Finite Group Databasebeta

← #331530

Subgroups — #331530

Order 512.

710 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 234 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#33310C211yesmaximal, characteristic
2#33289C212yesmaximal
3#33289C212yesmaximal
4#33225C211yesmaximal, characteristic
5#10686C218yesmaximal
6#8131C211yesmaximal, characteristic
7C2×#498—22no—
8C2×#487—22no—
9#2868C2212yes—
10#2868C2211yescharacteristic
11#2868C2211yescharacteristic
12#1990—24no—
13#1990—24no—
14#1990—24no—
15#1990C2218yes—
16#1956—22no—
17#1956C2214yes—
18#1844—22no—
19#1844C2214yes—
20#1834—24no—
21#1834C2218yes—
22C42×C8C2212yes—
23C42×C8C2211yescharacteristic
24#1273C2214yes—
25C2×C4⋊Q8—21no—
26C2×C4⋊Q8—22no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8C2312yes—
33#498—48no—
34#487—48no—
35C2×#106—28no—
36C2×#103—28no—
37C2×C4×C8D414yes—
38C2×C4×C8D412yes—
39C2×C4×C8D414yes—
40C2×C4×C8D412yes—
41C2×C4×C8C2314yes—
42C2×C4×C8C2312yes—
43#384—22no—
44#384—22no—
45#384—22no—
46#384—21no—
47#384—22no—
48#384—22no—
49#384—22no—
50#384C2312yes—
51#384C2314yes—
52C43C2311yescharacteristic
53#340—216no—
54#340—216no—
55C22×Q8—22no—
56C22×Q8—41no—
57C22×Q8—42no—
58C22×C8C2×D418yes—
59C22×C8C2×D414yes—
60C4⋊Q8—28no—
61C4⋊Q8—44no—
62C4⋊Q8—44no—
63C4⋊Q8—48no—
64C4⋊Q8—48no—
65C4⋊Q8—48no—
66C4⋊Q8—44no—
67C4⋊Q8—48no—
68C2×C4⋊C4—24no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—42no—
78C2×C4⋊C4—42no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C42C2×D411yescharacteristic
83C2×C42C2×D412yes—
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D412yes—
86C2×C42C2411yescharacteristic
87#106—416no—
88#103—416no—
89C4×C8—28no—
90C4×C8—24no—
91C4×C8—28no—
92C4×C8—24no—
93C4×C8—28no—
94C4×C8SD1618yes—
95C2×Q8—44no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—82no—
99C2×Q8—84no—
100C2×Q8—84no—
101C2×Q8—84no—
102C2×Q8—84no—
103C22×C4—42no—
104C22×C4—44no—
105C22×C4—42no—
106C22×C4#12711yescharacteristic
107C22×C4#12711yescharacteristic
108C22×C4#12712yes—
109C22×C4C22×D411yescharacteristic
110C22×C4C22×D412yes—
111C2×C8—24no—
112C2×C8—24no—
113C2×C8—28no—
114C2×C8—28no—
115C2×C8—28no—
116C2×C8C2×SD1618yes—
117C4⋊C4—48no—
118C4⋊C4—44no—
119C4⋊C4—48no—
120C4⋊C4—48no—
121C4⋊C4—48no—
122C4⋊C4—48no—
123C4⋊C4—82no—
124C4⋊C4—82no—
125C4⋊C4—82no—
126C4⋊C4—84no—
127C4⋊C4—84no—
128C4⋊C4—84no—
129C4⋊C4—84no—
130C4⋊C4—84no—
131C42C2×SD1614yes—
132C42#13518yes—
133C42AGL(1,ℤ/8)14yes—
134C42#13718yes—
135C422−1+414yes—
136C23C2×#12711yescharacteristic
137Q8—88no—
138Q8—164no—
139Q8—168no—
140C2×C4—82no—
141C2×C4—82no—
142C2×C4—84no—
143C2×C4—84no—
144C2×C4—84no—
145C2×C4—82no—
146C2×C4—84no—
147C2×C4—82no—
148C2×C4#49212yes—
149C2×C4#49212yes—
150C2×C4#49514yes—
151C2×C4#49514yes—
152C2×C4#49614yes—
153C2×C4#49714yes—
154C2×C4#49714yes—
155C2×C4#53612yes—
156C2×C4#53614yes—
157C2×C4C22×SD1612yes—
158C2×C4C2×#13514yes—
159C2×C4C2×AGL(1,ℤ/8)12yes—
160C2×C4C2×#13714yes—
161C8—28no—
162C8—48no—
163C22#265811yescharacteristic
164C22C2×#49211yescharacteristic
165C22C2×#49512yes—
166C22C2×#49611yescharacteristic
167C22C2×#49712yes—
168C4—162no—
169C4—162no—
170C4—164no—
171C4#170814yes—
172C4#171614yes—
173C4#171918yes—
174C4#322714yes—
175C4#323118yes—
176C2C2×#170811yescharacteristic
177C2C2×#171611yescharacteristic
178C2C2×#171912yes—
179C2#3027711yescharacteristic
180C2#3028112yes—
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#333102
#332892
#332252
#106862
#81312