Finite Group Databasebeta

← #304002

Subgroups — #304002

Order 512.

1049 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1049 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#28716C212yesmaximal, complemented
3#28512C211yesmaximal, characteristic, complemented
4#10432C214yesmaximal, complemented
5#9674C212yesmaximal, complemented
6#8452C214yesmaximal, complemented
7C4×#333C211yesmaximal, characteristic, complemented
8C2×#555C2212yes—
9C2×#486—22no—
10C2×#475—22no—
11#2868C2211yescharacteristic
12#2733—22no—
13#2548—22no—
14#2541—22no—
15#2537C2212yes—
16#2414—24no—
17#2354—22no—
18C4×C4⋊Q8C2211yescharacteristic
19C4×#127C2212yes—
20#2096—22no—
21#2071—24no—
22#1990—22no—
23#1956C2212yes—
24#1844C2212yes—
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#486—48no—
47#475—48no—
48C2×#106—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8C2314yes—
51C2×#103—24no—
52C2×#102—28no—
53C2×C4×C8—22no—
54C2×C4×C8C2312yes—
55#394—24no—
56#394—24no—
57#392—28no—
58#391D412yes—
59#390—24no—
60#390—24no—
61#390—22no—
62#390—22no—
63#390D414yes—
64#389—22no—
65#388—28no—
66#384—22no—
67#384—21no—
68#384D412yes—
69#384C2312yes—
70#384C2311yescharacteristic
71#380—24no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C43C2311yescharacteristic
75#340—28no—
76#336—22no—
77#336—21no—
78#333—24no—
79D4⋊C8—48no—
80C22×Q8—41no—
81C22×Q8C2×D412yes—
82C22×D4—24no—
83C22×D4—24no—
84C22×D4—22no—
85C22×D4—22no—
86C22×D4C2×D414yes—
87C23×C4—24no—
88C22×C8—22no—
89C22×C8—24no—
90C4⋊Q8—22no—
91C4⋊Q8—44no—
92C4⋊Q8—44no—
93C4⋊Q8—44no—
94C4⋊Q8#4114yes—
95#127—24no—
96#127—24no—
97#127—84no—
98#124—48no—
99#124—48no—
100#121—416no—
101#121—88no—
102C4×Q8—44no—
103C4×D4—28no—
104C4×D4—28no—
105C4×D4—44no—
106C4×D4—44no—
107C4×D4—84no—
108#117—416no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C2×D414yes—
119C2×C4⋊C4C2×D412yes—
120C2×C22⋊C4—28no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—28no—
124C2×C22⋊C4—24no—
125C2×C42—24no—
126C2×C42—22no—
127C2×C42C2×D412yes—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D412yes—
131C2×C42C2411yescharacteristic
132#106—48no—
133C4⋊C8—24no—
134C4⋊C8—24no—
135C4⋊C8—24no—
136#103—48no—
137#102—416no—
138C4×C8—24no—
139C4×C8—44no—
140#95—24no—
141#95—24no—
142C24—24no—
143C2×Q8—22no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—84no—
147C2×Q8—82no—
148C2×Q8#12414yes—
149C2×D4—28no—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—28no—
153C2×D4—48no—
154C2×D4—44no—
155C2×D4—44no—
156C2×D4—44no—
157C2×D4—44no—
158C2×D4—82no—
159C2×D4—82no—
160C2×D4—88no—
161C2×D4—84no—
162C2×D4—84no—
163C2×D4—84no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—48no—
168C22×C4—44no—
169C22×C4—44no—
170C22×C4—48no—
171C22×C4—42no—
172C22×C4#12011yescharacteristic
173C22×C4#12012yes—
174C22×C4#12011yescharacteristic
175C22×C4C22×D411yescharacteristic
176C22×C4C22×D412yes—
177C2×C8—24no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8—44no—
183C4⋊C4—28no—
184C4⋊C4—28no—
185C4⋊C4—42no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—44no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—84no—
195C22⋊C4—416no—
196C22⋊C4—44no—
197C22⋊C4—44no—
198C22⋊C4—48no—
199C22⋊C4—84no—
200C22⋊C4—88no—
201C42—24no—
202C42—22no—
203C42—41no—
204C42—48no—
205C42—44no—
206C42#13514yes—
207C42AGL(1,ℤ/8)14yes—
208C42C2×#4112yes—
209C42H(5,𝔽2)12yes—
210C23—44no—
211C23—44no—
212C23—44no—
213C23—44no—
214C23—44no—
215C23—44no—
216C23—44no—
217C23C2×#12011yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—164no—
221D4—48no—
222D4—48no—
223D4—88no—
224D4—88no—
225D4—168no—
226D4—168no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—44no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—88no—
242C2×C4—88no—
243C2×C4—84no—
244C2×C4—82no—
245C2×C4#44914yes—
246C2×C4#44914yes—
247C2×C4#45312yes—
248C2×C4C2×#12411yescharacteristic
249C2×C4#53511yescharacteristic
250C2×C4#54614yes—
251C2×C4C2×#13512yes—
252C2×C4C2×AGL(1,ℤ/8)12yes—
253C2×C4#57714yes—
254C8—44no—
255C8—84no—
256C22—84no—
257C22—84no—
258C22—84no—
259C22—84no—
260C22—84no—
261C22—84no—
262C22—84no—
263C22#243511yescharacteristic
264C22C2×#44912yes—
265C22#300912yes—
266C22C2×#45311yescharacteristic
267C22#302211yescharacteristic
268C4—21no—
269C4—21no—
270C4—24no—
271C4—44no—
272C4—41no—
273C4—82no—
274C4—84no—
275C4—162no—
276C4#162414yes—
277C4#330414yes—
278C4#333214yes—
279C2—164no—
280C2C2×#162411yescharacteristic
281C2#1871511yescharacteristic
282C2#1892812yes—
283C2#2587011yescharacteristic
284C2#2588712yes—
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. #28716C2semidirect
3. #28512C2semidirect
4. #10432C2semidirect
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
#287162
#285122
#104322
#96742
#84522
C4×#3332