Finite Group Databasebeta

← #6500795

Subgroups — #6500795

Order 512.

2372 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2416C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic, complemented
3#27613C212yesmaximal, complemented
4#27601C212yesmaximal, complemented
5#27585C212yesmaximal, complemented
6#26202C218yesmaximal, complemented
7#26130C218yesmaximal, complemented
8#24005C212yesmaximal, complemented
9#21758C212yesmaximal, complemented
10#21738C211yesmaximal, characteristic, complemented
11C2×#1853C212yesmaximal, complemented
12C22×#121—21no—
13C22×#121C2211yescharacteristic
14C22×#121C2211yescharacteristic
15C2×#518—21no—
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#481C2214yescomplemented
19C22×#106C2214yescomplemented
20C22×C22⋊C8C2212yes—
21#2416—24no—
22#2416C2218yes—
23C2×#396—21no—
24C2×#394—21no—
25#2349—24no—
26D4×C4⋊C4C2214yescomplemented
27D4×C4⋊C4C2214yescomplemented
28#2316—21no—
29#2315—21no—
30#2315—21no—
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#386—21no—
34#2285C2212yes—
35C2×#380C2212yes—
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38#2100C2214yescomplemented
39#2094C2214yescomplemented
40#2084C2214yescomplemented
41#2077C2218yescomplemented
42#2071C2214yescomplemented
43#2034C2218yescomplemented
44#1926C2218yescomplemented
45#1919C2218yescomplemented
46#1893C2218yescomplemented
47#1874C2214yescomplemented
48#1874C2214yescomplemented
49#1874C2214yescomplemented
50#1869C2218yescomplemented
51#1853C2214yes—
52#1853C2214yes—
53#1853C2214yescomplemented
54#1853C2218yescomplemented
55#1824C2214yescomplemented
56#1823C2214yes—
57#1817C2212yes—
58#1786C2214yes—
59#1779C2212yes—
60C2×#340C2214yescomplemented
61C23×D4—21no—
62C23×D4—21no—
63C23×D4D411yescharacteristic
64C23×D4C2311yescharacteristic
65C24×C4—21no—
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C23×C8C2314yes—
69C2×#121—28no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121—28no—
73C2×#121—44no—
74C2×#121—48no—
75C2×#121C2314yes—
76C2×#121C2314yes—
77#518—48no—
78C2×C4×D4—28no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4D414yes—
82C2×C4×D4C2314yes—
83C2×#117—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C22⋊C4—22no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—21no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4—21no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4D411yescharacteristic
105C22×C22⋊C4D411yescharacteristic
106C22×C42C2311yescharacteristic
107#481—48no—
108C2×#106—48no—
109C2×#106C2318yes—
110C2×#106C2318yes—
111C2×#102C2318yes—
112C2×#102C2318yes—
113C2×C22⋊C8—28no—
114C2×C22⋊C8C2314yes—
115C2×C22⋊C8C2314yes—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118#396—28no—
119#394—28no—
120#390—22no—
121#390C2314yes—
122#389—24no—
123#389—24no—
124#389—22no—
125#389C2314yes—
126#386—28no—
127#385—24no—
128#385—24no—
129#385—24no—
130#385—24no—
131#384C2312yes—
132#384C2312yes—
133#382—22no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380—22no—
139#380—22no—
140#380C2314yes—
141#380C2314yes—
142#380C2314yes—
143#380C2314yes—
144C4×C4⋊C4—22no—
145C4×C4⋊C4C2314yes—
146C2×#95—22no—
147C2×#95—22no—
148C2×#95—21no—
149C2×#95—21no—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#336C2314yes—
155#336C2314yes—
156C25—21no—
157C25C2×D411yescharacteristic
158C25C2×D411yescharacteristic
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—21no—
164C22×D4—24no—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—82no—
173C22×D4SD1618yes—
174C22×D4C2×D414yes—
175C22×D4C2412yes—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—82no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4#4118yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C22×C8—28no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8C2418yes—
206#121—28no—
207#121—28no—
208#121—28no—
209#121—44no—
210#121—48no—
211#121—432no—
212#121—48no—
213C4×D4—216no—
214C4×D4—28no—
215C4×D4—28no—
216C4×D4—416no—
217C4×D4—416no—
218#117—48no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—48no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2414yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2412yes—
250C2×C4⋊C4C2412yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—82no—
268C2×C22⋊C4C2×D414yes—
269C2×C22⋊C4C2×D414yes—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42C2×D412yes—
