Finite Group Databasebeta

← #6564396

Subgroups — #6564396

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3329C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic
3#34334C211yesmaximal, characteristic, complemented
4#29869C211yesmaximal, characteristic, complemented
5#29713C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28580C212yesmaximal, complemented
8#28157C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25319C212yesmaximal, complemented
11#25222C214yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#24541C214yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23629C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22503C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×D4×Q8C2211yescharacteristic
20#3329—216no—
21C2×#487C2212yes—
22C2×#485C2214yes—
23C2×#477C2212yes—
24C2×#468C2214yes—
25C2×#461C2212yes—
26C2×#448C2214yes—
27C2×C4×SD16C2212yes—
28C2×#436C2212yes—
29C2×#419C2214yes—
30#2868C2211yescharacteristic
31#2856C2212yes—
32#2563C2212yes—
33#2537—21no—
34#2537—21no—
35#2500C2214yes—
36#2491—22no—
37#2491—22no—
38#2369—21no—
39#2368—22no—
40#2365C2211yescharacteristic
41#2360C2212yes—
42#2355C2211yescharacteristic
43#2354—21no—
44#2354—21no—
45#2349—21no—
46D4×C4⋊C4C2211yescharacteristic
47Q8×C22⋊C4C2212yes—
48#2331—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#391C2212yes—
55#2285C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2084C2211yescharacteristic
59#2083C2212yes—
60#2083C2212yes—
61#2074C2214yes—
62#2074C2214yes—
63#2072C2212yes—
64#2072C2212yes—
65#2072C2212yes—
66#2072C2212yes—
67#2072C2211yescharacteristic
68#2072C2212yes—
69#2065C2212yes—
70#2055C2214yes—
71#2055C2214yes—
72#2055C2212yes—
73#2041C2211yescharacteristic
74#2014C2211yescharacteristic
75#2014C2212yes—
76#1990C2212yes—
77#1987C2212yes—
78#1949C2214yes—
79#1939C2212yes—
80#1925C2214yes—
81#1895C2214yes—
82#1880C2214yes—
83#1880C2214yes—
84#1874C2214yes—
85#1851C2214yes—
86#1831C2211yescharacteristic
87#1825C2214yes—
88#1805C2214yes—
89#1803C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C23×Q8C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99C22×M16C2314yes—
100#555—28no—
101#555—28no—
102D4×Q8—24no—
103D4×Q8D418yescomplemented
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×#127C2311yescharacteristic
110C2×#124—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121—22no—
114C2×#121—22no—
115C2×#121—22no—
116C2×#121C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4D411yescharacteristic
125C2×C4×D4D411yescharacteristic
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117—22no—
129C2×#117C2314yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2312yes—
132#487—28no—
133#485—216no—
134#477—28no—
135#468—216no—
136#461—28no—
137#448—216no—
138C4×SD16—28no—
139#436—28no—
140C2×#107C2314yes—
141C2×#106C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145#419—216no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157#394—24no—
158#394—24no—
159#393—24no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391C2312yes—
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2312yes—
168#391C2311yescharacteristic
169#390—21no—
170#390—21no—
171#390—22no—
172#390—22no—
173#390C2312yes—
174#390C2314yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#388—24no—
183#388—24no—
184#387—21no—
185#387—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—21no—
200#382C2314yes—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—22no—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226#376—21no—
227C2×#95—22no—
228C2×#95—22no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2314yes—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2×D418yes—
248C22×Q8C2412yes—
249C22×Q8C2411yescharacteristic
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C2×SD16—24no—
262C2×SD16—28no—
263C2×SD16—42no—
264C2×SD16—42no—
265C2×SD16—44no—
266C2×SD16—44no—
267C2×M16—28no—
268C2×M16—44no—
269C2×M16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—28no—
274C4⋊Q8—24no—
