Finite Group Databasebeta

← #173982

Subgroups — #173982

Order 512.

1013 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1013 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#7810C218yesmaximal, complemented
5#7571C211yesmaximal, characteristic, complemented
6#7569C212yesmaximal, complemented
7C2×#555—21no—
8C2×#486—22no—
9C2×#475—22no—
10#2868C2211yescharacteristic
11#2733C2214yes—
12#2548—22no—
13#2541—22no—
14#2537—21no—
15#2414—24no—
16#2354—22no—
17C4×C4⋊Q8C2211yescharacteristic
18C4×#127C2212yes—
19#2096C2214yes—
20#2071C2218yes—
21#1990—22no—
22#1956—21no—
23#1844—21no—
24#1802—24no—
25C4×C4⋊C8C2212yes—
26C4×C4⋊C8C2211yescharacteristic
27#1272C2214yes—
28#1270C2218yes—
29#555—44no—
30#555—44no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—22no—
33C2×C4⋊Q8D411yescharacteristic
34C2×#127—22no—
35C2×#127D412yes—
36C2×#124—24no—
37C2×#121—24no—
38C2×#121—24no—
39C2×C4×Q8D412yes—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4D414yes—
43C2×#117—24no—
44#486—48no—
45#475—48no—
46C2×#106—24no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—22no—
49C2×#103—24no—
50C2×#102—28no—
51C2×C4×C8—21no—
52C2×C4×C8—22no—
53#394—24no—
54#394—24no—
55#392—28no—
56#391C2312yes—
57#390—24no—
58#390—24no—
59#390—22no—
60#390—22no—
61#390C2314yes—
62#389—22no—
63#388—28no—
64#384—22no—
65#384—21no—
66#384D412yes—
67#384D411yescharacteristic
68#384C2312yes—
69#380—24no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C43C2311yescharacteristic
73#340—28no—
74#336C2314yes—
75#336C2312yes—
76C22×Q8—41no—
77C22×Q8C2×D412yes—
78C22×D4—24no—
79C22×D4—24no—
80C22×D4—22no—
81C22×D4—22no—
82C22×D4C2×D414yes—
83C23×C4—24no—
84C22×C8—22no—
85C22×C8—24no—
86C4⋊Q8—22no—
87C4⋊Q8—44no—
88C4⋊Q8—44no—
89C4⋊Q8—44no—
90C4⋊Q8SD1614yes—
91#127—24no—
92#127—24no—
93#127—84no—
94#124—48no—
95#124—48no—
96#121—416no—
97#121—88no—
98C4×Q8—44no—
99C4×D4—48no—
100C4×D4—44no—
101C4×D4—44no—
102C4×D4—44no—
103C4×D4—44no—
104#117—416no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4C2×D414yes—
115C2×C4⋊C4C2×D412yes—
116C2×C22⋊C4—28no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—28no—
120C2×C22⋊C4—24no—
121C2×C42—24no—
122C2×C42—22no—
123C2×C42C2×D412yes—
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D412yes—
127C2×C42C2411yescharacteristic
128#106—48no—
129C4⋊C8—24no—
130C4⋊C8—44no—
131#103—48no—
132#102—416no—
133C4×C8—42no—
134C4×C8—44no—
135#95—24no—
136#95—24no—
137C24—24no—
138C2×Q8—22no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—84no—
142C2×Q8—82no—
143C2×Q8C2×SD1614yes—
144C2×D4—24no—
145C2×D4—24no—
146C2×D4—44no—
147C2×D4—44no—
148C2×D4—48no—
149C2×D4—44no—
150C2×D4—44no—
151C2×D4—44no—
152C2×D4—44no—
153C2×D4—44no—
154C2×D4—44no—
155C2×D4—88no—
156C2×D4—84no—
157C2×D4—84no—
158C2×D4—84no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—48no—
163C22×C4—48no—
164C22×C4—48no—
165C22×C4—42no—
166C22×C4#12011yescharacteristic
167C22×C4#12011yescharacteristic
168C22×C4#12012yes—
169C22×C4C22×D411yescharacteristic
170C22×C4C22×D412yes—
171C2×C8—24no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—44no—
175C2×C8—44no—
176C2×C8—44no—
177C4⋊C4—28no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—44no—
186C4⋊C4—82no—
187C4⋊C4—82no—
188C4⋊C4—84no—
189C22⋊C4—416no—
190C22⋊C4—44no—
191C22⋊C4—44no—
192C22⋊C4—44no—
193C22⋊C4—44no—
194C22⋊C4—84no—
195C22⋊C4—88no—
196C42—22no—
197C42—24no—
198C42—41no—
199C42—48no—
200C42—44no—
201C42C2×SD1612yes—
202C42#13514yes—
203C42AGL(1,ℤ/8)14yes—
204C42AGL(1,ℤ/8)12yes—
205C23—44no—
206C23—44no—
207C23—44no—
208C23—44no—
209C23—44no—
210C23—44no—
211C23—44no—
212C23C2×#12011yescharacteristic
213Q8—44no—
214Q8—82no—
215Q8—164no—
216D4—48no—
217D4—84no—
218D4—88no—
219D4—88no—
220D4—168no—
221D4—168no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—816no—
237C2×C4—84no—
238C2×C4—82no—
239C2×C4#44914yes—
240C2×C4#44914yes—
241C2×C4#45014yes—
242C2×C4#45312yes—
243C2×C4#45514yes—
244C2×C4C22×SD1611yescharacteristic
245C2×C4C2×#13512yes—
246C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
247C2×C4C2×AGL(1,ℤ/8)12yes—
248C8—44no—
249C8—84no—
250C22—84no—
251C22—84no—
252C22—84no—
253C22—84no—
254C22—84no—
255C22—84no—
256C22—84no—
257C22C2×#45011yescharacteristic
258C22C2×#44912yes—
259C22C2×#45311yescharacteristic
260C22C2×#45512yes—
261C22#302811yescharacteristic
262C4—21no—
263C4—21no—
264C4—22no—
265C4—22no—
266C4—41no—
267C4—44no—
268C4—84no—
269C4—82no—
270C4—162no—
271C4#162414yes—
272C4#164218yes—
273C2—164no—
274C2#1839511yescharacteristic
275C2#1841212yes—
276C2C2×#162411yescharacteristic
277C2C2×#164212yes—
278C2#1907411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#59822C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28716C2semidirect
3. #28512C2semidirect
4. #7810C2semidirect
5. #7571C2semidirect
6. #7569C2semidirect

Maximal subgroups

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

subgroupindex
#598222
#287162
#285122
#78102
#75712
#75692