Finite Group Databasebeta

← #451127

Subgroups — #451127

Order 512.

1750 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1750 conjugacy classes of subgroups — too many to draw. Only its 198 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#35456C211yesmaximal, characteristic
2#14336C213yesmaximal
3#11197C213yesmaximal
4#11070C216yesmaximal
5#10936C212yesmaximal
6C24×Q8C2211yescharacteristic
7#2493—212no—
8#2315C2213yes—
9C2×#381C2216yes—
10C2×#381C2216yes—
11C2×#381C2213yes—
12C2×#381C2213yes—
13#2273C2211yescharacteristic
14C2×#342C2216yes—
15C2×#342C2216yes—
16#1446—212no—
17C23×Q8—26no—
18C23×Q8—212no—
19C23×Q8—46no—
20C24×C4C2312yes—
21C24×C4C2311yescharacteristic
22C22×C22⋊C4C2316yes—
23C22×C22⋊C4C2313yes—
24C22×C22⋊C4C2313yes—
25#401—24no—
26#400—212no—
27#387—212no—
28#385—26no—
29#385—26no—
30#383—24no—
31#381—26no—
32#381—26no—
33#381—212no—
34#381—212no—
35#381—46no—
36#381—46no—
37#381—46no—
38#380—26no—
39C4×C22⋊C4—26no—
40C2×#95—212no—
41C2×#95—212no—
42C2×#95—26no—
43C2×#95D4112yes—
44#342—46no—
45#342—46no—
46C25C2411yescharacteristic
47C22×Q8—224no—
48C22×Q8—424no—
49C22×Q8—424no—
50C22×Q8—424no—
51C22×Q8—412no—
52C22×Q8—424no—
53C22×Q8—48no—
54C22×Q8—44no—
55C22×Q8—48no—
56C23×C4—26no—
57C23×C4—26no—
58C23×C4—26no—
59C23×C4—26no—
60C23×C4—26no—
61C23×C4—26no—
62C23×C4—26no—
63C23×C4C2×D416yes—
64C23×C4C2×D416yes—
65C23×C4C2×D416yes—
66C23×C4#41112yes—
67C2×C4⋊C4—212no—
68C2×C22⋊C4—23no—
69C2×C22⋊C4—23no—
70C2×C22⋊C4—23no—
71C2×C22⋊C4—23no—
72C2×C22⋊C4—26no—
73C2×C22⋊C4—26no—
74C2×C22⋊C4—26no—
75C2×C22⋊C4—26no—
76C2×C22⋊C4—26no—
77C2×C22⋊C4—26no—
78C2×C22⋊C4—26no—
79C2×C22⋊C4—26no—
80C2×C22⋊C4—26no—
81C2×C22⋊C4—26no—
82C2×C42—24no—
83#95—24no—
84#95—212no—
85#95—212no—
86#95—212no—
87#95—424no—
88#95—424no—
89#95—412no—
90C24—44no—
91C24C22×D413yes—
92C24C2×#4116yes—
93C24H(5,𝔽2)16yes—
94C2×Q8—448no—
95C2×Q8—496no—
96C2×Q8—496no—
97C22×C4—26no—
98C22×C4—212no—
99C22×C4—26no—
100C22×C4—212no—
101C22×C4—212no—
102C22×C4—212no—
103C22×C4—212no—
104C22×C4—212no—
105C22×C4—212no—
106C22×C4—212no—
107C22×C4—26no—
108C22×C4—26no—
109C22×C4—212no—
110C22×C4—212no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—46no—
115C22×C4—46no—
116C22×C4—46no—
117C22×C4—46no—
118C22×C4—46no—
119C22×C4—46no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—412no—
124C22×C4—412no—
125C22×C4—424no—
126C22×C4—46no—
127C22×C4—46no—
128C22×C4—46no—
129C22×C4#12014yes—
130C22×C4#12618yes—
131C4⋊C4—812no—
132C22⋊C4—43no—
133C22⋊C4—43no—
134C22⋊C4—43no—
135C22⋊C4—43no—
136C22⋊C4—46no—
137C22⋊C4—46no—
138C22⋊C4—46no—
139C22⋊C4—46no—
140C22⋊C4—46no—
141C22⋊C4—46no—
142C42—84no—
143C23—212no—
144C23—412no—
145C23—412no—
146C23C2×#12011yescharacteristic
147C23#52513yes—
148C23C2×#12612yes—
149C23#53513yes—
150C23#53816yes—
151C23#54616yes—
152C23#54616yes—
153C23#55116yes—
154C23#56012yes—
155Q8—464no—
156C2×C4—212no—
157C2×C4—224no—
158C2×C4—412no—
159C2×C4—412no—
160C2×C4—412no—
161C2×C4—412no—
162C2×C4—412no—
163C2×C4—412no—
164C2×C4—46no—
165C2×C4—46no—
166C2×C4—412no—
167C2×C4—412no—
168C2×C4—812no—
169C2×C4—812no—
170C2×C4—812no—
171C2×C4—812no—
172C2×C4—812no—
173C2×C4—86no—
174C22—26no—
175C22—212no—
176C22—212no—
177C22—26no—
178C22—42no—
179C22—46no—
180C22—46no—
181C22—42no—
182C22#240413yes—
183C22#248016yes—
184C22#268516yes—
185C22#302814yes—
186C4—44no—
187C4—48no—
188C4—86no—
189C4—86no—
190C4—166no—
191C2—26no—
192C2—26no—
193C2#1598611yescharacteristic
194C2#2398916yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#354562
#143362
#111972
#110702
#109362