275C4⋊Q8—22no—
276C4⋊Q8—24no—
277C4⋊Q8—82no—
278C4⋊Q8—82no—
279C4⋊Q8C2×D414yes—
280#127—42no—
281#124—416no—
282C22⋊Q8—24no—
283C22⋊Q8—216no—
284C22⋊Q8C2×D418yes—
285#121—28no—
286#121—28no—
287#121—48no—
288#121—84no—
289#121—84no—
290C4×Q8—22no—
291C4×Q8—44no—
292C4×Q8—44no—
293C4×Q8C2×D414yes—
294C4×D4—28no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—42no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4C2×D414yes—
303C4×D4#4114yes—
304#117—28no—
305#117—28no—
306#117—48no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C42—21no—
348C2×C42—24no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D412yes—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—28no—
365#106—24no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368C4⋊C8—42no—
369#103—24no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—28no—
374#102—24no—
375#102—42no—
376#102—44no—
377#102—42no—
378C22⋊C8—28no—
379C4×C8—24no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—24no—
392#95—24no—
393#95—21no—
394#95—22no—
395#95—22no—
396#95—21no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24—22no—
401C24C22×D412yes—
402C2×Q8—28no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—416no—
410C2×Q8—44no—
411C2×Q8—82no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—88no—
416C2×Q8C22×D412yes—
417C2×Q8C22×D414yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—48no—
431C2×D4—42no—
432C2×D4—42no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4#12114yes—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D414yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4114yes—
472C22×C4C2511yescharacteristic
473SD16—48no—
474SD16—88no—
475M16—88no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4#12114yes—
514C4⋊C4C22×D412yes—
515C22⋊C4—28no—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4#12118yes—
527C42—24no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—22no—
532C42—22no—
533C42—42no—
534C42—42no—
535C42—41no—
536C42—48no—
537C42—42no—
538C42—82no—
539C42—82no—
540C42—82no—
541C42#13714yes—
542C42C22×D412yes—
543C42C2×#4112yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23C2×#12114yes—
558C23C23×D411yescharacteristic
559Q8—24no—
560Q8—28no—
561Q8—44no—
562Q8—42no—
563Q8—88no—
564Q8—164no—
565D4—28no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—42no—
570D4—84no—
571D4—84no—
572D4—168no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4—164no—
623C2×C4C2×#12114yes—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4C2×#13712yes—
629C2×C4#57512yes—
630C2×C4#57712yes—
631C2×C4C23×D411yescharacteristic
632C2×C4C22×#4111yescharacteristic
633C2×C4#58512yes—
634C2×C4#58512yes—
635C8—42no—
636C8—44no—
637C8—42no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22C22×#12111yescharacteristic
653C22#348511yescharacteristic
654C22#350011yescharacteristic
655C22C22×#13711yescharacteristic
656C22C2×#57511yescharacteristic
657C22C2×#57711yescharacteristic
658C22#358511yescharacteristic
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—41no—
668C4—42no—
669C4—44no—
670C4—84no—
671C4—81no—
672C4—82no—
673C4—81no—
674C4—82no—
675C4#303714yes—
676C4#348512yes—
677C4#352112yes—
678C2—44no—
679C2—162no—
680C2#3600911yescharacteristic
681C2C2×#303711yescharacteristic
682C2#6043111yescharacteristic
683C2#6172111yescharacteristic
684C2#6173111yescharacteristic
685C2#6254011yescharacteristic
686C2#6259211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
1. C2×#3329C2semidirect
3. #34334C2semidirect
4. #29869C2semidirect
5. #29713C2semidirect
6. #28717C2semidirect
7. #28580C2semidirect
8. #28157C2semidirect
9. #27652C2semidirect
10. #25319C2semidirect
11. #25222C2semidirect
12. #24679C2semidirect
13. #24541C2semidirect
14. #23678C2semidirect
15. #23629C2semidirect
16. #22700C2semidirect
17. #22503C2semidirect
103. D4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33292
#361872
#343342
#298692
#297132
#287172
#285802
#281572
#276522
#253192
#252222
#246792
#245412
#236782
#236292
#227002
#225032