Finite Group Databasebeta

← #330196

Subgroups — #330196

Order 512.

678 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 678 conjugacy classes of subgroups — too many to draw. Only its 162 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#33302C211yesmaximal, characteristic
2#29866C212yesmaximal
3#21341C211yesmaximal, characteristic
4#10538C214yesmaximal
5#10276C214yesmaximal
6#9705C212yesmaximal
7#8785C211yesmaximal, characteristic
8C2×#494—22no—
9C2×#494—21no—
10C2×#494—21no—
11C2×#494C2214yes—
12C2×#475C2214yes—
13#2868C2212yes—
14#2295C2211yescharacteristic
15#2084—24no—
16#2072—24no—
17#1988C2214yes—
18#1984C2212yes—
19#1939C2212yes—
20#1916—22no—
21#1831C2212yes—
22#1768—21no—
23#1767C2211yescharacteristic
24#1746—21no—
25C42×C8C2211yescharacteristic
26C2×#332C2218yes—
27#1277C2214yes—
28C22×Q16—24no—
29C22×Q16—24no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8—22no—
32C2×C4⋊Q8C2314yes—
33C2×C4×Q8C2311yescharacteristic
34#494—48no—
35#494—48no—
36#494—44no—
37#494—44no—
38#475—48no—
39C2×#106—24no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8C2314yes—
42C2×C4⋊C8C2312yes—
43C2×#103—24no—
44C2×#97—21no—
45C2×C4×C8D412yes—
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8C2312yes—
49#391D412yes—
50#389D412yes—
51#387—22no—
52#384—24no—
53#384—24no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2311yescharacteristic
56#376—21no—
57C43C2311yescharacteristic
58#340—28no—
59#336—22no—
60#336—22no—
61#336—22no—
62#336—24no—
63#336—21no—
64#332—48no—
65C22×Q8—24no—
66C22×Q8C2×D411yescharacteristic
67C2×Q16—48no—
68C2×Q16—48no—
69C2×Q16—48no—
70C2×Q16—48no—
71C2×Q16—48no—
72C2×Q16—48no—
73C22×C8—22no—
74C22×C8—24no—
75C22×C8C2×D412yes—
76C22×C8C2×D412yes—
77C4⋊Q8—28no—
78C4⋊Q8—28no—
79C4⋊Q8—48no—
80C4⋊Q8—44no—
81C4⋊Q8—44no—
82C4×Q8—24no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4C2×D411yescharacteristic
90C2×C4⋊C4C2×D412yes—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C2×D411yescharacteristic
94C2×C42C2×D412yes—
95C2×C42C2×D411yescharacteristic
96C2×C42#4112yes—
97C2×C42C2411yescharacteristic
98#106—48no—
99C4⋊C8—24no—
100C4⋊C8—28no—
101C4⋊C8—44no—
102#103—48no—
103#97—42no—
104C4×C8—24no—
105C4×C8—22no—
106C4×C8—24no—
107C4×C8SD1614yes—
108#95—22no—
109#95—22no—
110#95—22no—
111#95—21no—
112C2×Q8—22no—
113C2×Q8—24no—
114C2×Q8—48no—
115C2×Q8—44no—
116C2×Q8—44no—
117C2×Q8—48no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—44no—
122C22×C4#12112yes—
123C22×C4#12112yes—
124C22×C4C22×D411yescharacteristic
125C22×C4C22×D411yescharacteristic
126C22×C4C2×#4111yescharacteristic
127Q16—816no—
128Q16—816no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—44no—
134C2×C8—42no—
135C2×C8—42no—
136C2×C8—42no—
137C2×C8—48no—
138C2×C8C2×SD1614yes—
139C2×C8AGL(1,ℤ/8)14yes—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—44no—
145C4⋊C4—44no—
146C4⋊C4—48no—
147C4⋊C4—88no—
148C4⋊C4—88no—
149C42—22no—
150C42—24no—
151C42—24no—
152C42—42no—
153C42—42no—
154C42C2×SD1612yes—
155C42AGL(1,ℤ/8)12yes—
156C42H(5,𝔽2)14yes—
157C23C2×#12111yescharacteristic
158Q8—44no—
159Q8—88no—
160Q8—88no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—84no—
174C2×C4—84no—
175C2×C4—84no—
176C2×C4#46014yes—
177C2×C4#47314yes—
178C2×C4#53512yes—
179C2×C4#53512yes—
180C2×C4#54012yes—
181C2×C4C22×SD1611yescharacteristic
182C2×C4C2×AGL(1,ℤ/8)12yes—
183C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C4#57814yes—
185C8—24no—
186C8—24no—
187C8—82no—
188C8—84no—
189C22#261911yescharacteristic
190C22C2×#46011yescharacteristic
191C22#303811yescharacteristic
192C22#305912yes—
193C22C2×#47312yes—
194C4—24no—
195C4—22no—
196C4—24no—
197C4—81no—
198C4—81no—
199C4—82no—
200C4—164no—
201C4#320812yes—
202C4#321712yes—
203C4#322314yes—
204C2#1920412yes—
205C2#1935412yes—
206C2#2979611yescharacteristic
207C2#2981211yescharacteristic
208C2#2982011yescharacteristic
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
#333022
#298662
#213412
#105382
#102762
#97052
#87852