Finite Group Databasebeta

← #303980

Subgroups — #303980

Order 512.

1029 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1029 conjugacy classes of subgroups — too many to draw. Only its 176 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#59822C211yesmaximal, characteristic
2#29139C212yesmaximal, complemented
3#28511C211yesmaximal, characteristic, complemented
4#10430C214yesmaximal, complemented
5#9674C212yesmaximal, complemented
6#8452C214yesmaximal, complemented
7C4×#333C211yesmaximal, characteristic, complemented
8C2×#555C2212yes—
9C2×#477—22no—
10#2868C2211yescharacteristic
11#2733—22no—
12#2548—22no—
13#2541—22no—
14#2537C2212yes—
15#2414—24no—
16#2354—22no—
17C4×C4⋊Q8C2211yescharacteristic
18C4×#127C2212yes—
19#2095—22no—
20#2023—24no—
21#1957—22no—
22#1956C2212yes—
23#1845C2212yes—
24#1834—22no—
25#1802C2218yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#333C2212yes—
29C2×D4⋊C8C2218yes—
30#1296C2212yes—
31#555—28no—
32#555—44no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8C2311yescharacteristic
36C2×#127—22no—
37C2×#127D412yes—
38C2×#124—24no—
39C2×#121—24no—
40C2×#121—24no—
41C2×C4×Q8D412yes—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4C2314yes—
45C2×#117—24no—
46#477—48no—
47C2×#107—24no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8C2314yes—
50C2×#103—24no—
51C2×#102—28no—
52C2×C4×C8—22no—
53C2×C4×C8C2312yes—
54#394—24no—
55#394—24no—
56#392—28no—
57#391D412yes—
58#390—24no—
59#390—24no—
60#390—22no—
61#390—22no—
62#390D414yes—
63#389—22no—
64#388—28no—
65#384—22no—
66#384—21no—
67#384D412yes—
68#384C2312yes—
69#384C2311yescharacteristic
70#380—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C43C2311yescharacteristic
74#340—28no—
75#336—21no—
76#336—22no—
77#333—24no—
78D4⋊C8—48no—
79C22×Q8—41no—
80C22×Q8C2×D412yes—
81C22×D4—24no—
82C22×D4—24no—
83C22×D4—22no—
84C22×D4—22no—
85C22×D4C2×D414yes—
86C23×C4—24no—
87C22×C8—22no—
88C22×C8—24no—
89C4⋊Q8—22no—
90C4⋊Q8—44no—
91C4⋊Q8—84no—
92C4⋊Q8#4114yes—
93#127—24no—
94#127—24no—
95#127—84no—
96#124—48no—
97#124—48no—
98#121—416no—
99#121—88no—
100C4×Q8—44no—
101C4×D4—28no—
102C4×D4—28no—
103C4×D4—44no—
104C4×D4—44no—
105C4×D4—84no—
106#117—416no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2×D414yes—
117C2×C4⋊C4C2×D412yes—
118C2×C22⋊C4—28no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—28no—
122C2×C22⋊C4—24no—
123C2×C42—24no—
124C2×C42—22no—
125C2×C42C2×D412yes—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D412yes—
129C2×C42C2411yescharacteristic
130#107—48no—
131C4⋊C8—24no—
132C4⋊C8—24no—
133C4⋊C8—24no—
134#103—48no—
135#102—88no—
136C4×C8—24no—
137C4×C8—44no—
138#95—24no—
139#95—24no—
140C24—24no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—82no—
145C2×Q8—84no—
146C2×Q8#12414yes—
147C2×D4—28no—
148C2×D4—24no—
149C2×D4—24no—
150C2×D4—28no—
151C2×D4—48no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—82no—
157C2×D4—82no—
158C2×D4—88no—
159C2×D4—84no—
160C2×D4—84no—
161C2×D4—84no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—48no—
166C22×C4—44no—
167C22×C4—44no—
168C22×C4—48no—
169C22×C4—42no—
170C22×C4#12011yescharacteristic
171C22×C4#12012yes—
172C22×C4#12011yescharacteristic
173C22×C4C22×D411yescharacteristic
174C22×C4C22×D412yes—
175C2×C8—24no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8—44no—
180C2×C8—44no—
181C4⋊C4—28no—
182C4⋊C4—28no—
183C4⋊C4—42no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—44no—
187C4⋊C4—48no—
188C4⋊C4—82no—
189C4⋊C4—82no—
190C4⋊C4—84no—
191C4⋊C4—82no—
192C22⋊C4—416no—
193C22⋊C4—44no—
194C22⋊C4—44no—
195C22⋊C4—48no—
196C22⋊C4—84no—
197C22⋊C4—88no—
198C42—24no—
199C42—22no—
200C42—41no—
201C42—48no—
202C42—44no—
203C42#13514yes—
204C42#13714yes—
205C42C2×#4112yes—
206C42H(5,𝔽2)12yes—
207C23—44no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23C2×#12011yescharacteristic
215Q8—44no—
216Q8—82no—
217Q8—164no—
218D4—48no—
219D4—48no—
220D4—88no—
221D4—88no—
222D4—168no—
223D4—168no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—88no—
239C2×C4—88no—
240C2×C4—84no—
241C2×C4—82no—
242C2×C4#45214yes—
243C2×C4#45214yes—
244C2×C4#45312yes—
245C2×C4C2×#12411yescharacteristic
246C2×C4#53511yescharacteristic
247C2×C4#54614yes—
248C2×C4C2×#13512yes—
249C2×C4C2×#13712yes—
250C2×C4#57614yes—
251C8—44no—
252C8—84no—
253C22—84no—
254C22—84no—
255C22—84no—
256C22—84no—
257C22—84no—
258C22—84no—
259C22—84no—
260C22#243511yescharacteristic
261C22C2×#45212yes—
262C22#300812yes—
263C22C2×#45311yescharacteristic
264C22#302211yescharacteristic
265C4—21no—
266C4—21no—
267C4—24no—
268C4—44no—
269C4—41no—
270C4—82no—
271C4—84no—
272C4—162no—
273C4#163414yes—
274C4#330514yes—
275C4#333314yes—
276C2—164no—
277C2C2×#163411yescharacteristic
278C2#1871411yescharacteristic
279C2#1892912yes—
280C2#2587011yescharacteristic
281C2#2588312yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#59822C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #29139C2semidirect
3. #28511C2semidirect
4. #10430C2semidirect
5. #9674C2semidirect
6. #8452C2semidirect
7. C4×#333C2semidirect

Maximal subgroups

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

subgroupindex
#598222
#291392
#285112
#104302
#96742
#84522
C4×#3332