Finite Group Databasebeta

← #6484531

Subgroups — #6484531

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic, complemented
2C2×#2367C211yesmaximal, characteristic
3#32960C212yesmaximal, complemented
4#32959C212yesmaximal
5#29957C212yesmaximal, complemented
6#29956C212yesmaximal
7#26545C214yesmaximal, complemented
8#26543C214yesmaximal, complemented
9#26479C214yesmaximal, complemented
10#26477C214yesmaximal, complemented
11#21955C211yesmaximal, characteristic, complemented
12#21808C211yesmaximal, characteristic
13#21808C211yesmaximal, characteristic
14#21365C211yesmaximal, characteristic, complemented
15#21365C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#479C2214yescomplemented
20C2×#478C2214yes—
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23#2851C2212yes—
24#2851C2212yes—
25#2646—21no—
26#2646—21no—
27#2571C2212yescomplemented
28#2571C2212yescomplemented
29#2570—22no—
30#2570—22no—
31#2552C2212yescomplemented
32#2552C2212yescomplemented
33#2367—22no—
34#2367—22no—
35#2367C2214yes—
36#2367C2214yes—
37#2317—21no—
38#2317—21no—
39C2×#391C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#388—21no—
42C2×#388—21no—
43C2×#385—21no—
44#2285C2211yescharacteristic
45#2285C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4—21no—
49#2097C2214yescomplemented
50#2092C2214yes—
51#2068C2214yescomplemented
52#2067C2214yes—
53#2066C2214yescomplemented
54#2065C2214yes—
55#2026C2214yescomplemented
56#2025C2214yes—
57#1983C2214yes—
58#1983C2214yes—
59#1959C2212yes—
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1948C2212yescomplemented
64#1948C2212yescomplemented
65#1947C2212yescomplemented
66#1947C2212yescomplemented
67#1940C2214yescomplemented
68#1940C2214yescomplemented
69#1865C2214yescomplemented
70#1864C2214yes—
71#1863C2214yescomplemented
72#1862C2214yes—
73#1853C2212yes—
74#1853C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1805C2214yescomplemented
78#1804C2214yes—
79#1797C2212yes—
80#1797C2212yes—
81#1787C2212yes—
82#1787C2212yes—
83#1786C2212yes—
84#1786C2212yes—
85#1776C2214yescomplemented
86#1776C2214yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C2×#125C2312yes—
91C2×#125C2312yes—
92C2×#124—21no—
93C2×#124—21no—
94C2×#124—24no—
95C2×#124Q818yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×C4×Q8—22no—
99C2×C4×Q8—44no—
100C2×C4×Q8C2314yes—
101C2×C4×D4—22no—
102C2×C4×D4—44no—
103C2×C4×D4C2314yes—
104C2×#117—21no—
105C2×#117—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C42—21no—
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#479—48no—
122#478—48no—
123#432—24no—
124#432—24no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×#97C2312yes—
144C2×#97C2312yes—
145#400—22no—
146#400—22no—
147#400—22no—
148#400—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#398—22no—
153#391—22no—
154#391C2314yes—
155#390—22no—
156#390C2314yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—24no—
166#388—24no—
167#388—24no—
168#388—24no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—24no—
178#385—24no—
179#384—22no—
180#384—22no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—22no—
186#382—22no—
187#382—21no—
188#382—21no—
189#382C2312yes—
190#382C2312yes—
191#380—21no—
192#380—21no—
193#380C2312yes—
194#380C2312yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C22⋊C4—24no—
208C4×C22⋊C4—24no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25—21no—
221C22×Q8—21no—
222C22×Q8—22no—
223C22×Q8—42no—
224C22×Q8—42no—
225C22×Q8C2×Q814yes—
226C22×Q8C2412yes—
227C22×D4—21no—
228C22×D4—22no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4C2×Q814yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—24no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—82no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C22×C8C2414yes—
248C22×C8C2414yes—
249#125—44no—
250#125—44no—
251#124—28no—
252#124—28no—
253#124—24no—
254#124—24no—
255#124C2×Q818yes—
256#124C2×Q818yes—
257C4×Q8—28no—
258C4×Q8—44no—
