Finite Group Databasebeta

← #305074

Subgroups — #305074

Order 512.

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

normal subgroups only

This group has 909 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#59869C211yesmaximal, characteristic, complemented
2#28749C212yesmaximal
3#28441C211yesmaximal, characteristic
4#10351C214yesmaximal, complemented
5#9865C212yesmaximal, complemented
6#9063C214yesmaximal, complemented
7C4×#333C211yesmaximal, characteristic, complemented
8C2×#492C2212yes—
9C2×#486—22no—
10C2×#460—22no—
11#2867C2211yescharacteristic
12#2809—22no—
13#2571C2212yes—
14#2568—22no—
15#2546—22no—
16#2508—24no—
17#2369C2212yes—
18#2355—22no—
19C4×C4⋊Q8C2212yes—
20C4×C4⋊Q8C2211yescharacteristic
21#2041—24no—
22#2014—22no—
23#1982C2218yes—
24#1957—22no—
25#1957C2212yes—
26C4×C4⋊C8C2212yes—
27C4×C4⋊C8C2211yescharacteristic
28C2×#333C2212yes—
29#1296C2212yes—
30#1279C2218yes—
31C22×SD16—24no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D412yes—
34C2×C4⋊Q8C2311yescharacteristic
35C2×#127—22no—
36C2×#127—21no—
37C2×#127—21no—
38C2×#127—22no—
39C2×#127C2312yes—
40C2×C4×Q8—22no—
41C2×C4×Q8D412yes—
42#492—28no—
43#486—48no—
44#460—48no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8C2314yes—
47C2×#102—28no—
48C2×#102—24no—
49C2×#102—28no—
50C2×C4×C8—22no—
51C2×C4×C8C2312yes—
52#400—24no—
53#400—24no—
54#398—28no—
55#391—24no—
56#391—24no—
57#391—22no—
58#391D412yes—
59#389—24no—
60#389—24no—
61#389—22no—
62#389—22no—
63#389D414yes—
64#387—28no—
65#384D412yes—
66#384C2311yescharacteristic
67#382—24no—
68#382—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4C2314yes—
74C43C2311yescharacteristic
75#336—21no—
76#336—22no—
77#333—24no—
78C22×Q8—22no—
79C22×Q8C2×D412yes—
80C22×D4—22no—
81C22×D4—24no—
82C22×D4—41no—
83C22×D4—42no—
84C22×D4—41no—
85C2×SD16—48no—
86C2×SD16—48no—
87C2×SD16—84no—
88C22×C8—22no—
89C22×C8—24no—
90C4⋊Q8—24no—
91C4⋊Q8—24no—
92C4⋊Q8—22no—
93C4⋊Q8—84no—
94C4⋊Q8#4114yes—
95#127—24no—
96#127—24no—
97#127—44no—
98#127—44no—
99#127—44no—
100#127—84no—
101#127—82no—
102C4×Q8—44no—
103C4×Q8—44no—
104C4×Q8—44no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D414yes—
116C2×C4⋊C4C2×D412yes—
117C2×C42—22no—
118C2×C42—24no—
119C2×C42—24no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D412yes—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D412yes—
124C2×C42C2411yescharacteristic
125C4⋊C8—24no—
126C4⋊C8—24no—
127C4⋊C8—24no—
128#102—416no—
129#102—48no—
130#102—88no—
131C4×C8—24no—
132C4×C8—44no—
133#95—24no—
134#95—28no—
135#95—28no—
136#95—24no—
137#95—24no—
138#95—24no—
139C24—42no—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—44no—
146C2×Q8#12414yes—
147C2×D4—44no—
148C2×D4—44no—
149C2×D4—42no—
150C2×D4—42no—
151C2×D4—48no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—48no—
155C2×D4—82no—
156C2×D4—84no—
157C2×D4—84no—
158C2×D4—84no—
159C2×D4—84no—
160C2×D4—84no—
161C2×D4—82no—
162C22×C4—22no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4#12011yescharacteristic
167C22×C4#12012yes—
168C22×C4#12011yescharacteristic
169C22×C4C22×D411yescharacteristic
170C22×C4C22×D412yes—
171SD16—816no—
172C2×C8—24no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—44no—
177C2×C8—44no—
178C4⋊C4—28no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—28no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—48no—
186C4⋊C4—44no—
187C4⋊C4—82no—
188C4⋊C4—82no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C42—24no—
192C42—22no—
193C42—41no—
194C42—44no—
195C42—44no—
196C42—44no—
197C42—84no—
198C42#13514yes—
199C42AGL(1,ℤ/8)14yes—
200C42C2×#4112yes—
201C422−1+412yes—
202C23—82no—
203C23—84no—
204C23—84no—
205C23—82no—
206C23—82no—
207C23C2×#12011yescharacteristic
208Q8—44no—
209Q8—82no—
210Q8—88no—
211D4—84no—
212D4—84no—
213D4—88no—
214D4—88no—
215D4—168no—
216D4—164no—
217D4—164no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4#44914yes—
236C2×C4#44914yes—
237C2×C4#45312yes—
238C2×C4C2×#12411yescharacteristic
239C2×C4#53611yescharacteristic
240C2×C4#54814yes—
241C2×C4C2×#13512yes—
242C2×C4C2×AGL(1,ℤ/8)12yes—
243C2×C4#57714yes—
244C8—44no—
245C8—84no—
246C22—162no—
247C22—162no—
248C22—162no—
249C22—164no—
250C22—164no—
251C22#243611yescharacteristic
252C22C2×#44912yes—
253C22#301012yes—
254C22C2×#45311yescharacteristic
255C22#302111yescharacteristic
256C4—21no—
257C4—21no—
258C4—24no—
259C4—41no—
260C4—44no—
261C4—82no—
262C4—84no—
263C4—84no—
264C4#162414yes—
265C4#336514yes—
266C4#336814yes—
267C2—322no—
268C2C2×#162411yescharacteristic
269C2#1871811yescharacteristic
270C2#1889812yes—
271C2#2591811yescharacteristic
272C2#2593512yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#2867C22
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
1. #59869C2semidirect
4. #10351C2semidirect
5. #9865C2semidirect
6. #9063C2semidirect
7. C4×#333C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#287492
#284412
#103512
#98652
#90632
C4×#3332