Finite Group Databasebeta

← #508450

Subgroups — #508450

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×#2225C212yesmaximal
2#33312C211yesmaximal, characteristic
3#21276C211yesmaximal, characteristic
4#13506C214yesmaximal
5#13481C214yesmaximal
6#13275C212yesmaximal
7#12879C211yesmaximal, characteristic
8C2×#500—21no—
9C2×#500C2212yes—
10C2×C4×Q16C2211yescharacteristic
11#2868—21no—
12#2295—21no—
13#2225—28no—
14C2×#367C2214yes—
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—41no—
30C2×C4⋊Q8—41no—
31C2×C4×Q8—21no—
32C22×C16C2314yes—
33#500—24no—
34#500—24no—
35#500—44no—
36C4×Q16—22no—
37C4×Q16D414yes—
38C2×#107—22no—
39C2×#107—24no—
40C2×#107C2314yes—
41C2×#107C2312yes—
42C2×#107C2311yescharacteristic
43C2×#103—21no—
44C2×#103—22no—
45C2×C4×C8D412yes—
46C2×C4×C8D411yescharacteristic
47C2×C4×C8C2312yes—
48C2×C4×C8C2311yescharacteristic
49#391—22no—
50#389—22no—
51#387—42no—
52#384—24no—
53#384—42no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—41no—
56#376—41no—
57C43D411yescharacteristic
58#367—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—84no—
74C4⋊Q8—84no—
75C4×Q8—24no—
76C4×Q8—42no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—44no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—42no—
85C2×C42—41no—
86C2×C42—41no—
87C2×C42D811yescharacteristic
88C2×C42D812yes—
89C2×C42D811yescharacteristic
90C2×C42C2×D412yes—
91C2×C42C2×D411yescharacteristic
92C2×C16—24no—
93C2×C16—28no—
94#107—21no—
95#107—24no—
96#107—24no—
97#107—24no—
98#107—48no—
99#107—44no—
100#107C2×D412yes—
101#103—24no—
102#103—44no—
103C4×C8—24no—
104C4×C8—24no—
105C4×C8—22no—
106C4×C8C2×D412yes—
107C4×C8#4112yes—
108#95—42no—
109#95—42no—
110#95—41no—
111#95—42no—
112C2×Q8—22no—
113C2×Q8—41no—
114C2×Q8—42no—
115C2×Q8—42no—
116C2×Q8—84no—
117C2×Q8—84no—
118C2×Q8—84no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4C2×D811yescharacteristic
124C22×C4C2×D812yes—
125C22×C4C2×D812yes—
126C22×C4C2×D811yescharacteristic
127C22×C4C22×D411yescharacteristic
128Q16—24no—
129Q16—42no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8#12112yes—
134C2×C8#13714yes—
135C2×C8C22×D411yescharacteristic
136C2×C8C2×#4111yescharacteristic
137C2×C82−1+414yes—
138C4⋊C4—22no—
139C4⋊C4—41no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—88no—
146C4⋊C4—84no—
147C4⋊C4—88no—
148C4⋊C4—84no—
149C42—22no—
150C42—22no—
151C42—22no—
152C42—24no—
153C42—44no—
154C42—82no—
155C42SD3214yes—
156C42#12112yes—
157C42AGL(1,ℤ/8)12yes—
158C16—44no—
159C23C22×D811yescharacteristic
160Q8—42no—
161Q8—42no—
162Q8—82no—
163Q8—168no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—84no—
174C2×C4—82no—
175C2×C4—84no—
176C2×C4—82no—
177C2×C4—82no—
178C2×C4—84no—
179C2×C4#45912yes—
180C2×C4#45912yes—
181C2×C4C2×SD3214yes—
182C2×C4C2×SD3212yes—
183C2×C4#50912yes—
184C2×C4#51014yes—
185C2×C4C2×#12111yescharacteristic
186C2×C4#53612yes—
187C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
188C2×C4C2×#13712yes—
189C8—24no—
190C8—22no—
191C8#53612yes—
192C8#54112yes—
193C22C2×#45911yescharacteristic
194C22#322612yes—
195C22C22×SD3211yescharacteristic
196C22C2×#50911yescharacteristic
197C22C2×#51012yes—
198C4—24no—
199C4—24no—
200C4—22no—
201C4—82no—
202C4—82no—
203C4—162no—
204C4—164no—
205C4#220714yes—
206C4#262111yescharacteristic
207C4#303911yescharacteristic
208C4#323912yes—
209C2#2986411yescharacteristic
210C2C2×#220711yescharacteristic
211C2#3358311yescharacteristic
212C2#3379512yes—
213C2#3380512yes—
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×#22252
#333122
#212762
#135062
#134812
#132752
#128792