259C4×Q8—48no—
260C4×D4—28no—
261C4×D4—44no—
262C4×D4—48no—
263#117—44no—
264#117—44no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—82no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42—82no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×Q814yes—
327C2×C42#4114yes—
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#107—28no—
333#107—44no—
334#106—28no—
335#106—44no—
336C4⋊C8—24no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339C4⋊C8—44no—
340#103—44no—
341#103—44no—
342#103—44no—
343#103—44no—
344#102—44no—
345#102—44no—
346#102—44no—
347#102—44no—
348C22⋊C8—24no—
349C22⋊C8—24no—
350#97—24no—
351#97—24no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—24no—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24C22×D411yescharacteristic
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—48no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—48no—
389C2×Q8C22×Q814yes—
390C2×D4—24no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—48no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—48no—
398C2×D4C22×Q814yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—84no—
436C22×C4—816no—
437C22×C4C22⋊Q814yes—
438C22×C4C22⋊Q814yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×Q812yes—
449C22×C4C2×#4112yes—
450C22×C4C2511yescharacteristic
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—48no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4H(5,𝔽2)14yes—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—84no—
497C42—24no—
498C42—24no—
499C42—42no—
500C42—42no—
501C42—48no—
502C42—44no—
503C42—44no—
504C42—82no—
505C42—82no—
506C42C22⋊Q814yes—
507C42C22⋊Q814yes—
508C42C22×Q812yes—
509C42C22×Q812yes—
510C42C2×#4112yes—
511C42C2×#4112yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—24no—
517C23—48no—
518C23—48no—
519C23—42no—
520C23—42no—
521C23—48no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23C2×C22⋊Q812yes—
531C23#57512yes—
532C23#57512yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—48no—
536Q8—44no—
537D4—28no—
538D4—48no—
539D4—44no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—816no—
594C2×C4—84no—
595C2×C4C2×C22⋊Q812yes—
596C2×C4C2×C22⋊Q812yes—
597C2×C4C2×C22⋊Q812yes—
598C2×C4C2×C22⋊Q812yes—
599C2×C4C2×C22⋊Q812yes—
600C2×C4C23×Q811yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)12yes—
603C2×C4#58512yes—
604C8—44no—
605C8—44no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—816no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22#308014yes—
621C22C22×C22⋊Q811yescharacteristic
622C22#348411yescharacteristic
623C22#348711yescharacteristic
624C22C2×#57511yescharacteristic
625C22C2×#57511yescharacteristic
626C22#358511yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—44no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#346012yes—
645C4#350112yes—
646C2—24no—
647C2—84no—
648C2#3517411yescharacteristic
649C2C2×#308011yescharacteristic
650C2#6049211yescharacteristic
651C2#6157611yescharacteristic
652C2#6157811yescharacteristic
653C2#6189411yescharacteristic
654C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35181C2semidirect
3. #32960C2semidirect
5. #29957C2semidirect
7. #26545C2semidirect
8. #26543C2semidirect
9. #26479C2semidirect
10. #26477C2semidirect
11. #21955C2semidirect
14. #21365C2semidirect
15. #21365C2semidirect
19. C2×#479C22semidirect
27. #2571C22semidirect
28. #2571C22semidirect
31. #2552C22semidirect
32. #2552C22semidirect
49. #2097C22semidirect
51. #2068C22semidirect
53. #2066C22semidirect
55. #2026C22semidirect
63. #1948C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1947C22semidirect
67. #1940C22semidirect
68. #1940C22semidirect
69. #1865C22semidirect
71. #1863C22semidirect
77. #1805C22semidirect
85. #1776C22semidirect
86. #1776C22semidirect

Maximal subgroups

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

subgroupindex
#351812
C2×#23672
#329602
#329592
#299572
#299562
#265452
#265432
#264792
#264772
#219552
#218082
#213652