Finite Group Databasebeta

← #150021

Subgroups — #150021

Order 512.

259 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 54 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#29199C211yesmaximal, characteristic
2#11535C211yesmaximal, characteristic
3#11480C211yesmaximal, characteristic
4#7371C212yesmaximal, complemented
5#7367C212yesmaximal, complemented
6#2647C411yescharacteristic
7#2326C2211yescharacteristic
8#2066—21no—
9#2065—21no—
10#1959C411yescharacteristic
11#1958C411yescharacteristic
12#1865—21no—
13#1864—21no—
14#1844C2211yescharacteristic
15#1836C2211yescharacteristic
16#1768C411yescharacteristic
17#1465—21no—
18#1461—21no—
19#1325C2212yes—
20#1325C2212yes—
21#1319—21no—
22#1318—21no—
23C2×#124C2×C411yescharacteristic
24C2×C4×Q8D411yescharacteristic
25C2×C4×D4D411yescharacteristic
26#431—41no—
27C2×#106—21no—
28C2×#106—21no—
29C2×C4⋊C8—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#102—21no—
33C2×#102—21no—
34#407—41no—
35C2×#97C2×C411yescharacteristic
36C2×C4×C8C2×C411yescharacteristic
37#400—41no—
38#398—41no—
39#391D411yescharacteristic
40#390D411yescharacteristic
41#389—21no—
42#389—21no—
43#388—41no—
44#385—41no—
45#384C2×C411yescharacteristic
46#384C2311yescharacteristic
47#382C2×C411yescharacteristic
48#376C2×C411yescharacteristic
49#340—21no—
50#340—21no—
51#336—21no—
52#324—22no—
53#324—22no—
54#324—41no—
55#323—41no—
56C22×Q8C22⋊C411yescharacteristic
57C22×D4C22⋊C411yescharacteristic
58C23×C4—21no—
59C2×M16—81no—
60C22×C8—21no—
61C22×C8—21no—
62#124—22no—
63#124—22no—
64C4×Q8—81no—
65C4×D4—81no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4C2×D411yescharacteristic
74C2×C4⋊C4C2×D411yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—41no—
77C2×C22⋊C4—41no—
78C2×C42—21no—
79C2×C42C22⋊C411yescharacteristic
80C2×C42C22⋊C411yescharacteristic
81C2×C42C22×C411yescharacteristic
82#106—81no—
83#106—81no—
84C4⋊C8—42no—
85C4⋊C8—81no—
86#103—42no—
87#103—81no—
88#102—42no—
89#102—81no—
90C22⋊C8—42no—
91C22⋊C8—42no—
92C22⋊C8—81no—
93#97—22no—
94C4×C8—22no—
95C4×C8—81no—
96#95—21no—
97#95—21no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102C24—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8#10112yes—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4—41no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—161no—
121C22×C4#9911yescharacteristic
122C22×C4#9911yescharacteristic
123C22×C4C2×C22⋊C411yescharacteristic
124M16—162no—
125C2×C8—22no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C2×C8—41no—
131C2×C8—82no—
132C2×C8—82no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4—81no—
140C4⋊C4—161no—
141C4⋊C4—161no—
142C4⋊C4—161no—
143C4⋊C4—161no—
144C22⋊C4—44no—
145C22⋊C4—81no—
146C22⋊C4—161no—
147C42—21no—
148C42—41no—
149C42—41no—
150C42—41no—
151C42—41no—
152C42#13112yes—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—81no—
157C23—81no—
158C23C2×#9911yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—42no—
162D4—81no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—82no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—161no—
190C2×C4#40811yescharacteristic
191C2×C4C2×#10111yescharacteristic
192C8—42no—
193C8—81no—
194C8—162no—
195C8—162no—
196C22—21no—
197C22—21no—
198C22—82no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—161no—
203C22#147211yescharacteristic
204C22#212711yescharacteristic
205C22#213211yescharacteristic
206C4—21no—
207C4—21no—
208C4—42no—
209C4—41no—
210C4—41no—
211C4—42no—
212C4—81no—
213C4—81no—
214C4—81no—
215C4—161no—
216C4—161no—
217C2—21no—
218C2—21no—
219C2—161no—
220C2#1157211yescharacteristic
221C2#1361011yescharacteristic
222C2#1361111yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2326 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7371C2semidirect
5. #7367C2semidirect

Maximal subgroups

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

subgroupindex
#291992
#115352
#114802
#73712
#73672