Finite Group Databasebeta

← #6532006

Subgroups — #6532006

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#34187C211yesmaximal, characteristic, complemented
3#27616C212yesmaximal, complemented
4#27613C211yesmaximal, characteristic, complemented
5#27610C211yesmaximal, characteristic, complemented
6#27588C212yesmaximal
7#27585C211yesmaximal, characteristic
8#27582C211yesmaximal, characteristic
9#25073C214yesmaximal, complemented
10#25072C214yesmaximal, complemented
11#25007C214yesmaximal, complemented
12#25006C214yesmaximal, complemented
13#21987C212yesmaximal, complemented
14#21822C211yesmaximal, characteristic, complemented
15#20739C212yesmaximal
16C22×C22⋊Q8—21no—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#432C2212yes—
20#2538—22no—
21#2538—22no—
22#2538C2214yes—
23#2538C2214yes—
24C2×#396—21no—
25C2×#396—21no—
26#2356—22no—
27#2356—22no—
28#2355C2212yescomplemented
29#2355C2212yescomplemented
30#2355C2212yescomplemented
31#2355C2212yescomplemented
32#2313—21no—
33#2313—21no—
34#2310—21no—
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39#2285C2211yescharacteristic
40#2285C2211yescharacteristic
41C2×#384C2211yescharacteristic
42#2283C2212yes—
43#2283C2212yes—
44C2×#380—21no—
45#2274—21no—
46#2101C2214yescomplemented
47#2100C2212yescomplemented
48#2099C2212yescomplemented
49#2094C2212yes—
50#2093C2212yes—
51#2092C2214yes—
52#2085C2214yescomplemented
53#2084C2212yescomplemented
54#2083C2212yescomplemented
55#2073C2214yes—
56#2071C2212yes—
57#2071C2212yes—
58#1985C2214yescomplemented
59#1985C2214yescomplemented
60#1948C2214yescomplemented
61#1947C2214yescomplemented
62#1926C2214yes—
63#1926C2214yes—
64#1925C2214yes—
65#1925C2214yes—
66#1921C2214yescomplemented
67#1920C2214yescomplemented
68#1919C2214yescomplemented
69#1918C2214yescomplemented
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1847C2214yescomplemented
75#1836C2214yescomplemented
76#1823C2212yes—
77#1823C2212yes—
78#1817C2211yescharacteristic
79#1817C2211yescharacteristic
80#1817C2212yes—
81#1804C2214yes—
82#1804C2214yes—
83#1786C2212yes—
84#1786C2212yes—
85#1762C2214yes—
86#1762C2214yes—
87C23×Q8D411yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—48no—
93C2×C4×Q8—28no—
94C2×C4×Q8D412yes—
95C2×C4×Q8D412yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4—22no—
99C2×C4×D4—28no—
100C2×C4×D4C2314yes—
101C2×#117—21no—
102C2×#117—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4D411yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#432—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×#97C2314yes—
134#396—24no—
135#396—24no—
136#396—24no—
137#396—24no—
138#391—22no—
139#391—22no—
140#391—22no—
141#391—22no—
142#390—22no—
143#390—21no—
144#390—21no—
145#390C2312yes—
146#390C2312yes—
147#390C2314yes—
148#389—22no—
149#389—21no—
150#389—21no—
151#389C2312yes—
152#389C2312yes—
153#389C2314yes—
154#387—22no—
155#387—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#384—22no—
163#384C2311yescharacteristic
164#384C2311yescharacteristic
165#384C2311yescharacteristic
166#384C2311yescharacteristic
167#384C2314yes—
168#383—22no—
169#383—22no—
170#382—21no—
171#382—21no—
172#382C2312yes—
173#382C2312yes—
174#382C2312yes—
175#382C2312yes—
176#381—22no—
177#381—22no—
178#380—21no—
179#380—21no—
180#380—24no—
181#380—24no—
182#380C2312yes—
183#380C2312yes—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198#376—22no—
199#376—22no—
200C2×#95—21no—
201C2×#95—21no—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#336C2312yes—
211#336C2312yes—
212#336C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2314yes—
216C25—21no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—42no—
220C22×Q8C2×D412yes—
221C22×Q8C2×D412yes—
222C22×D4—24no—
223C22×D4—21no—
224C22×D4—24no—
225C22×D4—42no—
226C22×D4C2412yes—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2411yescharacteristic
