Finite Group Databasebeta

← #347830

Subgroups — #347830

Order 512.

1084 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 182 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×#2975C211yesmaximal, characteristic
2#32905C212yesmaximal
3#29876C211yesmaximal, characteristic, complemented
4C2×#1693C212yesmaximal, complemented
5#9225C214yesmaximal, complemented
6#9130C212yesmaximal, complemented
7#8656C212yesmaximal, complemented
8#8598C211yesmaximal, characteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#500C2211yescharacteristic, complemented
11C2×#461C2214yes—
12C2×#450—22no—
13#2975—28no—
14C2×#436—22no—
15C2×#436C2212yes—
16C2×#436C2212yes—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#407—22no—
20#2857C2212yes—
21#2086—22no—
22#2075—22no—
23#1990C2212yescomplemented
24#1877—24no—
25#1872—22no—
26#1834C2211yescharacteristic, complemented
27#1802C2214yes—
28#1797C2212yes—
29#1693—28no—
30C2×#334C2212yescomplemented
31C2×#332C2214yescomplemented
32#1296C2214yescomplemented
33#1277C2212yescomplemented
34C23×D4D411yescharacteristic
35C24×C4—21no—
36C22×SD16—24no—
37C22×M16—22no—
38C22×M16—22no—
39C2×C4⋊Q8C2312yes—
40C2×C22⋊Q8—22no—
41C2×C4×D4—24no—
42C2×C4×D4—24no—
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4D412yes—
46C2×C4×D4C2312yes—
47C22×C4⋊C4D411yescharacteristic
48C22×C22⋊C4—21no—
49C22×C42C2311yescharacteristic
50#500—24no—
51#461—28no—
52#461—28no—
53#450—48no—
54#436—216no—
55#436—24no—
56#436—24no—
57#436—24no—
58#436D418yes—
59#431—24no—
60#431—24no—
61C2×#107—22no—
62C2×C4⋊C8D412yes—
63C2×C4⋊C8C2314yes—
64C2×C4⋊C8C2312yes—
65C2×#103—24no—
66C2×#102—24no—
67#407—216no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8D412yes—
73C2×C4×C8—21no—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—22no—
78#389—22no—
79#384C2312yes—
80#380—22no—
81#380—22no—
82#340—22no—
83#340—24no—
84#340—24no—
85#336—22no—
86#334—24no—
87#332—28no—
88C25—21no—
89C22×Q8—22no—
90C22×D4—24no—
91C22×D4—21no—
92C22×D4—24no—
93C22×D4—42no—
94C22×D4C2×D412yes—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—21no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—84no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C2×SD16—48no—
105C2×SD16—48no—
106C2×SD16—48no—
107C2×M16—28no—
108C2×M16—28no—
109C2×M16—84no—
110C2×M16—84no—
111C22×C8—22no—
112C22×C8—24no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—24no—
116C4⋊Q8—24no—
117C22⋊Q8—44no—
118C22⋊Q8—44no—
119C4×D4—24no—
120C4×D4—28no—
121C4×D4—24no—
122C4×D4—24no—
123C4×D4—24no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4C2×D414yes—
127C4×D4#4114yes—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C2×D412yes—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—42no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42C2411yescharacteristic
147#107—44no—
148C4⋊C8—24no—
149C4⋊C8—22no—
150C4⋊C8—24no—
151C4⋊C8—24no—
152C4⋊C8C2×D414yes—
153#103—48no—
154#102—48no—
155C22⋊C8—24no—
156C22⋊C8—28no—
157C22⋊C8—28no—
158C22⋊C8—24no—
159C22⋊C8—44no—
160C4×C8—24no—
161C4×C8—22no—
162C4×C8C2×D414yes—
163#95—42no—
164#95—42no—
165C24—24no—
166C24—22no—
167C24—42no—
168C24—42no—
169C24—82no—
170C24—82no—
171C24#12111yescharacteristic
172C2×Q8—44no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—44no—
176C2×D4—24no—
177C2×D4—22no—
178C2×D4—24no—
179C2×D4—24no—
180C2×D4—416no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4#12114yes—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—24no—
188C22×C4—28no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—84no—
201C22×C4—88no—
202C22×C4—88no—
203C22×C4—82no—
204C22×C4#12111yescharacteristic
205C22×C4#12111yescharacteristic
206C22×C4#12111yescharacteristic
207C22×C4AGL(1,ℤ/8)14yes—
208C22×C4C22×D411yescharacteristic
209C22×C4C22×D411yescharacteristic
210C22×C4C2×#4111yescharacteristic
211SD16—816no—
212M16—88no—
213M16—88no—
214C2×C8—24no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—24no—
218C2×C8—44no—
219C2×C8—44no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8#12014yes—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C4⋊C4#12114yes—
234C22⋊C4—28no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—84no—
238C42—21no—
239C42—22no—
240C42—21no—
241C42—22no—
242C42C22×D412yes—
243C42C2×#4112yes—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—22no—
248C23—24no—
249C23—48no—
250C23—44no—
251C23—42no—
252C23—42no—
253C23—44no—
254C23—88no—
255C23—84no—
256C23—84no—
257C23—88no—
258C23#46912yes—
259C23C2×#12111yescharacteristic
260Q8—84no—
261Q8—84no—
262D4—28no—
263D4—48no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—24no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—44no—
275C2×C4—42no—
276C2×C4—48no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—82no—
281C2×C4—88no—
282C2×C4—84no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4#46912yes—
290C2×C4C2×#12011yescharacteristic
291C2×C4C2×#12112yes—
292C2×C4#52512yes—
293C2×C4#52511yescharacteristic
294C2×C4C2×AGL(1,ℤ/8)12yes—
295C2×C4#57512yes—
296C2×C4#57714yes—
297C8—24no—
298C8—42no—
299C8—44no—
300C8—84no—
301C22—22no—
302C22—24no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—44no—
307C22—44no—
308C22—44no—
309C22—816no—
310C22—84no—
311C22—84no—
312C22#284711yescharacteristic
313C22#304212yes—
314C22C2×#46911yescharacteristic
315C22#305411yescharacteristic
316C22#306912yes—
317C4—22no—
318C4—22no—
319C4—41no—
320C4—44no—
321C4—41no—
322C4—162no—
323C4—162no—
324C4#300912yes—
325C4#305514yes—
326C4#312312yes—
327C2—42no—
328C2—84no—
329C2#1942812yes—
330C2#1943112yes—
331C2#3291411yescharacteristic
332C2#3293611yescharacteristic
333C2#3294611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
3. #29876C2semidirect
4. C2×#1693C2semidirect
5. #9225C2semidirect
6. #9130C2semidirect
7. #8656C2semidirect
8. #8598C2semidirect
10. C2×#500C22semidirect
23. #1990C22semidirect
26. #1834C22semidirect
30. C2×#334C22semidirect
31. C2×#332C22semidirect
32. #1296C22semidirect
33. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#329052
#298762
C2×#16932
#92252
#91302
#86562
#85982