Finite Group Databasebeta

← #318406

Subgroups — #318406

Order 512.

678 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 154 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#28201C212yesmaximal
2#22270C211yesmaximal, characteristic, complemented
3#22264C211yesmaximal, characteristic
4C2×#1689C212yesmaximal, complemented
5#10485C214yesmaximal, complemented
6#8811C211yesmaximal, characteristic, complemented
7#8656C212yesmaximal, complemented
8#8593C212yesmaximal, complemented
9C2×#494C2211yescharacteristic
10C2×#474C2214yes—
11C2×#436C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×#127C2211yescharacteristic
14#2083—22no—
15#2041—22no—
16#1956C2212yes—
17#1954C2212yes—
18#1925—24no—
19#1878—24no—
20#1870—24no—
21#1834C2211yescharacteristic
22#1831C2212yes—
23#1831C2211yescharacteristic
24#1802C2214yes—
25#1744C2211yescharacteristic
26#1689—28no—
27C2×#335C2212yes—
28C2×#334C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31C22×Q16—22no—
32C22×M16—22no—
33C2×C4⋊Q8C2312yes—
34C2×#127D411yescharacteristic
35C2×C4×Q8—22no—
36C2×C4×D4—22no—
37C2×C4×D4C2312yes—
38#494—22no—
39#494D414yescomplemented
40#474—28no—
41#474—44no—
42#436—24no—
43#436—24no—
44C2×#107—24no—
45C2×C4⋊C8C2312yes—
46C2×C4⋊C8C2314yes—
47C2×#103—24no—
48C2×#102—24no—
49C2×#102—24no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—24no—
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#391—22no—
55#390—22no—
56#390D412yes—
57#389—22no—
58#389—22no—
59#384D411yescharacteristic
60#384C2312yes—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C43C2311yescharacteristic
64#340—22no—
65#340—24no—
66#336—22no—
67#336—22no—
68#336—22no—
69#336D414yes—
70#335—42no—
71#334—24no—
72#332—28no—
73C22×Q8—22no—
74C22×D4—22no—
75C22×D4C2×D412yes—
76C23×C4—22no—
77C2×Q16—28no—
78C2×Q16—84no—
79C2×M16—44no—
80C2×M16—84no—
81C22×C8—22no—
82C22×C8—24no—
83C22×C8C2×D412yes—
84C22×C8C2×D412yes—
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88C4⋊Q8C2×D414yes—
89#127—42no—
90C4×Q8—84no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—84no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4C2×D412yes—
104C2×C22⋊C4—22no—
105C2×C22⋊C4—24no—
106C2×C42—42no—
107C2×C42C2×D412yes—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4112yes—
111C2×C42C2411yescharacteristic
112#107—48no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—22no—
116C4⋊C8—24no—
117#103—84no—
118#102—48no—
119#102—48no—
120C22⋊C8—44no—
121C22⋊C8—84no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8C2×D412yes—
125C4×C8#4112yes—
126#95—42no—
127#95—42no—
128C24—22no—
129C2×Q8—24no—
130C2×Q8—42no—
131C2×Q8—44no—
132C2×Q8—44no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—44no—
138C2×D4—82no—
139C2×D4—82no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—44no—
145C22×C4—44no—
146C22×C4—42no—
147C22×C4#12112yes—
148C22×C4#12112yes—
149C22×C4C22×D411yescharacteristic
150C22×C4C22×D411yescharacteristic
151C22×C4C2×#4111yescharacteristic
152Q16—88no—
153M16—88no—
154C2×C8—24no—
155C2×C8—22no—
156C2×C8—24no—
157C2×C8—42no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8#12114yes—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—44no—
170C4⋊C4—44no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—84no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—84no—
177C42—22no—
178C42—24no—
179C42—22no—
180C42—42no—
181C42—84no—
182C42C22×D411yescharacteristic
183C42C2×#4111yescharacteristic
184C42H(5,𝔽2)11yescharacteristic
185C422−1+411yescharacteristic
186C23—44no—
187C23—42no—
188C23—42no—
189C23—42no—
190C23—44no—
191C23C2×#12111yescharacteristic
192Q8—48no—
193Q8—84no—
194D4—44no—
195D4—44no—
196D4—88no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—44no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—84no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4C2×#12111yescharacteristic
216C2×C4#53711yescharacteristic
217C2×C4D4211yescharacteristic
218C2×C4#54711yescharacteristic
219C2×C4#54711yescharacteristic
220C2×C4#55511yescharacteristic
221C2×C4#57512yes—
222C2×C4#57614yes—
223C2×C4#57812yes—
224C8—24no—
225C8—42no—
226C8—84no—
227C8—82no—
228C22—84no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22—84no—
233C22#253711yescharacteristic
234C22#304112yes—
235C22#305411yescharacteristic
236C22#305911yescharacteristic
237C22#306912yes—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—82no—
243C4—84no—
244C4—162no—
245C4#304112yes—
246C4#325912yes—
247C4#327712yes—
248C4#335212yes—
249C2—162no—
250C2#1942412yes—
251C2#1950712yes—
252C2#2823411yescharacteristic
253C2#2824411yescharacteristic
254C2#2825911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
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
2. #22270C2semidirect
4. C2×#1689C2semidirect
5. #10485C2semidirect
6. #8811C2semidirect
7. #8656C2semidirect
8. #8593C2semidirect
39. #494D4semidirect

Maximal subgroups

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

subgroupindex
#282012
#222702
#222642
C2×#16892
#104852
#88112
#86562
#85932