241C22×C8C2414yes—
242C22×C8C2414yes—
243C22⋊Q8—416no—
244C22⋊Q8—416no—
245C4×Q8—28no—
246C4×Q8—28no—
247C4×Q8—24no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×D4—28no—
252C4×D4—28no—
253C4×D4—48no—
254#117—44no—
255#117—44no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4—82no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D412yes—
321C2×C42#4118yes—
322C2×C42C2412yes—
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2412yes—
326#107—44no—
327#107—44no—
328#106—44no—
329#106—44no—
330C4⋊C8—28no—
331#102—28no—
332#102—28no—
333#102—44no—
334#102—44no—
335C22⋊C8—24no—
336C22⋊C8—24no—
337#97—28no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357C24—24no—
358C24—22no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—42no—
363C24—42no—
364C24—42no—
365C24C22×D411yescharacteristic
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—416no—
373C2×D4—24no—
374C2×D4—24no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—416no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D42−1+414yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—48no—
417C22×C4—816no—
418C22×C4#12314yes—
419C22×C4#12314yes—
420C22×C4AGL(1,ℤ/8)14yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C2×#4114yes—
431C22×C4C2511yescharacteristic
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—84no—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—84no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C42—22no—
478C42—22no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—44no—
485C42—44no—
486C42—82no—
487C42—82no—
488C42—82no—
489C42—82no—
490C42#12318yes—
491C42C2×#4114yes—
492C42C2×#4114yes—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—24no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23C2×#12312yes—
512C23C2×AGL(1,ℤ/8)12yes—
513C23#57512yes—
514C23C23×D411yescharacteristic
515Q8—48no—
516Q8—48no—
517D4—28no—
518D4—48no—
519D4—44no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—816no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4C2×#12312yes—
575C2×C4C2×#12314yes—
576C2×C4C2×#12314yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C22×#4112yes—
580C2×C4C2×2−1+412yes—
581C2×C4#58512yes—
582C8—44no—
583C8—44no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—816no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22#309414yes—
599C22C22×#12311yescharacteristic
600C22#348611yescharacteristic
601C22#348711yescharacteristic
602C22C22×AGL(1,ℤ/8)11yescharacteristic
603C22C2×#57511yescharacteristic
604C22#358511yescharacteristic
605C22#358611yescharacteristic
606C4—22no—
607C4—24no—
608C4—22no—
609C4—22no—
610C4—24no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#349514yes—
623C2—24no—
624C2—84no—
625C2#3567211yescharacteristic
626C2C2×#309411yescharacteristic
627C2#6052011yescharacteristic
628C2#6182711yescharacteristic
629C2#6183111yescharacteristic
630C2#6189011yescharacteristic
631C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34187C2semidirect
3. #27616C2semidirect
4. #27613C2semidirect
5. #27610C2semidirect
9. #25073C2semidirect
10. #25072C2semidirect
11. #25007C2semidirect
12. #25006C2semidirect
13. #21987C2semidirect
14. #21822C2semidirect
28. #2355C22semidirect
29. #2355C22semidirect
30. #2355C22semidirect
31. #2355C22semidirect
46. #2101C22semidirect
47. #2100C22semidirect
48. #2099C22semidirect
52. #2085C22semidirect
53. #2084C22semidirect
54. #2083C22semidirect
58. #1985C22semidirect
59. #1985C22semidirect
60. #1948C22semidirect
61. #1947C22semidirect
66. #1921C22semidirect
67. #1920C22semidirect
68. #1919C22semidirect
69. #1918C22semidirect
74. #1847C22semidirect
75. #1836C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#341872
#276162
#276132
#276102
#275882
#275852
#275822
#250732
#250722
#250072
#250062
#219872
#218222
#207392