Finite Group Databasebeta

← #150418

Subgroups — #150418

Order 512.

354 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 56 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#12583C211yesmaximal, characteristic
3#11508C211yesmaximal, characteristic
4#7435C212yesmaximal, complemented
5#7412C212yesmaximal, complemented
6C2×#410—21no—
7#2314C411yescharacteristic, complemented
8#2311C2211yescharacteristic
9#2096C411yescharacteristic, complemented
10#2094C2211yescharacteristic
11#2074C411yescharacteristic, complemented
12#2073C2211yescharacteristic
13#1927—21no—
14#1926—21no—
15#1853—21no—
16#1818C411yescharacteristic, complemented
17#1788—21no—
18#1463—21no—
19#1393—21no—
20#1393C2212yes—
21#1319—21no—
22#1318C2212yes—
23C22×#41D411yescharacteristic
24C2×#117—41no—
25C2×#117C2×C411yescharacteristic
26C22×C4⋊C4D411yescharacteristic
27C2×#106—21no—
28C2×#106—21no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#102—21no—
32C2×#102—21no—
33#410—24no—
34#410—42no—
35C2×#99—22no—
36C2×#99—41no—
37#407—41no—
38C2×C22⋊C8—21no—
39#391C2×C411yescharacteristic
40#391D411yescharacteristic
41#390D411yescharacteristic
42#390C2311yescharacteristic
43#388—41no—
44#387—41no—
45#386—41no—
46#384C2×C411yescharacteristic
47#382C2×C411yescharacteristic
48#380—21no—
49#380—41no—
50C4×C4⋊C4—21no—
51#340—21no—
52#340—21no—
53#336—21no—
54#336C2×C411yescharacteristic
55#336C2×C411yescharacteristic
56#324—22no—
57#324—41no—
58C2×#41—22no—
59C2×#41—21no—
60C2×#41—21no—
61C2×#41—42no—
62C2×#41—81no—
63C22×Q8C22⋊C411yescharacteristic
64C22×D4—21no—
65C22×D4C2×D411yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4C22⋊C411yescharacteristic
69C2×M16—81no—
70C22×C8—21no—
71C22×C8—21no—
72#117—42no—
73#117—44no—
74#117—82no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—81no—
81C2×C4⋊C4—81no—
82C2×C4⋊C4C22⋊C411yescharacteristic
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C22⋊C4—21no—
85C2×C22⋊C4—42no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42—41no—
90C2×C42—81no—
91C2×C42C22⋊C411yescharacteristic
92C2×C42C22×C411yescharacteristic
93#106—81no—
94#106—81no—
95#103—42no—
96#103—81no—
97#102—41no—
98#102—41no—
99#102—81no—
100#99—24no—
101#99—44no—
102#99—42no—
103C22⋊C8—42no—
104C22⋊C8—42no—
105C22⋊C8—81no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111C24—21no—
112C24#9911yescharacteristic
113#41—82no—
114#41—82no—
115#41—82no—
116#41—82no—
117C2×Q8—21no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8#10412yes—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—81no—
127C2×D4—81no—
128C2×D4AGL(1,ℤ/8)11yescharacteristic
129C2×D4#13711yescharacteristic
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—42no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—82no—
149C22×C4—161no—
150C22×C4#9911yescharacteristic
151C22×C4C2×C22⋊C411yescharacteristic
152M16—162no—
153C2×C8—41no—
154C2×C8—41no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—82no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—81no—
168C4⋊C4—84no—
169C4⋊C4—161no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—82no—
173C22⋊C4—161no—
174C22⋊C4—161no—
175C22⋊C4—161no—
176C42—41no—
177C42—41no—
178C42—84no—
179C42—161no—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—81no—
190C23—81no—
191C23C2×#9911yescharacteristic
192Q8—42no—
193Q8—81no—
194D4—41no—
195D4—41no—
196D4—81no—
197D4—82no—
198D4—82no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—82no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—84no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—161no—
231C2×C4—162no—
232C2×C4—162no—
233C2×C4#35212yes—
234C2×C4#41711yescharacteristic
235C2×C4C2×#10411yescharacteristic
236C8—81no—
237C8—81no—
238C8—162no—
239C22—21no—
240C22—21no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22#150411yescharacteristic
254C22C2×#35211yescharacteristic
255C22#212711yescharacteristic
256C4—21no—
257C4—21no—
258C4—22no—
259C4—42no—
260C4—81no—
261C4—81no—
262C4—81no—
263C4—81no—
264C4—161no—
265C4—164no—
266C4—161no—
267C2—21no—
268C2—21no—
269C2—81no—
270C2—81no—
271C2—81no—
272C2#1175511yescharacteristic
273C2#1360611yescharacteristic
274C2#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
7. #2314C4semidirect
9. #2096C4semidirect
11. #2074C4semidirect
16. #1818C4semidirect

Maximal subgroups

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

subgroupindex
#275972
#125832
#115082
#74352
#74122