Finite Group Databasebeta

← #6478172

Subgroups — #6478172

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35014C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic, complemented
4#33108C211yesmaximal, characteristic, complemented
5#32982C212yesmaximal
6#29968C211yesmaximal, characteristic, complemented
7#29893C212yesmaximal
8#29756C211yesmaximal, characteristic, complemented
9#27456C214yesmaximal, complemented
10#27392C214yesmaximal, complemented
11#27328C214yesmaximal, complemented
12#27160C214yesmaximal, complemented
13#22078C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic
15C4×#483C212yesmaximal, complemented
16C22×#117—21no—
17C23×C4⋊C4C2211yescharacteristic
18C2×#483C2212yes—
19C2×#483C2212yes—
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2848C2212yes—
23#2646—21no—
24#2626—21no—
25#2589—22no—
26#2587—22no—
27#2497C2212yes—
28#2492C2212yes—
29#2429C2212yes—
30#2423C2212yes—
31#2344—22no—
32#2344—22no—
33#2344C2214yes—
34#2344C2214yes—
35#2329—21no—
36#2312—21no—
37C4×C22⋊Q8C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#387—21no—
41C2×#387—21no—
42C2×#381—21no—
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47#2100C2212yescomplemented
48#2088C2214yescomplemented
49#2084C2212yescomplemented
50#2077C2214yes—
51#2077C2214yes—
52#2066C2214yescomplemented
53#2058C2212yescomplemented
54#2043C2212yescomplemented
55#2035C2214yes—
56#2035C2214yes—
57#1987C2212yescomplemented
58#1981C2212yescomplemented
59#1948C2212yescomplemented
60#1948C2212yescomplemented
61#1941C2214yescomplemented
62#1933C2214yescomplemented
63#1927C2214yescomplemented
64#1927C2214yescomplemented
65#1923C2214yes—
66#1923C2214yes—
67#1920C2214yescomplemented
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1872C2214yes—
71#1872C2214yes—
72#1872C2212yes—
73#1872C2212yes—
74#1865C2214yescomplemented
75#1853C2212yes—
76#1853C2214yes—
77#1853C2212yes—
78#1853C2214yes—
79#1823C2212yes—
80#1823C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1779C2212yes—
84C4×#106C2214yescomplemented
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C24×C4—21no—
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8C2312yes—
93C2×C22⋊Q8C2312yes—
94C2×C22⋊Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—48no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4—21no—
111C22×C22⋊C4D411yescharacteristic
112C22×C22⋊C4D411yescharacteristic
113C22×C42—21no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#483—28no—
118#483—28no—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2314yes—
133C2×C4×C8C2314yes—
134#400—22no—
135#400—22no—
136#399—22no—
137#398—22no—
138#398—22no—
139#396—22no—
140#396—22no—
141#393—22no—
142#391—21no—
143#391—22no—
144#391C2312yes—
145#391C2312yes—
146#389—21no—
147#389—21no—
148#389—21no—
149#389—22no—
150#389—21no—
151#389—21no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2312yes—
156#389C2314yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—24no—
166#387—24no—
167#387—24no—
168#387—24no—
169#385—22no—
170#385—22no—
171#385—22no—
172#385—22no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#381—24no—
176#381—24no—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—22no—
181#380C2312yes—
182#380C2312yes—
183#380C2314yes—
184#380C2312yes—
185#379—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2314yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—22no—
195C4×C22⋊C4D412yes—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4C2312yes—
198C4×C22⋊C4C2312yes—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95D411yescharacteristic
205C2×#95D411yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217C25C2×D411yescharacteristic
218C22×Q8—21no—
219C22×Q8C2412yes—
220C23×C4—28no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—21no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C22×C8C2414yes—
241C22×C8C2414yes—
242C4⋊Q8—42no—
243C4⋊Q8—42no—
244C22⋊Q8—24no—
245C22⋊Q8—24no—
246C22⋊Q8—44no—
247C22⋊Q8—44no—
248C22⋊Q8—44no—
249C4×Q8—44no—
250#117—416no—
251#117—416no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
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—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4#4114yes—
284C2×C4⋊C4#4114yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4SD1618yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D412yes—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#106—28no—
322#106—28no—
323#106—44no—
324#106—44no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327#103—28no—
328#103—28no—
329#103—44no—
330#103—44no—
331C22⋊C8—24no—
332C22⋊C8—24no—
333C22⋊C8—44no—
334C22⋊C8—44no—
335C4×C8—44no—
336#95—24no—
337#95—22no—
338#95—22no—
339#95—24no—
340#95—24no—
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#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356#95C2×D412yes—
357C24—22no—
358C24—22no—
359C24—41no—
360C24—41no—
361C24—41no—
362C24—41no—
363C24C2×SD1614yes—
364C24C22×D412yes—
365C24C22×D411yescharacteristic
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—42no—
371C2×Q8—81no—
372C2×Q8—81no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—432no—
409C22×C4—48no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C2×SD1614yes—
422C22×C4C2×SD1614yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2511yescharacteristic
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C4⋊C4—216no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4C2×#4114yes—
471C4⋊C4C2×#4114yes—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4C2×SD1618yes—
482C22⋊C4C2×SD1618yes—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—42no—
487C42—41no—
488C42—41no—
489C42—48no—
490C42—44no—
491C42—44no—
492C42—82no—
493C42—82no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—28no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23#52512yes—
512C23C22×SD1614yes—
513C23C22×SD1612yes—
514C23#57512yes—
515C23C23×D411yescharacteristic
516Q8—44no—
517Q8—44no—
518Q8—162no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—44no—
547C2×C4—432no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
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—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4#52512yes—
579C2×C4C22×SD1614yes—
580C2×C4C22×SD1612yes—
581C2×C4C22×SD1612yes—
582C2×C4C22×#4111yescharacteristic
583C2×C4C22×#4111yescharacteristic
584C2×C4#58512yes—
585C2×C4#58512yes—
586C8—44no—
587C8—44no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—416no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22#310114yes—
603C22C2×#52511yescharacteristic
604C22#348511yescharacteristic
605C22#348511yescharacteristic
606C22C23×SD1611yescharacteristic
607C22C2×#57511yescharacteristic
608C22#358611yescharacteristic
609C22#358611yescharacteristic
610C4—22no—
611C4—24no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—41no—
616C4—41no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4#345512yes—
627C4#349312yes—
628C2—24no—
629C2—44no—
630C2#3509311yescharacteristic
631C2C2×#310111yescharacteristic
632C2#6056611yescharacteristic
633C2#6169511yescharacteristic
634C2#6169511yescharacteristic
635C2#6171111yescharacteristic
636C2#6171611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33148C2semidirect
4. #33108C2semidirect
6. #29968C2semidirect
8. #29756C2semidirect
9. #27456C2semidirect
10. #27392C2semidirect
11. #27328C2semidirect
12. #27160C2semidirect
13. #22078C2semidirect
15. C4×#483C2semidirect
47. #2100C22semidirect
48. #2088C22semidirect
49. #2084C22semidirect
52. #2066C22semidirect
53. #2058C22semidirect
54. #2043C22semidirect
57. #1987C22semidirect
58. #1981C22semidirect
59. #1948C22semidirect
60. #1948C22semidirect
61. #1941C22semidirect
62. #1933C22semidirect
63. #1927C22semidirect
64. #1927C22semidirect
67. #1920C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
74. #1865C22semidirect
84. C4×#106C22semidirect
85. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#350142
C2×#23442
#331482
#331082
#329822
#299682
#298932
#297562
#274562
#273922
#273282
#271602
#220782
#219022
C4×#4832