Finite Group Databasebeta

← #318350

Subgroups — #318350

Order 512.

676 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 158 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#28202C212yesmaximal
2#22260C211yesmaximal, characteristic
3#22248C211yesmaximal, characteristic, complemented
4C2×#1675C212yesmaximal, complemented
5#10480C214yesmaximal, complemented
6#8811C211yesmaximal, characteristic, complemented
7#8648C212yesmaximal, complemented
8#8594C212yesmaximal, complemented
9C2×#500C2211yescharacteristic
10C2×#476C2214yes—
11C2×C8×D4C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×#127C2211yescharacteristic
14#2084—22no—
15#2041—22no—
16#1990C2212yes—
17#1954C2212yes—
18#1926—24no—
19#1921—24no—
20#1869—24no—
21#1834C2211yescharacteristic
22#1831C2212yes—
23#1831C2211yescharacteristic
24#1802C2214yes—
25#1744C2211yescharacteristic
26#1675—28no—
27C2×#335C2212yes—
28C2×#334C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31C23×C8—22no—
32C2×C4⋊Q8C2312yes—
33C2×#127D411yescharacteristic
34C2×C4×Q8—22no—
35C2×C4×D4—22no—
36C2×C4×D4C2312yes—
37#500—22no—
38#500D414yescomplemented
39#476—28no—
40#476—44no—
41C8×D4—24no—
42C8×D4—24no—
43C2×#107—22no—
44C2×#106—24no—
45C2×C4⋊C8C2312yes—
46C2×C4⋊C8C2314yes—
47C2×#102—24no—
48C2×#102—24no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—24no—
51C2×C4×C8C2311yescharacteristic
52C2×C4×C8C2311yescharacteristic
53#391—22no—
54#390—22no—
55#390D412yes—
56#389—22no—
57#389—22no—
58#384D411yescharacteristic
59#384C2312yes—
60C4×C4⋊C4—22no—
61C4×C4⋊C4—22no—
62C43C2311yescharacteristic
63#340—22no—
64#340—24no—
65#340—24no—
66#336—22no—
67#336—22no—
68#336—22no—
69#336D414yes—
70#335—24no—
71#334—42no—
72#333—28no—
73C22×Q8—22no—
74C22×D4—22no—
75C22×D4C2×D412yes—
76C23×C4—22no—
77C22×C8—22no—
78C22×C8—24no—
79C22×C8—44no—
80C22×C8—84no—
81C22×C8C2×D412yes—
82C22×C8C2×D412yes—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8C2×D414yes—
87#127—24no—
88C4×Q8—84no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—84no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4C2×D412yes—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—24no—
104C2×C42—42no—
105C2×C42C2×D412yes—
106C2×C42C2×D411yescharacteristic
107C2×C42C2×D411yescharacteristic
108C2×C42#4112yes—
109C2×C42C2411yescharacteristic
110#107—28no—
111#106—48no—
112C4⋊C8—22no—
113C4⋊C8—24no—
114C4⋊C8—22no—
115C4⋊C8—24no—
116#102—48no—
117#102—48no—
118C22⋊C8—44no—
119C22⋊C8—84no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8C2×D412yes—
123C4×C8#4112yes—
124#95—42no—
125#95—42no—
126C24—22no—
127C2×Q8—24no—
128C2×Q8—42no—
129C2×Q8—82no—
130C2×Q8—82no—
131C2×D4—28no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—82no—
138C2×D4—82no—
139C2×D4#13514yes—
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
152C2×C8—24no—
153C2×C8—22no—
154C2×C8—24no—
155C2×C8—42no—
156C2×C8—44no—
157C2×C8—44no—
158C2×C8—44no—
159C2×C8—44no—
160C2×C8—88no—
161C2×C8#12114yes—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—84no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—84no—
176C42—22no—
177C42—24no—
178C42—22no—
179C42—42no—
180C42—84no—
181C42C22×D411yescharacteristic
182C42C2×#4111yescharacteristic
183C42H(5,𝔽2)11yescharacteristic
184C422−1+411yescharacteristic
185C23—24no—
186C23—44no—
187C23—42no—
188C23—42no—
189C23—44no—
190C23C2×#12111yescharacteristic
191Q8—84no—
192Q8—84no—
193D4—24no—
194D4—24no—
195D4—42no—
196D4—42no—
197D4—88no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—84no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4C2×#12111yescharacteristic
217C2×C4#53711yescharacteristic
218C2×C4D4211yescharacteristic
219C2×C4#54711yescharacteristic
220C2×C4#54711yescharacteristic
221C2×C4#55511yescharacteristic
222C2×C4C2×#13512yes—
223C2×C4#57714yes—
224C2×C4#57812yes—
225C8—24no—
226C8—42no—
227C8—84no—
228C8—82no—
229C22—44no—
230C22—48no—
231C22—84no—
232C22—82no—
233C22—82no—
234C22#253711yescharacteristic
235C22#304212yes—
236C22#305011yescharacteristic
237C22#305911yescharacteristic
238C22#306812yes—
239C4—21no—
240C4—21no—
241C4—42no—
242C4—42no—
243C4—84no—
244C4—82no—
245C4—162no—
246C4#304312yes—
247C4#326012yes—
248C4#328512yes—
249C4#333612yes—
250C2—84no—
251C2#1939412yes—
252C2#1951512yes—
253C2#2822611yescharacteristic
254C2#2824311yescharacteristic
255C2#2826011yescharacteristic
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
3. #22248C2semidirect
4. C2×#1675C2semidirect
5. #10480C2semidirect
6. #8811C2semidirect
7. #8648C2semidirect
8. #8594C2semidirect
38. #500D4semidirect

Maximal subgroups

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

subgroupindex
#282022
#222602
#222482
C2×#16752
#104802
#88112
#86482
#85942