Finite Group Databasebeta

← #150748

Subgroups — #150748

Order 512.

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

normal subgroups only

This group has 313 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#27597C211yesmaximal, characteristic
2#12859C211yesmaximal, characteristic
3#11432C211yesmaximal, characteristic
4#7435C212yesmaximal, complemented
5#7412C212yesmaximal, complemented
6#2314C411yescharacteristic, complemented
7#2311C2211yescharacteristic
8#2096C411yescharacteristic, complemented
9#2094C2211yescharacteristic
10#2074C411yescharacteristic, complemented
11#2073C2211yescharacteristic
12#1927—21no—
13#1926—21no—
14#1853—21no—
15#1818C411yescharacteristic, complemented
16#1788—21no—
17#1782—21no—
18C2×#324—21no—
19#1393—21no—
20#1393C2212yes—
21#1319—21no—
22#1318C2212yes—
23C22×#41D411yescharacteristic
24C2×#117C2×C411yescharacteristic
25C22×C4⋊C4D411yescharacteristic
26C2×#106—21no—
27C2×#106—21no—
28C2×#103—21no—
29C2×#103—21no—
30C2×#102—21no—
31C2×#102—21no—
32C2×#99—22no—
33C2×#99—41no—
34C2×C22⋊C8—21no—
35C2×C22⋊C8—41no—
36#391C2×C411yescharacteristic
37#391D411yescharacteristic
38#390D411yescharacteristic
39#390C2311yescharacteristic
40#388—41no—
41#387—41no—
42#386—41no—
43#384C2×C411yescharacteristic
44#382C2×C411yescharacteristic
45#381—41no—
46#380—21no—
47#380—41no—
48C4×C4⋊C4—21no—
49#340—21no—
50#340—21no—
51#336—21no—
52#336C2×C411yescharacteristic
53#336C2×C411yescharacteristic
54#324—22no—
55#324—41no—
56C2×#41—22no—
57C2×#41—41no—
58C2×#41—42no—
59C2×#41—81no—
60C22×Q8C22⋊C411yescharacteristic
61C22×D4—21no—
62C22×D4C2×D411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C22⋊C411yescharacteristic
66C22×C8—21no—
67C22×C8—21no—
68C22×C8—81no—
69#117—42no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—41no—
75C2×C4⋊C4—81no—
76C2×C4⋊C4C22⋊C411yescharacteristic
77C2×C4⋊C4C2×D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—41no—
81C2×C22⋊C4—41no—
82C2×C22⋊C4—41no—
83C2×C42—41no—
84C2×C42C22⋊C411yescharacteristic
85C2×C42C22×C411yescharacteristic
86#106—81no—
87#106—81no—
88#103—42no—
89#103—81no—
90#102—42no—
91#102—81no—
92#99—42no—
93#99—42no—
94#99—82no—
95C22⋊C8—42no—
96C22⋊C8—42no—
97C22⋊C8—81no—
98#95—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—41no—
103#95—82no—
104C24—21no—
105C24#9911yescharacteristic
106#41—84no—
107#41—84no—
108C2×Q8—21no—
109C2×Q8—21no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×D4—21no—
113C2×D4—41no—
114C2×D4—41no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4—81no—
118C2×D4#13512yes—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—81no—
135C22×C4—82no—
136C22×C4—161no—
137C22×C4#9911yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—82no—
146C2×C8—162no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—42no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—81no—
155C4⋊C4—161no—
156C22⋊C4—42no—
157C22⋊C4—82no—
158C22⋊C4—82no—
159C22⋊C4—161no—
160C22⋊C4—161no—
161C22⋊C4—161no—
162C42—41no—
163C42—41no—
164C42—161no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—81no—
173C23—81no—
174C23#35212yes—
175C23C2×#9911yescharacteristic
176Q8—42no—
177Q8—81no—
178D4—42no—
179D4—81no—
180D4—84no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—84no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—161no—
210C2×C4—162no—
211C2×C4—162no—
212C2×C4—162no—
213C2×C4#41611yescharacteristic
214C2×C4#42111yescharacteristic
215C8—81no—
216C8—81no—
217C8—162no—
218C22—21no—
219C22—21no—
220C22—22no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—81no—
226C22—82no—
227C22—81no—
228C22—81no—
229C22—161no—
230C22#151611yescharacteristic
231C22C2×#35211yescharacteristic
232C22#212711yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4—81no—
240C4—81no—
241C4—161no—
242C4—161no—
243C4—322no—
244C2—21no—
245C2—21no—
246C2—42no—
247C2—82no—
248C2#1184111yescharacteristic
249C2#1360611yescharacteristic
250C2#1361011yescharacteristic
subgroupG/H
center, socleC22#2127
commutatorC2×C42C22×C4
Frattini, cosocle#390C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ #2311 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7435C2semidirect
5. #7412C2semidirect
6. #2314C4semidirect
8. #2096C4semidirect
10. #2074C4semidirect
15. #1818C4semidirect

Maximal subgroups

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

subgroupindex
#275972
#128592
#114322
#74352
#74122