Finite Group Databasebeta

← #510048

Subgroups — #510048

Order 512.

501 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 144 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
1C2×#2229C212yesmaximal
2#26483C212yesmaximal
3#13475C212yesmaximal
4#13475C212yesmaximal
5#13453C212yesmaximal
6#13240C211yesmaximal, characteristic
7#13167C211yesmaximal, characteristic
8#13160C211yesmaximal, characteristic
9#13015C212yesmaximal
10C22×Q32C2211yescharacteristic
11C2×#499C2212yes—
12C2×C4×Q16C2212yes—
13#2453—22no—
14#2229—28no—
15C2×#367C2211yescharacteristic
16C2×#367C2211yescharacteristic
17C2×#366C2212yes—
18C2×#366C2211yescharacteristic
19C2×C4⋊C16C2212yes—
20C2×C4⋊C16C2211yescharacteristic
21C2×#358C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#346C2211yescharacteristic
25#2066—22no—
26#2026—22no—
27#1969C2212yes—
28#1969C2212yes—
29#1942C2212yes—
30#1942C2212yes—
31#1865—22no—
32C8×C4⋊C4C2211yescharacteristic
33#1380C2212yes—
34#1380C2212yes—
35#1374C2212yes—
36C22×Q16C2312yes—
37C2×#125—22no—
38C2×C4×Q8—22no—
39C2×Q32—41no—
40C2×Q32—41no—
41C2×Q32Q814yes—
42C22×C16C2312yes—
43C22×C16C2311yescharacteristic
44C22×C16C2311yescharacteristic
45#499—24no—
46#499—42no—
47C4×Q16—24no—
48C4×Q16—42no—
49C2×#107C2312yes—
50C2×#107C2312yes—
51C2×#107C2312yes—
52C2×#106—22no—
53C2×C4⋊C8D411yescharacteristic
54C2×#103—22no—
55C2×#103—22no—
56C2×#103—22no—
57C2×C4×C8C2311yescharacteristic
58C2×C4×C8C2312yes—
59#400—42no—
60#398—42no—
61#391—22no—
62#391—22no—
63#389—22no—
64#389—22no—
65#389—42no—
66#387—42no—
67C4×C4⋊C4D411yescharacteristic
68#367—41no—
69#367D414yes—
70#366—24no—
71#366—41no—
72C4⋊C16—24no—
73C4⋊C16—41no—
74#358—24no—
75#358—24no—
76#358—42no—
77#346—41no—
78#340—22no—
79#340—22no—
80#336D412yes—
81C22×Q8—22no—
82C2×Q16—22no—
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—22no—
86C2×Q16—22no—
87C2×Q16C2×Q814yes—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D412yes—
90C22×C8C2411yescharacteristic
91#125—44no—
92#125—82no—
93C4×Q8—44no—
94C4×Q8—82no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—42no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4D812yes—
104C2×C42—42no—
105C2×C42C2×D412yes—
106C2×C42C2×D411yescharacteristic
107Q32—44no—
108C2×C16—21no—
109C2×C16—21no—
110C2×C16—21no—
111C2×C16—21no—
112C2×C16—22no—
113C2×C16—22no—
114C2×C16—22no—
115C2×C16C2×D412yes—
116C2×C16C2×Q812yes—
117#107—22no—
118#107—22no—
119#107—22no—
120#107—22no—
121#107—22no—
122#107C2×D414yes—
123#106—44no—
124C4⋊C8—41no—
125#103—44no—
126#103—44no—
127#103—82no—
128C4×C8—22no—
129C4×C8—21no—
130C4×C8—22no—
131C4×C8—21no—
132#95—42no—
133#95—42no—
134#95—42no—
135#95—42no—
136#95—42no—
137#95—42no—
138#95D811yescharacteristic
139#95D811yescharacteristic
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4C2×D811yescharacteristic
151C22×C4C2×D812yes—
152C22×C4C2×D811yescharacteristic
153C22×C4C2×D812yes—
154C22×C4C22×D411yescharacteristic
155Q16—24no—
156Q16—24no—
157Q16—42no—
158Q16—42no—
159C2×C8—21no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—22no—
165C2×C8C22×D411yescharacteristic
166C2×C8C22×Q811yescharacteristic
167C2×C8H(5,𝔽2)11yescharacteristic
168C2×C8H(5,𝔽2)11yescharacteristic
169C2×C82−1+411yescharacteristic
170C2×C82−1+411yescharacteristic
171C4⋊C4—24no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—84no—
179C4⋊C4—84no—
180C4⋊C4—84no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C42—21no—
184C42—22no—
185C42—21no—
186C42—22no—
187C42—84no—
188C16—22no—
189C16—22no—
190C16—42no—
191C23C22×D811yescharacteristic
192Q8—44no—
193Q8—44no—
194Q8—82no—
195Q8—82no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—44no—
207C2×C4—44no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4#53711yescharacteristic
219C2×C4D4211yescharacteristic
220C2×C4D4×Q811yescharacteristic
221C2×C4#57511yescharacteristic
222C2×C4#57611yescharacteristic
223C2×C4#57811yescharacteristic
224C8—41no—
225C8—42no—
226C8#53711yescharacteristic
227C8#54311yescharacteristic
228C8D4×Q811yescharacteristic
229C8#56311yescharacteristic
230C22#324511yescharacteristic
231C22D4×D811yescharacteristic
232C22Q8×D811yescharacteristic
233C22#341412yes—
234C22#341712yes—
235C4—41no—
236C4—41no—
237C4—42no—
238C4—42no—
239C4—84no—
240C4—84no—
241C4—162no—
242C4—162no—
243C4#268311yescharacteristic
244C4#326211yescharacteristic
245C4#327711yescharacteristic
246C4#338911yescharacteristic
247C2#3123411yescharacteristic
248C2#3382211yescharacteristic
249C2#3382711yescharacteristic
250C2#3383512yes—
251C2#3395712yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#22292
#264832
#134752
#134532
#132402
#131672
#131602
#130152