Finite Group Databasebeta

← #318363

Subgroups — #318363

Order 512.

658 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 170 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#28255C212yesmaximal
2#22247C211yesmaximal, characteristic
3#22238C211yesmaximal, characteristic
4C2×#1674C212yesmaximal
5#10478C214yesmaximal
6#8810C211yesmaximal, characteristic
7#8649C212yesmaximal
8#8589C212yesmaximal
9C2×#500C2211yescharacteristic
10C2×#462—22no—
11C2×#462C2214yes—
12C2×C8×Q8C2212yes—
13C4×C4⋊Q8C2212yes—
14C4×C4⋊Q8C2211yescharacteristic
15#2056—24no—
16#1990C2212yes—
17#1984C2212yes—
18#1918—24no—
19#1916—24no—
20#1842C2212yes—
21#1834C2211yescharacteristic
22#1831C2211yescharacteristic
23#1803—22no—
24#1803C2214yes—
25#1744C2211yescharacteristic
26#1674—28no—
27C2×#335C2212yes—
28C2×#335C2212yes—
29C2×#332C2214yes—
30#1296C2214yes—
31C22×Q16—24no—
32C2×C4⋊Q8D411yescharacteristic
33C2×C4⋊Q8C2312yes—
34C2×C4×Q8—22no—
35C2×C4×Q8C2312yes—
36#500—22no—
37#500D414yes—
38#462—28no—
39#462—44no—
40#462—48no—
41C8×Q8—24no—
42C8×Q8—24no—
43C2×#107—22no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2314yes—
49C2×#103—24no—
50C2×#103—24no—
51C2×C4×C8—22no—
52C2×C4×C8C2311yescharacteristic
53C2×C4×C8C2311yescharacteristic
54#391—22no—
55#391D412yes—
56#389—22no—
57#389—22no—
58#389—22no—
59#384D411yescharacteristic
60#384C2312yes—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C43C2311yescharacteristic
65#340—22no—
66#340—24no—
67#340—24no—
68#336—24no—
69#336—22no—
70#336D414yes—
71#335—24no—
72#335—42no—
73#332—28no—
74C22×Q8—22no—
75C22×Q8C2×D412yes—
76C2×Q16—48no—
77C2×Q16—48no—
78C2×Q16—84no—
79C22×C8—24no—
80C22×C8—22no—
81C22×C8C2×D412yes—
82C22×C8C2×D412yes—
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87C4⋊Q8D814yes—
88C4⋊Q8C2×D414yes—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—44no—
94C4×Q8—44no—
95C2×C4⋊C4—22no—
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—22no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4C2×D412yes—
105C2×C42—22no—
106C2×C42—42no—
107C2×C42C2×D412yes—
108C2×C42C2×D411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42#4112yes—
111C2×C42C2411yescharacteristic
112#107—28no—
113C4⋊C8—22no—
114C4⋊C8—24no—
115C4⋊C8—22no—
116C4⋊C8—24no—
117C4⋊C8—44no—
118C4⋊C8—44no—
119C4⋊C8—82no—
120#103—48no—
121#103—48no—
122C4×C8—21no—
123C4×C8—21no—
124C4×C8—44no—
125C4×C8C2×D412yes—
126C4×C8#4112yes—
127#95—24no—
128#95—42no—
129#95—42no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—24no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8C2×D814yes—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4#12112yes—
147C22×C4#12112yes—
148C22×C4C22×D411yescharacteristic
149C22×C4C22×D411yescharacteristic
150C22×C4C2×#4111yescharacteristic
151Q16—816no—
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#12114yes—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—84no—
176C42—21no—
177C42—24no—
178C42—21no—
179C42—42no—
180C42—42no—
181C42—42no—
182C42—84no—
183C42C2×D812yes—
184C42#13512yes—
185C42C22×D411yescharacteristic
186C42C2×#4111yescharacteristic
187C42H(5,𝔽2)11yescharacteristic
188C422−1+411yescharacteristic
189C23C2×#12111yescharacteristic
190Q8—24no—
191Q8—24no—
192Q8—42no—
193Q8—42no—
194Q8—44no—
195Q8—44no—
196Q8—82no—
197Q8—82no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4#46614yes—
216C2×C4C2×#12111yescharacteristic
217C2×C4#53711yescharacteristic
218C2×C4D4211yescharacteristic
219C2×C4#54711yescharacteristic
220C2×C4#54711yescharacteristic
221C2×C4#55511yescharacteristic
222C2×C4C22×D811yescharacteristic
223C2×C4C2×#13511yescharacteristic
224C2×C4#57611yescharacteristic
225C2×C4#57811yescharacteristic
226C2×C4#57814yes—
227C8—24no—
228C8—42no—
229C8—84no—
230C8—82no—
231C22#253711yescharacteristic
232C22#304412yes—
233C22C2×#46611yescharacteristic
234C22#306111yescharacteristic
235C22#306912yes—
236C4—21no—
237C4—21no—
238C4—24no—
239C4—44no—
240C4—42no—
241C4—84no—
242C4—82no—
243C4—162no—
244C4#304412yes—
245C4#326112yes—
246C4D4×D811yescharacteristic
247C4#328311yescharacteristic
248C4#333411yescharacteristic
249C4#334111yescharacteristic
250C2#1940412yes—
251C2#1953412yes—
252C2#2822711yescharacteristic
253C2#2824411yescharacteristic
254C2#2826111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#282552
#222472
#222382
C2×#16742
#104782
#88102
#86492
#85892