Finite Group Databasebeta

← #512081

Subgroups — #512081

Order 512.

1226 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 186 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
1C22×C4×D8C211yesmaximal, characteristic
2C2×#2235C211yesmaximal, characteristic, complemented
3C2×#2187C211yesmaximal, characteristic
4#33130C211yesmaximal, characteristic, complemented
5#13014C214yesmaximal, complemented
6#12937C214yesmaximal, complemented
7#12866C211yesmaximal, characteristic, complemented
8#12844C211yesmaximal, characteristic, complemented
9#12731C211yesmaximal, characteristic
10C23×D8C2211yescharacteristic
11C22×C4×D4—21no—
12C22×SD32C2212yes—
13C2×#467C2211yescharacteristic
14C2×C4×D8—24no—
15C2×C4×D8—22no—
16C2×C4×D8—24no—
17C2×C4×D8C2214yes—
18C22×#107C2211yescharacteristic
19C22×#102—21no—
20C22×C4×C8C2211yescharacteristic
21#2855—21no—
22#2235—24no—
23#2187—24no—
24C2×#367C2212yescomplemented
25C2×#358C2212yescomplemented
26C2×#357C2212yes—
27C2×#357C2214yes—
28C2×C22⋊C16C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#1969C2212yescomplemented
31#1942C2212yescomplemented
32#1933—22no—
33#1892—21no—
34#1808C2211yescharacteristic
35#1380C2214yescomplemented
36#1374C2214yescomplemented
37C23×D4—21no—
38C24×C4—41no—
39C22×Q16C2311yescharacteristic
40C22×D8—28no—
41C22×D8—21no—
42C22×D8—22no—
43C22×D8D414yes—
44C22×D8C2312yes—
45C23×C8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C4×D4—28no—
49C2×C4×D4—24no—
50C2×C4×D4—44no—
51C2×C4×D4—42no—
52C22×C4⋊C4—21no—
53C22×C22⋊C4—41no—
54C22×C42D411yescharacteristic
55C2×SD32—24no—
56C2×SD32—28no—
57C22×C16C2312yes—
58C22×C16C2312yes—
59#467—22no—
60#467—22no—
61C4×D8—216no—
62C4×D8—28no—
63C4×D8—28no—
64C2×#107—21no—
65C2×#107—22no—
66C2×#107D414yes—
67C2×#107C2312yes—
68C2×#107C2311yescharacteristic
69C2×#107C2312yes—
70C2×#103—22no—
71C2×#103—21no—
72C2×#102—28no—
73C2×#102—24no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8D412yes—
77C2×C4×C8C2312yes—
78#391—22no—
79#389—22no—
80#387—42no—
81#385—41no—
82#381—41no—
83#380—21no—
84#367—24no—
85#358—24no—
86#357—24no—
87#357—28no—
88C22⋊C16—22no—
89C22⋊C16—22no—
90#340—21no—
91#340—22no—
92C25—41no—
93C22×Q8—21no—
94C22×D4—28no—
95C22×D4—24no—
96C22×D4—22no—
97C22×D4—44no—
98C22×D4—41no—
99C22×D4—42no—
100C23×C4—48no—
101C23×C4—41no—
102C23×C4—44no—
103C23×C4—84no—
104C23×C4—84no—
105C23×C4D811yescharacteristic
106C23×C4D811yescharacteristic
107C23×C4C2×D411yescharacteristic
108C2×Q16—21no—
109C2×Q16—21no—
110C2×Q16—24no—
111C2×D8—28no—
112C2×D8—216no—
113C2×D8—216no—
114C2×D8—24no—
115C2×D8—22no—
116C2×D8—24no—
117C2×D8C2×D414yes—
118C2×D8#4118yes—
119C22×C8—24no—
120C22×C8—22no—
121C22×C8—42no—
122C22×C8C2×D412yes—
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D412yes—
125C22×C8C2×D412yes—
126C22×C8C2411yescharacteristic
127C22⋊Q8—42no—
128C22⋊Q8—42no—
129C4×D4—216no—
130C4×D4—48no—
131C4×D4—416no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C22⋊C4—48no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42D812yes—
147C2×C42C2×D412yes—
148SD32—48no—
149C2×C16—22no—
150C2×C16—22no—
151C2×C16—24no—
152C2×C16—24no—
153#107—24no—
154#107—22no—
155#107—21no—
156#107—21no—
157#107—24no—
158#107C2×D414yes—
