Finite Group Databasebeta

← #319008

Subgroups — #319008

Order 512.

1233 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33042C211yesmaximal, characteristic, complemented
2#27950C211yesmaximal, characteristic, complemented
3#27906C211yesmaximal, characteristic, complemented
4#25721C211yesmaximal, characteristic
5#25717C211yesmaximal, characteristic, complemented
6C2×#1957C211yesmaximal, characteristic, complemented
7#10672C212yesmaximal, complemented
8#10305C212yesmaximal, complemented
9#9824C212yesmaximal, complemented
10#9820C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#493C2212yes—
14C2×#492C2212yes—
15C2×#486C2211yescharacteristic, complemented
16C2×#481—22no—
17C2×#450—22no—
18C2×#447—22no—
19C22×#102—22no—
20C22×C4×C8C2211yescharacteristic
21#2857C2211yescharacteristic
22#2856C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2094—22no—
25#2083—22no—
26#2014—22no—
27#2011—22no—
28#1990C2211yescharacteristic, complemented
29#1990C2212yescomplemented
30#1957—22no—
31#1957C2212yes—
32#1957C2212yes—
33#1957C2212yes—
34#1957C2212yes—
35#1957C2212yescomplemented
36#1957C2211yescharacteristic, complemented
37#1934—22no—
38#1893—22no—
39#1852—22no—
40#1845C2212yescomplemented
41#1844C2212yescomplemented
42#1834C2211yescharacteristic, complemented
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×C82C2212yescomplemented
46#1273C2212yescomplemented
47C23×D4—21no—
48C23×D4—21no—
49C23×D4—21no—
50C22×SD16—24no—
51C22×D8—24no—
52C23×C8D412yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×#127—24no—
55C2×#127—21no—
56C2×#127—21no—
57C2×#127—21no—
58C2×#127—42no—
59C2×#127C2311yescharacteristic
60C2×#127C2312yes—
61C2×C22⋊Q8—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#493—24no—
70#493—42no—
71#492—24no—
72#492—42no—
73#486—42no—
74#481—48no—
75#450—48no—
76#447—48no—
77C2×#107—24no—
78C2×#106—24no—
79C2×#102—24no—
80C2×#102—22no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—48no—
87C2×C22⋊C8D412yes—
88C2×C22⋊C8D412yes—
89C2×C4×C8—42no—
90C2×C4×C8D412yes—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2312yes—
93C2×C4×C8C2312yes—
94C2×C4×C8C2311yescharacteristic
95#391—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#389—21no—
100#389—21no—
101#384—21no—
102#384—21no—
103#384—21no—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2312yes—
107#380—21no—
108#380—21no—
109#340—24no—
110#340—22no—
111#340—22no—
112#336D412yes—
113#336D412yes—
114C82—42no—
115C25—41no—
116C22×Q8—21no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—44no—
125C22×D4—44no—
126C22×D4—44no—
127C22×D4—42no—
128C22×D4—41no—
129C22×D4—41no—
130C22×D4—41no—
131C22×D4—41no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—82no—
136C22×D4—82no—
137C23×C4—41no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×SD16—48no—
142C2×SD16—48no—
143C2×SD16—84no—
144C2×D8—48no—
145C2×D8—48no—
146C2×D8—84no—
147C22×C8—24no—
148C22×C8—24no—
149C22×C8—42no—
150C22×C8C2×D412yes—
151C22×C8C2×D414yes—
152C22×C8C2×D414yes—
153C22×C8C2×D412yes—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156#127—24no—
157#127—22no—
158#127—22no—
159#127—22no—
160#127—22no—
161#127—22no—
162#127—44no—
163#127—44no—
164#127—42no—
165#127#4114yes—
166C22⋊Q8—44no—
167#121—44no—
168#121—44no—
169#121—44no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—82no—
188C2×C22⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2411yescharacteristic
198#107—84no—
199#106—48no—
200#102—48no—
201#102—44no—
202#102—48no—
203#102—48no—
204#102—48no—
205#102—44no—
206#102—48no—
207C22⋊C8D818yescomplemented
208C22⋊C8SD1618yescomplemented
209C4×C8—22no—
210C4×C8—22no—
211C4×C8—22no—
212C4×C8—22no—
213C4×C8—22no—
214C4×C8—22no—
215C4×C8—42no—
216#95—41no—
217#95—41no—
218C24—44no—
219C24—42no—
220C24—41no—
