Finite Group Databasebeta

← #6470486

Subgroups — #6470486

Order 512.

2356 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37477C211yesmaximal, characteristic, complemented
2#34172C211yesmaximal, characteristic
3#32966C212yesmaximal, complemented
4#32962C212yesmaximal, complemented
5#29963C212yesmaximal, complemented
6#29960C212yesmaximal, complemented
7#25980C214yesmaximal, complemented
8#25978C214yesmaximal, complemented
9#25972C214yesmaximal, complemented
10#25970C214yesmaximal, complemented
11#21944C212yesmaximal, complemented
12#21935C211yesmaximal, characteristic, complemented
13#20601C212yesmaximal
14C23×#41C2211yescharacteristic
15C2×#518C2211yescharacteristic
16C2×#518C2211yescharacteristic
17C2×#478C2214yescomplemented
18C2×#476C2212yescomplemented
19C2×#474C2212yescomplemented
20C2×#417C2214yes—
21C2×#417C2214yes—
22C2×#407C2212yes—
23#2841—22no—
24#2659—22no—
25#2646—22no—
26#2629C2212yes—
27#2628C2212yes—
28#2442C2214yescomplemented
29#2440C2214yescomplemented
30#2437—22no—
31#2339—22no—
32#2339—22no—
33#2339C2214yes—
34#2339C2214yes—
35#2310C2211yescharacteristic
36#2310C2211yescharacteristic
37#2308—22no—
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40C2×#381—22no—
41C2×#380C2212yes—
42#2097C2214yescomplemented
43#2096C2212yescomplemented
44#2095C2212yescomplemented
45#2068C2214yescomplemented
46#2068C2214yescomplemented
47#2066C2214yescomplemented
48#2066C2214yescomplemented
49#2026C2212yescomplemented
50#2026C2212yescomplemented
51#2024C2214yescomplemented
52#1864C2214yescomplemented
53#1864C2214yescomplemented
54#1862C2214yescomplemented
55#1862C2214yescomplemented
56#1854C2214yes—
57#1854C2214yes—
58#1853C2212yescomplemented
59#1853C2212yescomplemented
60#1852C2212yescomplemented
61#1852C2212yescomplemented
62#1851C2214yescomplemented
63#1851C2214yescomplemented
64#1809C2214yescomplemented
65#1809C2214yescomplemented
66#1804C2212yescomplemented
67#1804C2212yescomplemented
68#1802C2214yescomplemented
69#1798C2212yes—
70#1798C2212yes—
71#1788C2214yes—
72#1788C2214yes—
73#1787C2214yes—
74#1786C2214yes—
75C22×#41—22no—
76C22×#41—24no—
77C22×#41—44no—
78C22×#41—44no—
79C22×#41Q818yescomplemented
80C22×#41C2314yes—
81C23×Q8C2311yescharacteristic
82C23×D4—21no—
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C22×M16C2314yes—
87C2×C4⋊Q8C2312yes—
88C2×#125C2312yes—
89C2×C22⋊Q8—24no—
90#518—44no—
91#518—44no—
92C2×C4×D4—21no—
93C2×C4×D4—22no—
94C2×C4×D4—21no—
95C2×C4×D4C2314yes—
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—22no—
107#478—48no—
108#476—44no—
109#474—44no—
110C2×#107C2314yes—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117#417—48no—
118#417—48no—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127#407—28no—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131#400—22no—
132#400—24no—
133#399—22no—
134#398—24no—
135#398—24no—
136#398—24no—
137#396—24no—
138#395—22no—
139#395—24no—
140#393—22no—
141#391—21no—
142#391—22no—
143#391—21no—
144#391C2312yes—
145#391C2314yes—
146#391C2312yes—
147#389C2314yes—
148#389C2314yes—
149#386—24no—
150#386—24no—
151#386—24no—
152#386—24no—
153#385—24no—
154#385—24no—
155#384C2312yes—
156#382C2312yes—
157#381—21no—
158#381—22no—
159#381—22no—
160#381—24no—
161#381—24no—
162#381D412yes—
163#380—22no—
164#380—22no—
165#380—24no—
166#380D412yes—
167#380D412yes—
168#380C2312yes—
169#380C2312yes—
170#380C2314yes—
171#380C2314yes—
172#379—21no—
173#379D412yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2314yes—
176C2×#95—22no—
177C2×#95—24no—
178C2×#95—24no—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2312yes—
184#336C2312yes—
185#336C2314yes—
186C25—21no—
187C25C2×D411yescharacteristic
188C2×#41—216no—
189C2×#41—28no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—44no—
193C2×#41—44no—
194C2×#41—432no—
195C2×#41—432no—
196C2×#41—48no—
197C2×#41—48no—
198C2×#41C2×Q8116yes—
199C22×Q8—21no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8C2×Q814yes—
205C22×Q8C2412yes—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—28no—
209C22×D4—24no—
210C22×D4—21no—
211C22×D4—22no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4C2×Q814yes—
215C22×D4C2412yes—
216C23×C4—21no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D412yes—
230C23×C4C2×Q814yes—
231C23×C4#4114yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C2×M16—28no—
235C2×M16—44no—
236C2×M16—44no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—42no—
240C4⋊Q8—42no—
241#125—44no—
242C22⋊Q8—88no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—28no—
246C4×D4—44no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—44no—
250C4×D4—48no—
251C4×D4—44no—
252#117—42no—
253#117—42no—
254#117—42no—
