Finite Group Databasebeta

← #324592

Subgroups — #324592

Order 512.

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

normal subgroups only

This group has 588 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#29150C211yesmaximal, characteristic
3#28440C211yesmaximal, characteristic
4#22242C211yesmaximal, characteristic
5#10478C214yesmaximal
6#9897C211yesmaximal, characteristic
7#9705C211yesmaximal, characteristic
8#9226C214yesmaximal
9#8278C211yesmaximal, characteristic
10C2×#494—21no—
11C2×#462C2214yes—
12C2×#446C2212yes—
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2099—24no—
17#2026—22no—
18#2016—22no—
19#1990C2212yes—
20#1990C2212yes—
21#1984C2212yes—
22#1960—21no—
23#1956—21no—
24#1917—24no—
25#1836—21no—
26#1831C2211yescharacteristic
27#1803C2214yes—
28#1767C2211yescharacteristic
29C4×C4⋊C8C2211yescharacteristic
30#1744C2211yescharacteristic
31C2×#335C2212yes—
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#462—28no—
44#462—44no—
45#446—44no—
46C2×C4⋊C8—22no—
47C2×C4⋊C8C2312yes—
48C2×C4⋊C8C2312yes—
49C2×C4⋊C8C2312yes—
50C2×#103—24no—
51C2×#103—24no—
52C2×#97—22no—
53C2×#97—21no—
54C2×C4×C8—21no—
55C2×C4×C8C2311yescharacteristic
56C2×C4×C8C2311yescharacteristic
57#391D412yes—
58#389—22no—
59#389—22no—
60#389—22no—
61#384—22no—
62#384—22no—
63#384D411yescharacteristic
64#384C2312yes—
65#382—21no—
66C4×C4⋊C4—22no—
67C43C2311yescharacteristic
68#340—24no—
69#340—24no—
70#340—24no—
71#336—24no—
72#336—22no—
73#336—21no—
74#336—21no—
75#336D414yes—
76#335—24no—
77#332—44no—
78C22×Q8—22no—
79C22×Q8C2×D412yes—
80C2×Q16—48no—
81C2×Q16—48no—
82C2×Q16—84no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8C2×D412yes—
87C22×C8C2×D412yes—
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—82no—
95#125—82no—
96C4×Q8—24no—
97C4×Q8—24no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
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—42no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4C2×D412yes—
110C2×C42—22no—
111C2×C42C2×D411yescharacteristic
112C2×C42C2×D412yes—
113C2×C42C2×D411yescharacteristic
114C2×C42#4112yes—
115C2×C42C2411yescharacteristic
116C4⋊C8—22no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119C4⋊C8—24no—
120C4⋊C8—44no—
121C4⋊C8#4114yes—
122#103—48no—
123#103—84no—
124#97—44no—
125#97—42no—
126C4×C8—22no—
127C4×C8—22no—
128C4×C8—42no—
129#95—24no—
130#95—42no—
131C2×Q8—24no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—24no—
135C2×Q8—44no—
136C2×Q8—42no—
137C2×Q8—42no—
138C2×Q8—44no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4#12112yes—
144C22×C4#12112yes—
145C22×C4C22×D411yescharacteristic
146C22×C4C22×D411yescharacteristic
147C22×C4C2×#4111yescharacteristic
148Q16—816no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—24no—
152C2×C8—24no—
153C2×C8—24no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—42no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—44no—
161C4⋊C4—24no—
162C4⋊C4—44no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C4⋊C4—44no—
169C4⋊C4—82no—
170C4⋊C4—82no—
171C4⋊C4—84no—
172C4⋊C4—84no—
173C4⋊C4—84no—
174C42—22no—
175C42—24no—
176C42—42no—
177C42—42no—
178C42—42no—
179C42AGL(1,ℤ/8)12yes—
180C42#13712yes—
181C42C2×#4112yes—
182C42H(5,𝔽2)12yes—
183C23C2×#12111yescharacteristic
184Q8—44no—
185Q8—44no—
186Q8—84no—
187Q8—84no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#46814yes—
206C2×C4#53211yescharacteristic
207C2×C4#53511yescharacteristic
208C2×C4#54612yes—
209C2×C4#55912yes—
210C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
211C2×C4C2×#13711yescharacteristic
212C2×C4#57612yes—
213C2×C4#57814yes—
214C8—42no—
215C8—42no—
216C8—44no—
217C8—82no—
218C8—82no—
219C22#254411yescharacteristic
220C22#304412yes—
221C22C2×#46811yescharacteristic
222C22#305811yescharacteristic
223C22#306912yes—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—82no—
228C4—82no—
229C4—162no—
230C4—162no—
231C4#317712yes—
232C4#322312yes—
233C4#332512yes—
234C4#333312yes—
235C2#1943812yes—
236C2#1953012yes—
237C2#2860511yescharacteristic
238C2#2862511yescharacteristic
239C2#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
#291502
#284402
#222422
#104782
#98972
#97052
#92262
#82782