221C24—82no—
222C24—82no—
223C24—82no—
224C24—82no—
225C24—82no—
226C24—82no—
227C24#12711yescharacteristic
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—416no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—42no—
247C2×D4—44no—
248C2×D4—42no—
249C2×D4—44no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—42no—
253C2×D4—44no—
254C2×D4—42no—
255C2×D4—44no—
256C2×D4—42no—
257C2×D4—816no—
258C2×D4—816no—
259C2×D4—82no—
260C2×D4—88no—
261C2×D4—82no—
262C2×D4—82no—
263C2×D4—82no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4—82no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4#12711yescharacteristic
284C22×C4#12711yescharacteristic
285C22×C4#12711yescharacteristic
286C22×C4C2×D814yes—
287C22×C4C2×SD1614yes—
288C22×C4#13514yes—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292SD16—816no—
293D8—816no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—48no—
304C2×C8C2×D818yes—
305C2×C8C2×SD1618yes—
306C2×C8AGL(1,ℤ/8)14yes—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—21no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42C2×#4112yes—
332C42H(5,𝔽2)12yes—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—44no—
338C23—88no—
339C23—88no—
340C23—88no—
341C23—82no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23—82no—
346C23—82no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—164no—
355C23—164no—
356C23—164no—
357C23—164no—
358C23#49212yes—
359C23#49312yes—
360C23#49512yes—
361C23C2×#12711yescharacteristic
362Q8—84no—
363D4—48no—
364D4—48no—
365D4—48no—
366D4—88no—
367D4—88no—
368D4—88no—
369D4—82no—
370D4—84no—
371D4—82no—
372D4—84no—
373D4—84no—
374D4—84no—
375D4—84no—
376D4—164no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—28no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—164no—
400C2×C4#49212yes—
401C2×C4#49312yes—
402C2×C4#49512yes—
403C2×C4#52511yescharacteristic
404C2×C4#53411yescharacteristic
405C2×C4#54612yes—
406C2×C4#54612yes—
407C2×C4C22×D812yes—
408C2×C4C22×SD1612yes—
409C2×C4C2×#13512yes—
410C2×C4C2×AGL(1,ℤ/8)12yes—
411C2×C4#57612yes—
412C2×C4#57712yes—
413C8—24no—
414C8—28no—
415C8—28no—
416C8—42no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—84no—
424C22—84no—
425C22—84no—
426C22—1616no—
427C22—162no—
428C22—162no—
429C22—162no—
430C22—162no—
431C22—161no—
432C22—161no—
433C22—161no—
434C22—161no—
435C22—161no—
436C22—161no—
437C22—161no—
438C22#170614yes—
439C22#253911yescharacteristic
440C22C2×#49211yescharacteristic
441C22C2×#49311yescharacteristic
442C22C2×#49511yescharacteristic
443C22#315111yescharacteristic
444C22#315411yescharacteristic
445C22#315511yescharacteristic
446C4—22no—
447C4—22no—
448C4—22no—
449C4—161no—
450C4—161no—
451C4—161no—
452C4—162no—
453C4#310512yes—
454C4#318812yes—
455C4#329414yes—
456C4#329514yes—
457C2—24no—
458C2—164no—
459C2—321no—
460C2C2×#170611yescharacteristic
461C2#1962111yescharacteristic
462C2#1962911yescharacteristic
463C2#1965611yescharacteristic
464C2#2830311yescharacteristic
465C2#2832911yescharacteristic
466C2#2833211yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#25721C2
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. #33042C2semidirect
2. #27950C2semidirect
3. #27906C2semidirect
5. #25717C2semidirect
6. C2×#1957C2semidirect
7. #10672C2semidirect
8. #10305C2semidirect
9. #9824C2semidirect
10. #9820C2semidirect
11. C8×C22⋊C8C2semidirect
15. C2×#486C22semidirect
28. #1990C22semidirect
29. #1990C22semidirect
35. #1957C22semidirect
36. #1957C22semidirect
40. #1845C22semidirect
41. #1844C22semidirect
42. #1834C22semidirect
45. C2×C82C22semidirect
46. #1273C22semidirect
207. C22⋊C8D8semidirect
208. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#330422
#279502
#279062
#257212
#257172
C2×#19572
#106722
#103052
#98242
#98202
C8×C22⋊C82