Finite Group Databasebeta

← #318349

Subgroups — #318349

Order 512.

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

normal subgroups only

This group has 708 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#28157C212yesmaximal
2#22260C211yesmaximal, characteristic
3#22247C211yesmaximal, characteristic, complemented
4C2×#1673C212yesmaximal, complemented
5#10477C214yesmaximal, complemented
6#8811C211yesmaximal, characteristic, complemented
7#8648C212yesmaximal, complemented
8#8593C212yesmaximal, complemented
9C2×#500C2211yescharacteristic
10C2×#461C2214yes—
11C2×C8×D4C2212yes—
12C4×C4⋊Q8C2212yes—
13C4×#127C2211yescharacteristic
14#2072—22no—
15#2014—22no—
16#1990C2212yes—
17#1954C2212yes—
18#1918—24no—
19#1880—24no—
20#1869—24no—
21#1834C2211yescharacteristic
22#1831C2212yes—
23#1831C2211yescharacteristic
24#1802C2214yes—
25#1744C2211yescharacteristic
26#1673—28no—
27C2×#335C2212yes—
28C2×#334C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31C22×SD16—24no—
32C23×C8—22no—
33C2×C4⋊Q8C2312yes—
34C2×#127D411yescharacteristic
35C2×C4×Q8—22no—
36C2×C4×D4—22no—
37C2×C4×D4C2312yes—
38#500—22no—
39#500D414yescomplemented
40#461—28no—
41#461—44no—
42C8×D4—24no—
43C8×D4—24no—
44C2×#107—22no—
45C2×C4⋊C8C2312yes—
46C2×C4⋊C8C2314yes—
47C2×#103—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#332—28no—
73C22×Q8—22no—
74C22×D4—22no—
75C22×D4C2×D412yes—
76C23×C4—22no—
77C2×SD16—48no—
78C2×SD16—48no—
79C2×SD16—84no—
80C22×C8—22no—
81C22×C8—24no—
82C22×C8—44no—
83C22×C8—84no—
84C22×C8C2×D412yes—
85C22×C8C2×D412yes—
86C4⋊Q8—22no—
87C4⋊Q8—22no—
88C4⋊Q8—22no—
89C4⋊Q8C2×D414yes—
90#127—24no—
91C4×Q8—84no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—84no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C2×D412yes—
105C2×C22⋊C4—22no—
106C2×C22⋊C4—24no—
107C2×C42—42no—
108C2×C42C2×D412yes—
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42#4112yes—
112C2×C42C2411yescharacteristic
113#107—28no—
114C4⋊C8—22no—
115C4⋊C8—24no—
116C4⋊C8—22no—
117C4⋊C8—24no—
118#103—48no—
119#102—84no—
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—28no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—82no—
140C2×D4—82no—
141C2×D4#13514yes—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4#12112yes—
150C22×C4#12112yes—
151C22×C4C22×D411yescharacteristic
152C22×C4C22×D411yescharacteristic
153C22×C4C2×#4111yescharacteristic
154SD16—816no—
155C2×C8—24no—
156C2×C8—22no—
157C2×C8—24no—
158C2×C8—42no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—44no—
163C2×C8—88no—
164C2×C8#12114yes—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—24no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—84no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—84no—
179C42—22no—
180C42—22no—
181C42—24no—
182C42—42no—
183C42—84no—
184C42C22×D411yescharacteristic
185C42C2×#4111yescharacteristic
186C42H(5,𝔽2)11yescharacteristic
187C422−1+411yescharacteristic
188C23—24no—
189C23—44no—
190C23—42no—
191C23—42no—
192C23—44no—
193C23C2×#12111yescharacteristic
194Q8—48no—
195Q8—84no—
196D4—24no—
197D4—24no—
198D4—42no—
199D4—42no—
200D4—88no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—44no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—84no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4C2×#12111yescharacteristic
220C2×C4#53711yescharacteristic
221C2×C4D4211yescharacteristic
222C2×C4#54711yescharacteristic
223C2×C4#54711yescharacteristic
224C2×C4#55511yescharacteristic
225C2×C4C2×#13512yes—
226C2×C4#57612yes—
227C2×C4#57714yes—
228C8—24no—
229C8—42no—
230C8—84no—
231C8—82no—
232C22—44no—
233C22—48no—
234C22—84no—
235C22—82no—
236C22—82no—
237C22#253711yescharacteristic
238C22#304212yes—
239C22#305011yescharacteristic
240C22#305811yescharacteristic
241C22#306912yes—
242C4—21no—
243C4—21no—
244C4—42no—
245C4—42no—
246C4—84no—
247C4—82no—
248C4—162no—
249C4#304312yes—
250C4#326012yes—
251C4#328012yes—
252C4#333912yes—
253C2—84no—
254C2#1939512yes—
255C2#1951412yes—
256C2#2822511yescharacteristic
257C2#2824411yescharacteristic
258C2#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. #22247C2semidirect
4. C2×#1673C2semidirect
5. #10477C2semidirect
6. #8811C2semidirect
7. #8648C2semidirect
8. #8593C2semidirect
39. #500D4semidirect

Maximal subgroups

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

subgroupindex
#281572
#222602
#222472
C2×#16732
#104772
#88112
#86482
#85932