273C2×C42C2412yes—
274#106—216no—
275#106—48no—
276#102—48no—
277#102—48no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280C22⋊C8—44no—
281C22⋊C8—44no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24#12314yes—
307C24C2×SD1614yes—
308C24C2×SD1614yes—
309C24AGL(1,ℤ/8)14yes—
310C24C22×D412yes—
311C24C22×D411yescharacteristic
312C24C22×D412yes—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—232no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—48no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—42no—
327C2×D4—88no—
328C2×D4—84no—
329C2×D4—816no—
330C2×D4—816no—
331C2×D4C2×SD16116yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—416no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—48no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—88no—
376C22×C4—84no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—816no—
386C22×C4#12314yes—
387C22×C4#12318yes—
388C22×C4C2×SD1614yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C2×#4114yes—
395C22×C4C2×#4114yes—
396C22×C4C2×#4114yes—
397C22×C4C2511yescharacteristic
398C2×C8—28no—
399C2×C8—28no—
400C2×C8—28no—
401C2×C8—416no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4H(5,𝔽2)14yes—
422C22⋊C4—28no—
423C22⋊C4—28no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—84no—
432C42—24no—
433C42—24no—
434C42—48no—
435C23—21no—
436C23—22no—
437C23—21no—
438C23—22no—
439C23—21no—
440C23—22no—
441C23—21no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—28no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23#481116yescomplemented
477C23C2×#12314yes—
478C23C2×#12312yes—
479C23C22×SD1614yes—
480C23C22×SD1612yes—
481C23C22×SD1614yes—
482C23C2×AGL(1,ℤ/8)12yes—
483C23C2×AGL(1,ℤ/8)14yes—
484C23C23×D411yescharacteristic
485D4—232no—
486D4—48no—
487D4—84no—
488D4—816no—
489D4—816no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—48no—
508C2×C4—44no—
509C2×C4—48no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—84no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—816no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4—164no—
532C2×C4—164no—
533C2×C4—164no—
534C2×C4—164no—
535C2×C4C2×#12312yes—
536C2×C4C2×#12314yes—
537C2×C4C22×SD1614yes—
538C2×C4C22×#4112yes—
539C2×C4C2×H(5,𝔽2)12yes—
540C2×C4#58512yes—
541C8—48no—
542C22—24no—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—22no—
547C22—24no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—22no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—416no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—88no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—168no—
571C22C2×#48114yes—
572C22C2×#48118yes—
573C22C22×#12311yescharacteristic
574C22#348411yescharacteristic
575C22#348511yescharacteristic
576C22C23×SD1611yescharacteristic
577C22C22×AGL(1,ℤ/8)11yescharacteristic
578C22#358511yescharacteristic
579C22#358611yescharacteristic
580C4—22no—
581C4—24no—
582C4—24no—
583C4—42no—
584C4—42no—
585C4—44no—
586C4—84no—
587C4—82no—
588C4—82no—
589C4—84no—
590C4#350514yes—
591C2—24no—
592C2—28no—
593C2—44no—
594C2—162no—
595C2#3535111yescharacteristic
596C2C22×#48111yescharacteristic
597C2#6052011yescharacteristic
598C2#6156011yescharacteristic
599C2#6157211yescharacteristic
600C2#6169511yescharacteristic
601C2#6170711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2416C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34181C2semidirect
3. #27613C2semidirect
4. #27601C2semidirect
5. #27585C2semidirect
6. #26202C2semidirect
7. #26130C2semidirect
8. #24005C2semidirect
9. #21758C2semidirect
10. #21738C2semidirect
11. C2×#1853C2semidirect
18. C2×#481C22semidirect
19. C22×#106C22semidirect
26. D4×C4⋊C4C22semidirect
27. D4×C4⋊C4C22semidirect
38. #2100C22semidirect
39. #2094C22semidirect
40. #2084C22semidirect
41. #2077C22semidirect
42. #2071C22semidirect
43. #2034C22semidirect
44. #1926C22semidirect
45. #1919C22semidirect
46. #1893C22semidirect
47. #1874C22semidirect
48. #1874C22semidirect
49. #1874C22semidirect
50. #1869C22semidirect
53. #1853C22semidirect
54. #1853C22semidirect
55. #1824C22semidirect
60. C2×#340C22semidirect
476. C23#481semidirect

Maximal subgroups

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

subgroupindex
C2×#24162
#341812
#276132
#276012
#275852
#262022
#261302
#240052
#217582
#217382
C2×#18532