255#117—42no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2414yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2411yescharacteristic
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2411yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—84no—
285C2×C22⋊C4C2×D412yes—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C2412yes—
292C2×C42C2412yes—
293#107—44no—
294#107—44no—
295#106—44no—
296#106—44no—
297C4⋊C8—42no—
298C4⋊C8—42no—
299C4⋊C8—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304#102—28no—
305#102—28no—
306#102—28no—
307#102—28no—
308C22⋊C8—28no—
309C22⋊C8—24no—
310C22⋊C8—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95C2×D414yes—
324#95C2×D414yes—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—24no—
329C24—42no—
330C24—41no—
331C24—41no—
332C24—42no—
333C24—82no—
334C24—82no—
335C24C22⋊Q814yes—
336C24C22×D411yescharacteristic
337C24C22×D412yes—
338#41—232no—
339#41—416no—
340#41—416no—
341#41—416no—
342#41—464no—
343C2×Q8—24no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—48no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—48no—
351C2×Q8—84no—
352C2×Q8—82no—
353C2×Q8C22×Q814yes—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—216no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—432no—
366C2×D4C22×Q814yes—
367C2×D4H(5,𝔽2)14yes—
368C2×D4H(5,𝔽2)14yes—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—44no—
390C22×C4—48no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—432no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4C22⋊Q814yes—
406C22×C4C22⋊Q818yes—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×Q812yes—
413C22×C4C22×Q814yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4114yes—
416C22×C4C2511yescharacteristic
417M16—88no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—82no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—48no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C22⋊C4—84no—
454C42—24no—
455C42—22no—
456C42—22no—
457C42—42no—
458C42—42no—
459C23—216no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—44no—
474C23—48no—
475C23—44no—
476C23—44no—
477C23—88no—
478C23—88no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—82no—
483C23—82no—
484C23C2×C22⋊Q812yes—
485C23C2×C22⋊Q814yes—
486C23C23×D411yescharacteristic
487Q8—28no—
488Q8—48no—
489Q8—44no—
490Q8—164no—
491D4—216no—
492D4—24no—
493D4—24no—
494D4—24no—
495D4—432no—
496D4#54318yes—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—432no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—168no—
533C2×C4—164no—
534C2×C4—164no—
535C2×C4—164no—
536C2×C4—164no—
537C2×C4C2×C22⋊Q814yes—
538C2×C4C2×C22⋊Q812yes—
539C2×C4C23×Q811yescharacteristic
540C2×C4C22×#4111yescharacteristic
541C2×C4C2×H(5,𝔽2)12yes—
542C2×C4C2×H(5,𝔽2)12yes—
543C8—44no—
544C8—44no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—22no—
549C22—24no—
550C22—28no—
551C22—48no—
552C22—42no—
553C22—48no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—41no—
559C22—44no—
560C22—42no—
561C22—41no—
562C22—816no—
563C22—88no—
564C22C22×C22⋊Q811yescharacteristic
565C22#348411yescharacteristic
566C22#348411yescharacteristic
567C22#358512yes—
568C22#358612yes—
569C4—24no—
570C4—22no—
571C4—44no—
572C4—44no—
573C4—42no—
574C4—42no—
575C4—84no—
576C4—82no—
577C4—81no—
578C4—84no—
579C4—81no—
580C4C2×#54312yes—
581C4#349012yes—
582C2—44no—
583C2—42no—
584C2—48no—
585C2#3498211yescharacteristic
586C2#6049312yes—
587C2#6164712yes—
588C2#6164811yescharacteristic
589C2#6164911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37477C2semidirect
3. #32966C2semidirect
4. #32962C2semidirect
5. #29963C2semidirect
6. #29960C2semidirect
7. #25980C2semidirect
8. #25978C2semidirect
9. #25972C2semidirect
10. #25970C2semidirect
11. #21944C2semidirect
12. #21935C2semidirect
17. C2×#478C22semidirect
18. C2×#476C22semidirect
19. C2×#474C22semidirect
28. #2442C22semidirect
29. #2440C22semidirect
42. #2097C22semidirect
43. #2096C22semidirect
44. #2095C22semidirect
45. #2068C22semidirect
46. #2068C22semidirect
47. #2066C22semidirect
48. #2066C22semidirect
49. #2026C22semidirect
50. #2026C22semidirect
51. #2024C22semidirect
52. #1864C22semidirect
53. #1864C22semidirect
54. #1862C22semidirect
55. #1862C22semidirect
58. #1853C22semidirect
59. #1853C22semidirect
60. #1852C22semidirect
61. #1852C22semidirect
62. #1851C22semidirect
63. #1851C22semidirect
64. #1809C22semidirect
65. #1809C22semidirect
66. #1804C22semidirect
67. #1804C22semidirect
68. #1802C22semidirect
79. C22×#41Q8semidirect

Maximal subgroups

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

subgroupindex
#374772
#341722
#329662
#329622
#299632
#299602
#259802
#259782
#259722
#259702
#219442
#219352
#206012