Finite Group Databasebeta

← #331529

Subgroups — #331529

Order 512.

782 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 266 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#33310C212yesmaximal
3#33289C212yesmaximal
4#33225C211yesmaximal, characteristic
5#10685C218yesmaximal
6#8131C211yesmaximal, characteristic
7C2×#498—22no—
8C2×#498—24no—
9C2×#487—22no—
10#2868C2212yes—
11#2868C2211yescharacteristic
12#2868C2211yescharacteristic
13#1990—24no—
14#1990C2218yes—
15#1956—22no—
16#1956C2214yes—
17#1844—24no—
18#1844—22no—
19#1844C2218yes—
20#1844C2214yes—
21#1834—24no—
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#498—416no—
35#487—48no—
36C2×#106—28no—
37C2×#106—216no—
38C2×#103—28no—
39C2×C4×C8D414yes—
40C2×C4×C8D412yes—
41C2×C4×C8D414yes—
42C2×C4×C8D412yes—
43C2×C4×C8C2312yes—
44C2×C4×C8C2314yes—
45#384—22no—
46#384—22no—
47#384—22no—
48#384—22no—
49#384—22no—
50#384—21no—
51#384—22no—
52#384C2312yes—
53#384C2314yes—
54C43C2311yescharacteristic
55#340—216no—
56C22×Q8—22no—
57C22×Q8—41no—
58C22×Q8—42no—
59C22×C8C2×D418yes—
60C22×C8C2×D414yes—
61C4⋊Q8—28no—
62C4⋊Q8—44no—
63C4⋊Q8—48no—
64C4⋊Q8—48no—
65C4⋊Q8—44no—
66C4⋊Q8—48no—
67C4⋊Q8—44no—
68C4⋊Q8—48no—
69C2×C4⋊C4—24no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—24no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4—24no—
74C2×C4⋊C4—24no—
75C2×C4⋊C4—42no—
76C2×C4⋊C4—42no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C42C2×D412yes—
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D412yes—
87C2×C42C2411yescharacteristic
88#106—416no—
89#106—432no—
90#103—416no—
91C4×C8—24no—
92C4×C8—28no—
93C4×C8—24no—
94C4×C8—28no—
95C4×C8SD1618yes—
96C4×C8SD16116yes—
97C2×Q8—44no—
98C2×Q8—44no—
99C2×Q8—44no—
100C2×Q8—82no—
101C2×Q8—84no—
102C2×Q8—84no—
103C2×Q8—84no—
104C2×Q8—84no—
105C22×C4—42no—
106C22×C4—44no—
107C22×C4—42no—
108C22×C4#12711yescharacteristic
109C22×C4#12712yes—
110C22×C4#12711yescharacteristic
111C22×C4C22×D411yescharacteristic
112C22×C4C22×D412yes—
113C2×C8—24no—
114C2×C8—28no—
115C2×C8—24no—
116C2×C8—28no—
117C2×C8C2×SD1618yes—
118C2×C8C2×SD16116yes—
119C4⋊C4—48no—
120C4⋊C4—44no—
121C4⋊C4—48no—
122C4⋊C4—48no—
123C4⋊C4—48no—
124C4⋊C4—48no—
125C4⋊C4—84no—
126C4⋊C4—84no—
127C4⋊C4—82no—
128C4⋊C4—82no—
129C4⋊C4—82no—
130C4⋊C4—84no—
131C4⋊C4—84no—
132C4⋊C4—84no—
133C42C2×SD1614yes—
134C42C2×SD1618yes—
135C42AGL(1,ℤ/8)14yes—
136C42#13718yes—
137C422−1+414yes—
138C23C2×#12711yescharacteristic
139Q8—88no—
140Q8—164no—
141Q8—168no—
142C2×C4—82no—
143C2×C4—84no—
144C2×C4—84no—
145C2×C4—82no—
146C2×C4—84no—
147C2×C4—82no—
148C2×C4—84no—
149C2×C4—82no—
150C2×C4#49212yes—
151C2×C4#49214yes—
152C2×C4#49214yes—
153C2×C4#49212yes—
154C2×C4#49614yes—
155C2×C4#49714yes—
156C2×C4#49714yes—
157C2×C4#53612yes—
158C2×C4#53614yes—
159C2×C4C22×SD1612yes—
160C2×C4C22×SD1614yes—
161C2×C4C2×AGL(1,ℤ/8)12yes—
162C2×C4C2×#13714yes—
163C8—28no—
164C8—216no—
165C22#265811yescharacteristic
166C22C2×#49211yescharacteristic
167C22C2×#49212yes—
168C22C2×#49611yescharacteristic
169C22C2×#49712yes—
170C4—162no—
171C4—162no—
172C4—164no—
173C4#170514yes—
174C4#171418yes—
175C4#171614yes—
176C4#322714yes—
177C4#322918yes—
178C2C2×#170511yescharacteristic
179C2C2×#171412yes—
180C2C2×#171611yescharacteristic
181C2#3027711yescharacteristic
182C2#3028012yes—
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
#106852
#81312