Finite Group Databasebeta

← #508452

Subgroups — #508452

Order 512.

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

normal subgroups only

This group has 582 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×#2226C212yesmaximal
2#33312C211yesmaximal, characteristic, complemented
3#21275C211yesmaximal, characteristic
4#13506C214yesmaximal, complemented
5#13480C214yesmaximal
6#13276C212yesmaximal, complemented
7#12879C211yesmaximal, characteristic, complemented
8C2×#500—21no—
9C2×#500C2212yes—
10C2×C4×D8C2211yescharacteristic
11#2868—21no—
12#2293—21no—
13#2226—28no—
14C2×#367C2214yes—
15C2×C4⋊C16C2212yes—
16C2×#357C2214yes—
17#1970C2212yes—
18#1942C2212yes—
19#1934—22no—
20#1845—22no—
21#1845C2214yes—
22C4×#107C2211yescharacteristic
23C4×#102—21no—
24C42×C8C2211yescharacteristic
25#1380C2218yes—
26#1374C2214yes—
27C22×D8C2311yescharacteristic
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—42no—
30C2×C4×D4—21no—
31C2×#117—41no—
32C22×C16C2314yes—
33#500—24no—
34#500—24no—
35#500—44no—
36C4×D8—22no—
37C4×D8D414yes—
38C2×#107—22no—
39C2×#107—24no—
40C2×#107C2314yes—
41C2×#107C2312yes—
42C2×#107C2311yescharacteristic
43C2×#102—21no—
44C2×#102—22no—
45C2×C4×C8D412yes—
46C2×C4×C8D411yescharacteristic
47C2×C4×C8C2312yes—
48C2×C4×C8C2311yescharacteristic
49#390—22no—
50#389—22no—
51#388—42no—
52#384—24no—
53#384—41no—
54#384—41no—
55C4×C4⋊C4—21no—
56C4×C22⋊C4—41no—
57C43D411yescharacteristic
58#367—28no—
59C4⋊C16—24no—
60#357—28no—
61#340—21no—
62#340—22no—
63C22×Q8—42no—
64C22×D4—21no—
65C23×C4—41no—
66C2×D8—21no—
67C2×D8—24no—
68C2×D8C2×D412yes—
69C22×C8C2×D411yescharacteristic
70C22×C8C2×D412yes—
71C22×C8C2411yescharacteristic
72C4⋊Q8—44no—
73C4⋊Q8—44no—
74C4⋊Q8—88no—
75C4×D4—24no—
76C4×D4—42no—
77#117—84no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—42no—
87C2×C22⋊C4—42no—
88C2×C42—41no—
89C2×C42D812yes—
90C2×C42D811yescharacteristic
91C2×C42D811yescharacteristic
92C2×C42C2×D412yes—
93C2×C42C2×D411yescharacteristic
94C2×C16—24no—
95C2×C16—28no—
96#107—21no—
97#107—24no—
98#107—24no—
99#107—24no—
100#107—48no—
101#107—44no—
102#107C2×D412yes—
103#102—24no—
104#102—44no—
105C4×C8—24no—
106C4×C8—24no—
107C4×C8—22no—
108C4×C8C2×D412yes—
109C4×C8#4112yes—
110#95—42no—
111#95—41no—
112C24—41no—
113C2×Q8—88no—
114C2×Q8—84no—
115C2×D4—22no—
116C2×D4—42no—
117C2×D4—41no—
118C2×D4—42no—
119C22×C4—41no—
120C22×C4—44no—
121C22×C4—42no—
122C22×C4—44no—
123C22×C4—82no—
124C22×C4—82no—
125C22×C4C2×D811yescharacteristic
126C22×C4C2×D812yes—
127C22×C4C2×D812yes—
128C22×C4C2×D811yescharacteristic
129C22×C4C22×D411yescharacteristic
130D8—24no—
131D8—42no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8#12112yes—
136C2×C8#13714yes—
137C2×C8C22×D411yescharacteristic
138C2×C8C2×#4111yescharacteristic
139C2×C82−1+414yes—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—44no—
143C4⋊C4—42no—
144C4⋊C4—44no—
145C4⋊C4—42no—
146C4⋊C4—88no—
147C4⋊C4—84no—
148C4⋊C4—84no—
149C4⋊C4—88no—
150C22⋊C4—44no—
151C22⋊C4—84no—
152C22⋊C4—84no—
153C42—22no—
154C42—22no—
155C42—24no—
156C42—22no—
157C42—82no—
158C42SD3214yes—
159C42#12112yes—
160C42AGL(1,ℤ/8)12yes—
161C16—44no—
162C23—42no—
163C23—81no—
164C23—82no—
165C23—81no—
166C23—82no—
167C23C22×D811yescharacteristic
168Q8—168no—
169D4—42no—
170D4—42no—
171D4—82no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—88no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4—88no—
184C2×C4—82no—
185C2×C4—84no—
186C2×C4#45912yes—
187C2×C4#45912yes—
188C2×C4C2×SD3214yes—
189C2×C4C2×SD3212yes—
190C2×C4#50914yes—
191C2×C4#51012yes—
192C2×C4C2×#12111yescharacteristic
193C2×C4#53612yes—
194C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
195C2×C4C2×#13712yes—
196C8—24no—
197C8—22no—
198C8#53612yes—
199C8#54112yes—
200C22—82no—
201C22—82no—
202C22—82no—
203C22—162no—
204C22—162no—
205C22C2×#45911yescharacteristic
206C22#322612yes—
207C22C22×SD3211yescharacteristic
208C22C2×#50912yes—
209C22C2×#51011yescharacteristic
210C4—24no—
211C4—24no—
212C4—22no—
213C4—82no—
214C4—162no—
215C4—164no—
216C4#220814yes—
217C4#262111yescharacteristic
218C4#303911yescharacteristic
219C4#323912yes—
220C2—162no—
221C2#2986411yescharacteristic
222C2C2×#220811yescharacteristic
223C2#3358211yescharacteristic
224C2#3379312yes—
225C2#3380412yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D8C23
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #33312C2semidirect
4. #13506C2semidirect
6. #13276C2semidirect
7. #12879C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#22262
#333122
#212752
#135062
#134802
#132762
#128792