Finite Group Databasebeta

← #6566106

Subgroups — #6566106

Order 512.

1861 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3357C212yesmaximal
2#36180C211yesmaximal, characteristic, complemented
3#34321C211yesmaximal, characteristic
4#29868C211yesmaximal, characteristic, complemented
5#29759C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28577C212yesmaximal, complemented
8#28206C211yesmaximal, characteristic, complemented
9#27644C212yesmaximal, complemented
10#25455C212yesmaximal, complemented
11#25223C214yesmaximal, complemented
12#24815C212yesmaximal, complemented
13#24596C214yesmaximal, complemented
14#23677C211yesmaximal, characteristic, complemented
15#23658C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22502C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C2×#554C2212yes—
20#3357—216no—
21C2×#500C2212yes—
22C2×#485C2214yes—
23C2×#480C2214yes—
24C2×#477C2212yes—
25C2×#476C2212yes—
26C2×#468C2214yes—
27C2×C4×SD16C2212yes—
28C2×#436C2212yes—
29C2×#428C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2567C2212yes—
33#2552—21no—
34#2552—21no—
35#2502C2214yes—
36#2440—22no—
37#2440—22no—
38#2369C2211yescharacteristic
39#2367—22no—
40#2363C2212yes—
41#2361—21no—
42#2354C2211yescharacteristic
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2211yescharacteristic
46#2347—21no—
47#2347C2212yes—
48#2337—22no—
49#2324—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#121C2212yes—
53#2296C2212yes—
54#2293C2211yescharacteristic
55C2×#384C2212yes—
56#2274—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2214yes—
60#2084C2212yes—
61#2084C2211yescharacteristic
62#2083C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2214yes—
66#2071C2211yescharacteristic
67#2071C2212yes—
68#2071C2212yes—
69#2068C2214yes—
70#2066C2212yes—
71#2066C2214yes—
72#2055C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2013C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1947C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1928C2214yes—
82#1928C2214yes—
83#1894C2214yes—
84#1877C2214yes—
85#1873C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1823C2214yes—
89#1803C2212yes—
90#1802C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×SD16C2312yes—
97C22×M16C2314yes—
98#555—24no—
99#555D418yes—
100#554—28no—
101#554—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—22no—
108C2×#124—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8C2314yes—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—22no—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×#117—21no—
123C2×#117—21no—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2314yes—
130C22×C42C2312yes—
131#500—28no—
132#485—216no—
133#480—216no—
134#477—28no—
135#476—28no—
136#468—216no—
137C4×SD16—28no—
138#436—28no—
139#428—216no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157#398—24no—
158#398—24no—
159#395—24no—
160#391—21no—
161#391—22no—
162#391C2312yes—
163#391C2314yes—
164#390—21no—
165#390—22no—
166#390C2311yescharacteristic
167#390C2311yescharacteristic
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389C2312yes—
177#389C2311yescharacteristic
178#389C2312yes—
179#389C2312yes—
180#389C2311yescharacteristic
181#388—22no—
182#388—22no—
183#388—22no—
184#388—21no—
185#388—22no—
186#388—21no—
187#388—22no—
188#386—22no—
189#386—22no—
190#385—22no—
191#385—22no—
192#385—24no—
193#385—24no—
194#385—22no—
195#384—21no—
196#384—21no—
197#384—22no—
198#384—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2314yes—
202#384C2312yes—
203#384C2312yes—
204#383—22no—
205#380—22no—
206#380—22no—
207#380—21no—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229C43C2311yescharacteristic
230#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2312yes—
240#336C2311yescharacteristic
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8—22no—
244C22×Q8C2412yes—
245C22×D4—22no—
246C22×D4C2×D411yescharacteristic
247C22×D4C2411yescharacteristic
248C23×C4—21no—
249C23×C4—24no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C2×M16—28no—
258C2×M16—28no—
259C2×M16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—42no—
267C4⋊Q8—82no—
268C4⋊Q8—82no—
269C4⋊Q8C2×D414yes—
270#125—48no—
271#124—216no—
272C22⋊Q8—28no—
273C22⋊Q8—28no—
274C22⋊Q8—416no—
275#121—28no—
276#121—24no—
277#121C2×D418yes—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—82no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—44no—
285C4×D4—48no—
286C4×D4—84no—
287C4×D4C2×D414yes—
288C4×D4C2×D414yes—
289C4×D4#4114yes—
290#117—24no—
291#117—28no—
292#117—44no—
293#117—44no—
294#117C2×D418yes—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—28no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—84no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2414yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363#107—28no—
364#107—28no—
365#107—28no—
366#106—24no—
367#106—28no—
368C4⋊C8—24no—
369C4⋊C8—24no—
370C4⋊C8—44no—
371#103—24no—
372#103—24no—
373#103—28no—
374#103—42no—
375#103—44no—
376#102—24no—
377#102—28no—
378C22⋊C8—28no—
379C4×C8—24no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—21no—
397C24C22×D412yes—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—44no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8—82no—
409C2×Q8—82no—
410C2×D4—28no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—21no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4#12114yes—
419C2×D4C22×D412yes—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4114yes—
456C22×C4C2511yescharacteristic
457SD16—48no—
458M16—416no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C82−1+414yes—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—82no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—81no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4#12114yes—
501C4⋊C4#13714yes—
502C4⋊C4C22×D412yes—
503C4⋊C4C22×D414yes—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—24no—
509C22⋊C4—48no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—81no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4#12118yes—
518C42—28no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—41no—
527C42—44no—
528C42—42no—
529C42—82no—
530C42—82no—
531C42—82no—
532C42#13714yes—
533C42C22×D412yes—
534C42C2×#4112yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23C2×#12114yes—
549C23C23×D411yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—84no—
555Q8—164no—
556D4—24no—
557D4—28no—
558D4—42no—
559D4—88no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—164no—
608C2×C4—168no—
609C2×C4C2×#12114yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#12112yes—
612C2×C4C2×#13712yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13714yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4#57612yes—
618C2×C4C23×D411yescharacteristic
619C2×C4C22×#4111yescharacteristic
620C2×C4C2×2−1+412yes—
621C2×C4#58512yes—
622C8—24no—
623C8—42no—
624C8—44no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22C22×#12111yescharacteristic
640C22#348511yescharacteristic
641C22#349911yescharacteristic
642C22C22×#13711yescharacteristic
643C22C22×#13711yescharacteristic
644C22C2×#57611yescharacteristic
645C22#358511yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—24no—
651C4—24no—
652C4—44no—
653C4—42no—
654C4—41no—
655C4—42no—
656C4—82no—
657C4—84no—
658C4—81no—
659C4—81no—
660C4—82no—
661C4—82no—
662C4#303914yes—
663C4#348912yes—
664C4#351512yes—
665C2—44no—
666C2—82no—
667C2#3602011yescharacteristic
668C2C2×#303911yescharacteristic
669C2#6042811yescharacteristic
670C2#6171911yescharacteristic
671C2#6172611yescharacteristic
672C2#6248011yescharacteristic
673C2#6251011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36180C2semidirect
4. #29868C2semidirect
5. #29759C2semidirect
6. #28689C2semidirect
7. #28577C2semidirect
8. #28206C2semidirect
9. #27644C2semidirect
10. #25455C2semidirect
11. #25223C2semidirect
12. #24815C2semidirect
13. #24596C2semidirect
14. #23677C2semidirect
15. #23658C2semidirect
16. #22700C2semidirect
17. #22502C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33572
#361802
#343212
#298682
#297592
#286892
#285772
#282062
#276442
#254552
#252232
#248152
#245962
#236772
#236582
#227002
#225022