Finite Group Databasebeta

← #324602

Subgroups — #324602

Order 512.

598 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 150 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#30228C211yesmaximal, characteristic
2#28725C211yesmaximal, characteristic
3#28440C211yesmaximal, characteristic
4#22242C211yesmaximal, characteristic
5#10479C214yesmaximal
6#9899C211yesmaximal, characteristic
7#9705C211yesmaximal, characteristic
8#9226C214yesmaximal
9#8277C211yesmaximal, characteristic
10C2×#494—21no—
11C2×#491—21no—
12C2×#462C2214yes—
13C2×#446C2212yes—
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16C4×C4⋊Q8C2211yescharacteristic
17#2099—24no—
18#2074—22no—
19#2016—22no—
20#1992—21no—
21#1990C2212yes—
22#1990C2212yes—
23#1984C2212yes—
24#1956—21no—
25#1917—24no—
26#1831C2211yescharacteristic
27#1803C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#334C2212yes—
32C2×#332C2214yes—
33#1296C2214yes—
34#1277C2212yes—
35C22×Q16—24no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8D411yescharacteristic
39C2×C4⋊Q8C2312yes—
40C2×#125—21no—
41C2×C4×Q8C2312yes—
42#494—44no—
43#491—44no—
44#462—28no—
45#462—44no—
46#446—44no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8C2312yes—
49C2×C4⋊C8C2312yes—
50C2×C4⋊C8C2312yes—
51C2×#103—24no—
52C2×#103—24no—
53C2×#97—22no—
54C2×#97—21no—
55C2×C4×C8—21no—
56C2×C4×C8C2311yescharacteristic
57C2×C4×C8C2311yescharacteristic
58#391D412yes—
59#389—22no—
60#389—22no—
61#389—22no—
62#384—22no—
63#384—22no—
64#384D411yescharacteristic
65#384C2312yes—
66#382—21no—
67C4×C4⋊C4—22no—
68C43C2311yescharacteristic
69#340—24no—
70#340—24no—
71#340—24no—
72#336—24no—
73#336—21no—
74#336—22no—
75#336—21no—
76#336D414yes—
77#334—24no—
78#332—44no—
79C22×Q8—22no—
80C22×Q8C2×D412yes—
81C2×Q16—48no—
82C2×Q16—48no—
83C2×Q16—84no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8C2×D412yes—
88C22×C8C2×D412yes—
89C4⋊Q8—22no—
90C4⋊Q8—22no—
91C4⋊Q8—24no—
92C4⋊Q8—24no—
93C4⋊Q8—42no—
94C4⋊Q8—42no—
95C4⋊Q8—82no—
96#125—42no—
97#125—42no—
98C4×Q8—24no—
99C4×Q8—24no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4C2×D412yes—
112C2×C42—22no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D411yescharacteristic
116C2×C42#4112yes—
117C2×C42C2411yescharacteristic
118C4⋊C8—22no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—24no—
122C4⋊C8—44no—
123C4⋊C8#4114yes—
124#103—48no—
125#103—48no—
126#97—44no—
127#97—42no—
128C4×C8—22no—
129C4×C8—22no—
130C4×C8—42no—
131#95—24no—
132#95—42no—
133C2×Q8—24no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—24no—
137C2×Q8—44no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—44no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4#12112yes—
146C22×C4#12112yes—
147C22×C4C22×D411yescharacteristic
148C22×C4C22×D411yescharacteristic
149C22×C4C2×#4111yescharacteristic
150Q16—816no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—24no—
154C2×C8—24no—
155C2×C8—24no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—44no—
163C4⋊C4—24no—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—44no—
169C4⋊C4—42no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—82no—
173C4⋊C4—82no—
174C4⋊C4—84no—
175C4⋊C4—84no—
176C4⋊C4—84no—
177C42—22no—
178C42—24no—
179C42—42no—
180C42—42no—
181C42—42no—
182C42AGL(1,ℤ/8)12yes—
183C42AGL(1,ℤ/8)12yes—
184C42C2×#4112yes—
185C42H(5,𝔽2)12yes—
186C23C2×#12111yescharacteristic
187Q8—44no—
188Q8—44no—
189Q8—84no—
190Q8—84no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—82no—
206C2×C4—82no—
207C2×C4—82no—
208C2×C4#46914yes—
209C2×C4#53211yescharacteristic
210C2×C4#53511yescharacteristic
211C2×C4#54612yes—
212C2×C4#55912yes—
213C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
215C2×C4#57712yes—
216C2×C4#57814yes—
217C8—42no—
218C8—42no—
219C8—44no—
220C8—82no—
221C8—82no—
222C22#254411yescharacteristic
223C22#304412yes—
224C22C2×#46911yescharacteristic
225C22#305711yescharacteristic
226C22#306912yes—
227C4—22no—
228C4—24no—
229C4—42no—
230C4—82no—
231C4—82no—
232C4—162no—
233C4—162no—
234C4#317712yes—
235C4#322312yes—
236C4#332412yes—
237C4#333212yes—
238C2#1944012yes—
239C2#1952812yes—
240C2#2860711yescharacteristic
241C2#2862511yescharacteristic
242C2#2863411yescharacteristic
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
#302282
#287252
#284402
#222422
#104792
#98992
#97052
#92262
#82772