Finite Group Databasebeta

← #326986

Subgroups — #326986

Order 512.

1134 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1134 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
1C2×#2582C211yesmaximal, characteristic, complemented
2#32903C212yesmaximal
3#27909C211yesmaximal, characteristic
4#10439C214yesmaximal, complemented
5#10230C214yesmaximal, complemented
6#8723C211yesmaximal, characteristic, complemented
7#8311C212yesmaximal, complemented
8C22×C4×Q8C2212yes—
9C2×#480—22no—
10C2×#460C2214yes—
11C2×#448—22no—
12#2856C2211yescharacteristic
13#2582—22no—
14#2582—24no—
15#2582C2214yes—
16C2×#398—22no—
17C2×#391—22no—
18C2×#389—22no—
19C2×#384C2211yescharacteristic
20#2093—22no—
21#1957C2212yes—
22#1880—24no—
23#1874—22no—
24#1845C2212yes—
25#1803C2214yes—
26#1797C2212yes—
27C4×C22⋊C8C2211yescharacteristic
28C2×Q8⋊C8C2218yes—
29#1296C2214yes—
30C23×Q8D412yes—
31C22×SD16—24no—
32C2×#127C2311yescharacteristic
33C2×#121—21no—
34C2×#121—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—48no—
38C2×C4×Q8D414yes—
39C2×C4×Q8C2314yes—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D412yes—
45C22×C42—22no—
46C22×C42C2311yescharacteristic
47#480—48no—
48#460—48no—
49#448—48no—
50C2×#107—24no—
51C2×C4⋊C8C2314yes—
52C2×#103—24no—
53C2×#102—24no—
54C2×#102—24no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—22no—
57C2×C4×C8C2312yes—
58#398—28no—
59#398—28no—
60#391—28no—
61#391—28no—
62#390—21no—
63#390—22no—
64#389—28no—
65#389—28no—
66#384—22no—
67#384D412yes—
68#384C2311yescharacteristic
69#384C2312yes—
70C2×#95—24no—
71C2×#95—22no—
72#340—24no—
73#336—22no—
74#336—21no—
75Q8⋊C8—48no—
76C22×Q8—28no—
77C22×Q8—22no—
78C22×Q8—24no—
79C22×Q8—24no—
80C22×Q8—44no—
81C22×Q8C2×D414yes—
82C22×D4—22no—
83C22×D4—21no—
84C22×D4—41no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C2×SD16—48no—
91C2×SD16—48no—
92C2×SD16—48no—
93C22×C8—22no—
94C22×C8—24no—
95#127—24no—
96#121—44no—
97#121—44no—
98#121—44no—
99C4×Q8—28no—
100C4×Q8—28no—
101C4×Q8—44no—
102C4×Q8—44no—
103C4×Q8—48no—
104C4×Q8—48no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—28no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—48no—
116C2×C4⋊C4—44no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—84no—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D414yes—
122C2×C22⋊C4—41no—
123C2×C22⋊C4—42no—
124C2×C42—24no—
125C2×C42—24no—
126C2×C42—24no—
127C2×C42—48no—
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D412yes—
130C2×C42C2411yescharacteristic
131#107—48no—
132C4⋊C8—28no—
133#103—48no—
134#102—48no—
135#102—48no—
136C22⋊C8—24no—
137C22⋊C8—44no—
138C4×C8—24no—
139#95—216no—
140#95—28no—
141C24—41no—
142C24#12011yescharacteristic
143C2×Q8—28no—
144C2×Q8—24no—
145C2×Q8—24no—
146C2×Q8—24no—
147C2×Q8—24no—
148C2×Q8—24no—
149C2×Q8—416no—
150C2×Q8—416no—
151C2×Q8—44no—
152C2×Q8—44no—
153C2×Q8#13718yes—
154C2×D4—42no—
155C2×D4—42no—
156C2×D4—44no—
157C2×D4—42no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—44no—
161C2×D4—84no—
162C2×D4—82no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—28no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—41no—
172C22×C4—44no—
173C22×C4—44no—
174C22×C4—44no—
175C22×C4—44no—
176C22×C4—48no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—41no—
180C22×C4#12011yescharacteristic
181C22×C4#12012yes—
182C22×C4#13514yes—
183C22×C4AGL(1,ℤ/8)14yes—
184C22×C4C22×D411yescharacteristic
185C22×C4C22×D412yes—
186SD16—816no—
187C2×C8—24no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—44no—
191C2×C8—44no—
192C2×C8—44no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—44no—
196C4⋊C4—44no—
197C4⋊C4—48no—
198C4⋊C4—42no—
199C4⋊C4—48no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—48no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—82no—
206C4⋊C4—88no—
207C22⋊C4—82no—
208C22⋊C4—84no—
209C42—24no—
210C42—22no—
211C42—48no—
212C42—48no—
213C42H(5,𝔽2)14yes—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—81no—
218C23—82no—
219C23—82no—
220C23—81no—
221C23—81no—
222C23#44914yes—
223C23#45312yes—
224C23C2×#12011yescharacteristic
225Q8—28no—
226Q8—28no—
227Q8—416no—
228Q8—44no—
229Q8—44no—
230D4—84no—
231D4—84no—
232D4—84no—
233D4—164no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—28no—
239C2×C4—28no—
240C2×C4—21no—
241C2×C4—28no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—416no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—48no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—88no—
254C2×C4—88no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4#44914yes—
258C2×C4#53414yes—
259C2×C4#53512yes—
260C2×C4C2×#13512yes—
261C2×C4C2×AGL(1,ℤ/8)12yes—
262C2×C4C2×#13714yes—
263C8—44no—
264C8—84no—
265C22—24no—
266C22—22no—
267C22—24no—
268C22—22no—
269C22—161no—
270C22—162no—
271C22—161no—
272C22—162no—
273C22—161no—
274C22#162014yes—
275C22#261311yescharacteristic
276C22C2×#44912yes—
277C22#300412yes—
278C22C2×#45311yescharacteristic
279C22#302011yescharacteristic
280C4—24no—
281C4—22no—
282C4—48no—
283C4—41no—
284C4—84no—
285C4—84no—
286C4—161no—
287C4#319014yes—
288C4#319314yes—
289C2—24no—
290C2—321no—
291C2C2×#162011yescharacteristic
292C2#1854611yescharacteristic
293C2#1876012yes—
294C2#2959512yes—
295C2#2961611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2582C2semidirect
4. #10439C2semidirect
5. #10230C2semidirect
6. #8723C2semidirect
7. #8311C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
#329032
#279092
#104392
#102302
#87232
#83112