Finite Group Databasebeta

← #508453

Subgroups — #508453

Order 512.

544 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 170 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×#2227C212yesmaximal
2#33312C211yesmaximal, characteristic
3#21276C211yesmaximal, characteristic
4#13505C214yesmaximal
5#13481C214yesmaximal
6#13276C212yesmaximal
7#12879C211yesmaximal, characteristic
8C2×#500—21no—
9C2×#500C2212yes—
10C2×C4×Q16C2211yescharacteristic
11#2868—21no—
12#2295—21no—
13#2227—28no—
14C2×#366C2214yes—
15C2×C4⋊C16C2212yes—
16C2×#358C2214yes—
17#1969C2212yes—
18#1942C2212yes—
19#1933—22no—
20#1845—22no—
21#1845C2214yes—
22C4×#107C2211yescharacteristic
23C4×#103—21no—
24C42×C8C2211yescharacteristic
25#1380C2218yes—
26#1374C2214yes—
27C22×Q16C2311yescharacteristic
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—42no—
30C2×C4×Q8—21no—
31C22×C16C2314yes—
32#500—24no—
33#500—24no—
34#500—44no—
35C4×Q16—22no—
36C4×Q16D414yes—
37C2×#107—22no—
38C2×#107—24no—
39C2×#107C2314yes—
40C2×#107C2312yes—
41C2×#107C2311yescharacteristic
42C2×#103—21no—
43C2×#103—22no—
44C2×C4×C8D412yes—
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2312yes—
47C2×C4×C8C2311yescharacteristic
48#391—22no—
49#389—22no—
50#387—42no—
51#384—24no—
52#384—41no—
53#384—41no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—41no—
56#376—41no—
57C43D411yescharacteristic
58#366—28no—
59C4⋊C16—24no—
60#358—28no—
61#340—21no—
62#340—22no—
63C22×Q8—21no—
64C22×Q8—42no—
65C2×Q16—21no—
66C2×Q16—24no—
67C2×Q16C2×D412yes—
68C22×C8C2×D411yescharacteristic
69C22×C8C2×D412yes—
70C22×C8C2411yescharacteristic
71C4⋊Q8—44no—
72C4⋊Q8—44no—
73C4⋊Q8—88no—
74C4×Q8—24no—
75C4×Q8—42no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—42no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—42no—
84C2×C42—41no—
85C2×C42—41no—
86C2×C42D812yes—
87C2×C42D811yescharacteristic
88C2×C42D811yescharacteristic
89C2×C42C2×D412yes—
90C2×C42C2×D411yescharacteristic
91C2×C16—24no—
92C2×C16—28no—
93#107—21no—
94#107—24no—
95#107—24no—
96#107—24no—
97#107—48no—
98#107—44no—
99#107C2×D412yes—
100#103—24no—
101#103—44no—
102C4×C8—24no—
103C4×C8—24no—
104C4×C8—22no—
105C4×C8C2×D412yes—
106C4×C8#4112yes—
107#95—42no—
108#95—42no—
109#95—41no—
110#95—42no—
111C2×Q8—22no—
112C2×Q8—41no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8—88no—
116C2×Q8—84no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—44no—
121C22×C4C2×D811yescharacteristic
122C22×C4C2×D812yes—
123C22×C4C2×D812yes—
124C22×C4C2×D811yescharacteristic
125C22×C4C22×D411yescharacteristic
126Q16—24no—
127Q16—42no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8#12112yes—
132C2×C8#13714yes—
133C2×C8C22×D411yescharacteristic
134C2×C8C2×#4111yescharacteristic
135C2×C82−1+414yes—
136C4⋊C4—22no—
137C4⋊C4—41no—
138C4⋊C4—44no—
139C4⋊C4—44no—
140C4⋊C4—42no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—88no—
144C4⋊C4—84no—
145C4⋊C4—84no—
146C4⋊C4—88no—
147C42—22no—
148C42—22no—
149C42—24no—
150C42—22no—
151C42—44no—
152C42—82no—
153C42Q3214yes—
154C42#12112yes—
155C42AGL(1,ℤ/8)12yes—
156C16—44no—
157C23C22×D811yescharacteristic
158Q8—42no—
159Q8—42no—
160Q8—82no—
161Q8—168no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—88no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—84no—
176C2×C4#45912yes—
177C2×C4#45912yes—
178C2×C4C2×Q3214yes—
179C2×C4C2×Q3212yes—
180C2×C4#50914yes—
181C2×C4#51012yes—
182C2×C4C2×#12111yescharacteristic
183C2×C4#53612yes—
184C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C4C2×#13712yes—
186C8—24no—
187C8—22no—
188C8#53612yes—
189C8#54112yes—
190C22C2×#45911yescharacteristic
191C22#322612yes—
192C22C22×Q3211yescharacteristic
193C22C2×#50912yes—
194C22C2×#51011yescharacteristic
195C4—24no—
196C4—24no—
197C4—22no—
198C4—82no—
199C4—82no—
200C4—162no—
201C4—164no—
202C4#220914yes—
203C4#262111yescharacteristic
204C4#303911yescharacteristic
205C4#323912yes—
206C2#2986411yescharacteristic
207C2C2×#220911yescharacteristic
208C2#3358211yescharacteristic
209C2#3379612yes—
210C2#3380412yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ 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×#22272
#333122
#212762
#135052
#134812
#132762
#128792