Finite Group Databasebeta

← #327000

Subgroups — #327000

Order 512.

1567 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1567 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×#2537C211yesmaximal, characteristic
2#32956C212yesmaximal, complemented
3#27910C211yesmaximal, characteristic, complemented
4#10430C214yesmaximal, complemented
5#10235C214yesmaximal, complemented
6#8725C211yesmaximal, characteristic, complemented
7#8310C212yesmaximal, complemented
8C22×#121—22no—
9C22×C4×D4C2212yes—
10C2×#484—22no—
11C2×#476C2214yescomplemented
12#2857C2211yescharacteristic
13#2537—22no—
14#2537—24no—
15#2537C2214yes—
16C2×#394—22no—
17C2×#384C2211yescharacteristic
18C2×#380—22no—
19#2095—22no—
20#1956C2212yescomplemented
21#1864—24no—
22#1853—22no—
23#1845C2212yescomplemented
24#1802C2214yescomplemented
25#1797C2212yes—
26#1786—22no—
27C4×C22⋊C8C2211yescharacteristic
28C2×D4⋊C8C2218yescomplemented
29#1296C2214yescomplemented
30C23×D4—22no—
31C23×D4D412yes—
32C24×C4—22no—
33C2×C4⋊Q8C2311yescharacteristic
34C2×C22⋊Q8—21no—
35C2×#121—28no—
36C2×#121—28no—
37C2×#121—416no—
38C2×C4×D4—22no—
39C2×C4×D4—22no—
40C2×C4×D4—48no—
41C2×C4×D4D414yes—
42C2×C4×D4C2314yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4D412yes—
45C22×C4⋊C4D412yes—
46C22×C22⋊C4—24no—
47C22×C22⋊C4—22no—
48C22×C42C2311yescharacteristic
49#484—48no—
50#476—48no—
51C2×#107—24no—
52C2×#106—24no—
53C2×C4⋊C8C2314yes—
54C2×#103—24no—
55C2×#102—24no—
56C2×#102—24no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—22no—
59C2×C4×C8C2312yes—
60#394—28no—
61#394—28no—
62#391—21no—
63#389—22no—
64#384—22no—
65#384D412yes—
66#384C2311yescharacteristic
67#384C2312yes—
68#380—28no—
69#380—28no—
70#380—22no—
71C2×#95—22no—
72#340—24no—
73#336—22no—
74#336—21no—
75D4⋊C8—48no—
76C25—22no—
77C22×Q8—21no—
78C22×D4—28no—
79C22×D4—28no—
80C22×D4—22no—
81C22×D4—24no—
82C22×D4—24no—
83C22×D4—44no—
84C22×D4—416no—
85C22×D4—84no—
86C22×D4C2×D414yes—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—24no—
90C23×C4—24no—
91C23×C4—48no—
92C23×C4—48no—
93C23×C4—44no—
94C23×C4—44no—
95C23×C4C2×D411yescharacteristic
96C23×C4C2×D412yes—
97C22×C8—22no—
98C22×C8—24no—
99C4⋊Q8—24no—
100C22⋊Q8—44no—
101#121—432no—
102#121—416no—
103#121—416no—
104C4×D4—28no—
105C4×D4—28no—
106C4×D4—44no—
107C4×D4—44no—
108C4×D4—48no—
109C4×D4—48no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—48no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4C2×D414yes—
122C2×C4⋊C4C2×D414yes—
123C2×C22⋊C4—216no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—48no—
127C2×C22⋊C4—416no—
128C2×C22⋊C4—41no—
129C2×C42—24no—
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D412yes—
132C2×C42C2411yescharacteristic
133#107—48no—
134#106—48no—
135C4⋊C8—28no—
136#103—48no—
137#102—48no—
138#102—48no—
139C22⋊C8—24no—
140C22⋊C8—44no—
141C4×C8—24no—
142#95—28no—
143#95—42no—
144C24—28no—
145C24—24no—
146C24—44no—
147C24—44no—
148C24—44no—
149C24—44no—
150C24—44no—
151C24—44no—
152C24#12011yescharacteristic
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×D4—28no—
157C2×D4—24no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—416no—
163C2×D4—416no—
164C2×D4—44no—
165C2×D4—432no—
166C2×D4—44no—
167C2×D4—88no—
168C2×D4—832no—
169C2×D4AGL(1,ℤ/8)18yes—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—41no—
177C22×C4—48no—
178C22×C4—48no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—48no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—48no—
185C22×C4—44no—
186C22×C4—48no—
187C22×C4—44no—
188C22×C4—48no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—816no—
192C22×C4—816no—
193C22×C4#12011yescharacteristic
194C22×C4#12012yes—
195C22×C4#13514yes—
196C22×C4#13714yes—
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D412yes—
199C2×C8—24no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—44no—
203C2×C8—44no—
204C2×C8—44no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—48no—
210C4⋊C4—42no—
211C4⋊C4—48no—
212C4⋊C4—44no—
213C4⋊C4—44no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C22⋊C4—48no—
217C22⋊C4—432no—
218C22⋊C4—48no—
219C22⋊C4—82no—
220C42—24no—
221C42—22no—
222C42H(5,𝔽2)14yes—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—28no—
227C23—416no—
228C23—416no—
229C23—416no—
230C23—44no—
231C23—44no—
232C23—44no—
233C23—44no—
234C23—44no—
235C23—44no—
236C23—88no—
237C23—88no—
238C23—88no—
239C23—88no—
240C23#45214yes—
241C23#45312yes—
242C23C2×#12011yescharacteristic
243Q8—84no—
244D4—28no—
245D4—28no—
246D4—416no—
247D4—44no—
248D4—44no—
249D4—832no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—28no—
255C2×C4—28no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—44no—
265C2×C4—816no—
266C2×C4—88no—
267C2×C4—88no—
268C2×C4—88no—
269C2×C4—88no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4#45214yes—
276C2×C4#53414yes—
277C2×C4#53512yes—
278C2×C4C2×#13512yes—
279C2×C4C2×AGL(1,ℤ/8)14yes—
280C2×C4C2×#13712yes—
281C8—44no—
282C8—84no—
283C22—22no—
284C22—24no—
285C22—24no—
286C22—22no—
287C22—48no—
288C22—48no—
289C22—48no—
290C22—832no—
291C22—84no—
292C22—84no—
293C22—84no—
294C22—84no—
295C22#163414yes—
296C22#261311yescharacteristic
297C22C2×#45212yes—
298C22#300312yes—
299C22C2×#45311yescharacteristic
300C22#302011yescharacteristic
301C4—24no—
302C4—22no—
303C4—41no—
304C4—84no—
305C4—84no—
306C4—161no—
307C4—161no—
308C4#318814yes—
309C4#319314yes—
310C2—24no—
311C2—88no—
312C2C2×#163411yescharacteristic
313C2#1852411yescharacteristic
314C2#1878312yes—
315C2#2959812yes—
316C2#2961611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#2537C2
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
2. #32956C2semidirect
3. #27910C2semidirect
4. #10430C2semidirect
5. #10235C2semidirect
6. #8725C2semidirect
7. #8310C2semidirect
11. C2×#476C22semidirect
20. #1956C22semidirect
23. #1845C22semidirect
24. #1802C22semidirect
28. C2×D4⋊C8C22semidirect
29. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
#329562
#279102
#104302
#102352
#87252
#83102