Finite Group Databasebeta

← #319603

Subgroups — #319603

Order 512.

1018 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 170 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#33210C211yesmaximal, characteristic, complemented
2#27892C212yesmaximal, complemented
3#25855C212yesmaximal
4#21203C211yesmaximal, characteristic
5#10673C211yesmaximal, characteristic, complemented
6#10671C211yesmaximal, characteristic, complemented
7#10312C211yesmaximal, characteristic, complemented
8#10312C211yesmaximal, characteristic, complemented
9#9838C212yesmaximal, complemented
10#9830C212yesmaximal, complemented
11#8147C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#498C2211yescharacteristic, complemented
14C2×#486C2212yes—
15C2×#486C2212yes—
16C2×#418—22no—
17C2×C4×M16C2211yescharacteristic
18#2857C2212yes—
19#2285C2211yescharacteristic
20#2041—24no—
21#2036—24no—
22#1990C2212yescomplemented
23#1961—22no—
24#1957C2211yescharacteristic
25#1957C2211yescharacteristic
26#1957C2211yescharacteristic
27#1957C2212yes—
28#1957C2211yescharacteristic
29#1957C2212yes—
30#1956C2212yescomplemented
31#1956C2212yescomplemented
32#1943—22no—
33#1928—24no—
34#1845C2212yescomplemented
35#1845C2211yescharacteristic, complemented
36#1834C2212yescomplemented
37#1809—22no—
38#1786—24no—
39C4×C22⋊C8C2212yes—
40C2×#322C2212yescomplemented
41#1273C2211yescharacteristic, complemented
42#1273C2211yescharacteristic, complemented
43C23×D4—21no—
44C23×D4—22no—
45C22×M16D412yes—
46C2×C4⋊Q8C2312yes—
47C2×#127—22no—
48C2×#127—22no—
49C2×#127—21no—
50C2×#127—22no—
51C2×#127—42no—
52C2×#127C2311yescharacteristic
53C2×#127C2311yescharacteristic
54C2×C22⋊Q8—22no—
55C2×#117—21no—
56C22×C42C2311yescharacteristic
57#498—42no—
58#486—24no—
59#486—24no—
60#486—42no—
61#486—42no—
62C2×#107—22no—
63C2×#107—24no—
64C2×#106—22no—
65#418—48no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×C22⋊C8D414yes—
77C4×M16—42no—
78C2×#97D412yes—
79C2×C4×C8C2312yes—
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2312yes—
82C2×C4×C8C2311yescharacteristic
83#391—22no—
84#389—22no—
85#389—22no—
86#384C2312yes—
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89#382—21no—
90#380—22no—
91#380—22no—
92#380—22no—
93C4×C4⋊C4—22no—
94#340—24no—
95#340—24no—
96#336D414yes—
97#322—42no—
98C25—41no—
99C22×Q8—22no—
100C22×D4—24no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—44no—
106C22×D4—44no—
107C22×D4—44no—
108C22×D4—42no—
109C22×D4—48no—
110C22×D4—42no—
111C22×D4—41no—
112C22×D4—44no—
113C22×D4—84no—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—42no—
119C22×C8C2×D412yes—
120C22×C8C2×D414yes—
121C22×C8C2×D414yes—
122C22×C8C2×D412yes—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127#127—24no—
128#127—21no—
129#127—21no—
130#127—22no—
131#127—22no—
132#127—44no—
133#127—44no—
134#127—42no—
135#127#4112yes—
136#127#4112yes—
137C22⋊Q8—84no—
138#117—44no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—42no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—41no—
155C2×C42C2×D412yes—
156C2×C42C2×D411yescharacteristic
157C2×C42C2411yescharacteristic
158#107—44no—
159#107—84no—
160#106—44no—
161#103—48no—
162#103—84no—
163#102—48no—
164#102—48no—
165#102—44no—
166#102—44no—
167#102—44no—
168#102—44no—
169#102—84no—
170#102—84no—
171C22⋊C8—44no—
172#97—42no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179#95—42no—
180#95—42no—
181#95—42no—
182C24—44no—
183C24—41no—
184C24—41no—
185C24—82no—
186C24—84no—
187C24—84no—
188C24—82no—
189C24#12711yescharacteristic
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×Q8—44no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—416no—
199C2×D4—48no—
200C2×D4—44no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—82no—
214C2×D4—88no—
215C2×D4—84no—
216C2×D4—84no—
217C2×D4—816no—
218C2×D4—816no—
219C2×D4—82no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4#12712yes—
232C22×C4#12711yescharacteristic
233C22×C4#13714yes—
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D412yes—
236M16—48no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8AGL(1,ℤ/8)14yes—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—84no—
261C22⋊C4—82no—
262C22⋊C4—84no—
263C42—21no—
264C42—21no—
265C42—22no—
266C42—22no—
267C42—82no—
268C42C2×#4111yescharacteristic
269C42C2×#4111yescharacteristic
270C42H(5,𝔽2)12yes—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—21no—
275C23—42no—
276C23—42no—
277C23—88no—
278C23—816no—
279C23—81no—
280C23—81no—
281C23—81no—
282C23—81no—
283C23—82no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23—168no—
288C23—164no—
289C23—164no—
290C23#49712yes—
291C23C2×#12711yescharacteristic
292Q8—84no—
293Q8—84no—
294D4—44no—
295D4—44no—
296D4—48no—
297D4—48no—
298D4—88no—
299D4—816no—
300D4—82no—
301D4—82no—
302D4—84no—
303D4—84no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4#49712yes—
324C2×C4#52511yescharacteristic
325C2×C4#53411yescharacteristic
326C2×C4#54612yes—
327C2×C4#54612yes—
328C2×C4C2×AGL(1,ℤ/8)12yes—
329C2×C4C2×#13712yes—
330C2×C4#57514yes—
331C2×C4#57614yes—
332C8—24no—
333C8—42no—
334C8—44no—
335C8—44no—
336C22—22no—
337C22—24no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—82no—
342C22—82no—
343C22—84no—
344C22—84no—
345C22—1616no—
346C22—161no—
347C22—161no—
348C22—161no—
349C22—161no—
350C22—162no—
351C22—162no—
352C22#253911yescharacteristic
353C22C2×#49711yescharacteristic
354C22#315111yescharacteristic
355C22#315312yes—
356C22#315512yes—
357C4—22no—
358C4—42no—
359C4—161no—
360C4—162no—
361C4—161no—
362C4—162no—
363C4#311012yes—
364C4#319312yes—
365C4#331914yes—
366C4#332214yes—
367C2—42no—
368C2—162no—
369C2—162no—
370C2#1968212yes—
371C2#1972711yescharacteristic
372C2#1974911yescharacteristic
373C2#2831111yescharacteristic
374C2#2835712yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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. #33210C2semidirect
2. #27892C2semidirect
5. #10673C2semidirect
6. #10671C2semidirect
7. #10312C2semidirect
8. #10312C2semidirect
9. #9838C2semidirect
10. #9830C2semidirect
11. #8147C2semidirect
13. C2×#498C22semidirect
22. #1990C22semidirect
30. #1956C22semidirect
31. #1956C22semidirect
34. #1845C22semidirect
35. #1845C22semidirect
36. #1834C22semidirect
40. C2×#322C22semidirect
41. #1273C22semidirect
42. #1273C22semidirect

Maximal subgroups

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

subgroupindex
#332102
#278922
#258552
#212032
#106732
#106712
#103122
#98382
#98302
#81472