159#103—44no—
160#103—42no—
161#102—216no—
162C4×C8—24no—
163C4×C8—22no—
164C4×C8—22no—
165#95—41no—
166#95—42no—
167#95—42no—
168#95—42no—
169C24—48no—
170C24—42no—
171C24—44no—
172C24—84no—
173C24—82no—
174C24—82no—
175C24C2×D811yescharacteristic
176C2×Q8—41no—
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—42no—
180C2×D4—216no—
181C2×D4—24no—
182C2×D4—28no—
183C2×D4—48no—
184C2×D4—416no—
185C2×D4—48no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4—44no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—416no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—88no—
202C22×C4—832no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—88no—
206C22×C4SD3214yescomplemented
207C22×C4#12012yes—
208C22×C4C2×D812yes—
209C22×C4C2×D811yescharacteristic
210C22×C4C2×D812yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4C22×D411yescharacteristic
213Q16—42no—
214Q16—42no—
215D8—216no—
216D8—24no—
217D8—28no—
218D8#12318yes—
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—44no—
226C2×C8#12012yes—
227C2×C8C22×D411yescharacteristic
228C2×C8C22×D411yescharacteristic
229C2×C8C2×#4114yes—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—84no—
238C4⋊C4—82no—
239C22⋊C4—416no—
240C22⋊C4—82no—
241C22⋊C4—82no—
242C42—22no—
243C42—22no—
244C42—24no—
245C16—42no—
246C16—42no—
247C23—22no—
248C23—22no—
249C23—416no—
250C23—48no—
251C23—44no—
252C23—816no—
253C23—816no—
254C23—88no—
255C23—82no—
256C23—82no—
257C23—84no—
258C23—88no—
259C23#44712yes—
260C23C2×SD3212yes—
261C23#50912yes—
262C23C22×D811yescharacteristic
263Q8—81no—
264Q8—82no—
265Q8—81no—
266D4—28no—
267D4—416no—
268D4—48no—
269D4—44no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—28no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—24no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—832no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—88no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4#44712yes—
294C2×C4C2×SD3214yes—
295C2×C4C2×#12011yescharacteristic
296C2×C4#52512yes—
297C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
298C2×C4#57512yes—
299C8—22no—
300C8—42no—
301C8C2×#12312yes—
302C8#52512yes—
303C22—24no—
304C22—24no—
305C22—22no—
306C22—22no—
307C22—48no—
308C22—832no—
309C22—88no—
310C22—88no—
311C22—84no—
312C22—84no—
313C22#218714yes—
314C22C2×#44711yescharacteristic
315C22#310012yes—
316C22C22×SD3211yescharacteristic
317C22C2×#50911yescharacteristic
318C22#341412yes—
319C4—24no—
320C4—22no—
321C4—42no—
322C4—84no—
323C4—162no—
324C4—161no—
325C4—161no—
326C4#284911yescharacteristic
327C4#300511yescharacteristic
328C4#311412yes—
329C2—24no—
330C2—88no—
331C2#3300211yescharacteristic
332C2C2×#218711yescharacteristic
333C2#3351911yescharacteristic
334C2#3366112yes—
335C2#3367412yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2235C2semidirect
4. #33130C2semidirect
5. #13014C2semidirect
6. #12937C2semidirect
7. #12866C2semidirect
8. #12844C2semidirect
24. C2×#367C22semidirect
25. C2×#358C22semidirect
30. #1969C22semidirect
31. #1942C22semidirect
35. #1380C22semidirect
36. #1374C22semidirect
206. C22×C4SD32semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#22352
C2×#21872
#331302
#130142
#129372
#128662
#128442
#127312