2×#46911yescharacteristic
236C22#305911yescharacteristic
237C22#306712yes—
238C4—22no—
239C4—24no—
240C4—42no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4—164no—
245C4C2×#48811yescharacteristic
246C4#314211yescharacteristic
247C4#323312yes—
248C4#335314yes—
249C2—161no—
250C2#1915712yes—
251C2#1922612yes—
252C2#2871211yescharacteristic
253C2#2872111yescharacteristic
254C2#2873411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
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
1. #33258C2semidirect
4. #10624C2semidirect
6. #9899C2semidirect
7. #8653C2semidirect
9. C4×#334C2semidirect
227. C8#488semidirect

Maximal subgroups

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

subgroupindex
#332582
#287212
#216312
#106242
#106222
#98992
#86532
#86522
C4×#3342