Finite Group Databasebeta

← #512261

Subgroups — #512261

Order 512.

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

normal subgroups only

This group has 598 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#33130C211yesmaximal, characteristic
2C2×C4×#107C211yesmaximal, characteristic
3#13132C214yesmaximal, complemented
4#13111C214yesmaximal, complemented
5#12993C212yesmaximal
6#12920C212yesmaximal
7#12844C211yesmaximal, characteristic
8C2×#467C2211yescharacteristic
9C22×#107C2212yes—
10C22×C4×C8C2211yescharacteristic
11#2855—21no—
12C2×C4×C4⋊C4—21no—
13C2×#367C2214yes—
14C2×#358C2214yes—
15C2×C22⋊C16C2212yes—
16#1969C2212yes—
17#1942C2212yes—
18#1933—22no—
19#1892—21no—
20#1808C2211yescharacteristic
21C4×#107—22no—
22C4×#107C2214yes—
23C2×#340—21no—
24#1380C2214yes—
25#1380C2214yes—
26#1374C2214yes—
27C22×Q16C2311yescharacteristic
28C23×C8D411yescharacteristic
29C23×C8C2311yescharacteristic
30C2×C22⋊Q8—21no—
31C22×C4⋊C4—22no—
32C22×C42—41no—
33C22×C42D411yescharacteristic
34C22×C16C2314yes—
35#467—24no—
36C2×#107—22no—
37C2×#107—28no—
38C2×#107C2312yes—
39C2×#107C2311yescharacteristic
40C2×#107C2314yes—
41C2×#103—22no—
42C2×#103—21no—
43C2×C4×C8—24no—
44C2×C4×C8D412yes—
45C2×C4×C8C2312yes—
46#391—22no—
47#389—22no—
48#387—42no—
49#385—41no—
50#381—41no—
51#380—21no—
52C4×C4⋊C4—24no—
53C4×C4⋊C4—42no—
54C2×#95—41no—
55#367—28no—
56#358—28no—
57C22⋊C16—24no—
58#340—21no—
59#340—22no—
60#340—24no—
61C22×Q8—21no—
62C23×C4—42no—
63C23×C4D811yescharacteristic
64C23×C4D811yescharacteristic
65C23×C4C2×D411yescharacteristic
66C2×Q16—24no—
67C2×Q16—22no—
68C22×C8—24no—
69C22×C8—22no—
70C22×C8—42no—
71C22×C8C2×D412yes—
72C22×C8C2×D411yescharacteristic
73C22×C8#4114yes—
74C22×C8C2411yescharacteristic
75C22⋊Q8—42no—
76C22⋊Q8—42no—
77C2×C4⋊C4—28no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—44no—
82C2×C4⋊C4—42no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—44no—
90C2×C42—84no—
91C2×C42D812yes—
92C2×C42C2×D412yes—
93C2×C16—24no—
94C2×C16—24no—
95C2×C16—24no—
96#107—28no—
97#107—24no—
98#107—24no—
99#107—22no—
100#107#4118yes—
101#103—44no—
102#103—42no—
103C4×C8—24no—
104C4×C8—24no—
105#95—44no—
106#95—41no—
107#95—42no—
108#95—42no—
109#95—42no—
110C24C2×D811yescharacteristic
111C2×Q8—41no—
112C2×Q8—42no—
113C2×Q8—41no—
114C2×Q8—42no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—42no—
121C22×C4—48no—
122C22×C4—44no—
123C22×C4—42no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—84no—
127C22×C4—84no—
128C22×C4SD3214yes—
129C22×C4#12312yes—
130C22×C4C2×D812yes—
131C22×C4C2×D811yescharacteristic
132C22×C4C2×D812yes—
133C22×C4#13712yes—
134C22×C4C22×D411yescharacteristic
135Q16—44no—
136C2×C8—22no—
137C2×C8—28no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—44no—
142C2×C8#12312yes—
143C2×C8C2×#4112yes—
144C2×C8C2×#4114yes—
145C4⋊C4—28no—
146C4⋊C4—48no—
147C4⋊C4—44no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—84no—
153C4⋊C4—82no—
154C22⋊C4—82no—
155C22⋊C4—82no—
156C42—22no—
157C42—22no—
158C42—24no—
159C42—88no—
160C16—44no—
161C23—22no—
162C23—22no—
163C23#48012yes—
164C23C2×SD3212yes—
165C23#51012yes—
166C23C22×D811yescharacteristic
167Q8—81no—
168Q8—81no—
169Q8—82no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—28no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—48no—
182C2×C4—816no—
183C2×C4—84no—
184C2×C4—84no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#48012yes—
193C2×C4C2×SD3214yes—
194C2×C4C2×#12311yescharacteristic
195C2×C4#52512yes—
196C2×C4C2×#13711yescharacteristic
197C2×C4#57512yes—
198C8—22no—
199C8—42no—
200C8#52914yes—
201C22—22no—
202C22—22no—
203C22—24no—
204C22—24no—
205C22#223514yes—
206C22C2×#48011yescharacteristic
207C22#310012yes—
208C22C22×SD3211yescharacteristic
209C22C2×#51011yescharacteristic
210C22#341412yes—
211C4—22no—
212C4—24no—
213C4—42no—
214C4—88no—
215C4—161no—
216C4—161no—
217C4—162no—
218C4#285311yescharacteristic
219C4#309311yescharacteristic
220C4#311712yes—
221C2—24no—
222C2#3309411yescharacteristic
223C2C2×#223511yescharacteristic
224C2#3364711yescharacteristic
225C2#3366112yes—
226C2#3367712yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13132C2semidirect
4. #13111C2semidirect

Maximal subgroups

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

subgroupindex
#331302
C2×C4×#1072
#131322
#131112
#129932
#129202
#128442