Finite Group Databasebeta

← #6564384

Subgroups — #6564384

Order 512.

1911 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3299C212yesmaximal
2#36180C211yesmaximal, characteristic
3#34334C211yesmaximal, characteristic
4#29869C211yesmaximal, characteristic, complemented
5#29715C212yesmaximal, complemented
6#28717C211yesmaximal, characteristic, complemented
7#28575C212yesmaximal, complemented
8#28161C211yesmaximal, characteristic, complemented
9#27648C212yesmaximal, complemented
10#25324C212yesmaximal, complemented
11#25206C214yesmaximal, complemented
12#24682C212yesmaximal, complemented
13#24532C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15C2×#2072C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic
17#22487C212yesmaximal
18C2×#554C2212yes—
19C2×D4×Q8C2211yescharacteristic
20#3299—216no—
21C2×#487C2212yes—
22C2×#484C2214yes—
23C2×#477C2212yes—
24C2×#467C2214yes—
25C2×#462C2212yes—
26C2×#451C2214yes—
27C2×C4×Q16C2212yes—
28C2×C8×D4C2212yes—
29C22×#103C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2567C2212yes—
33#2552—21no—
34#2552—21no—
35#2502C2214yes—
36#2440—22no—
37#2440—22no—
38#2369—21no—
39#2368—22no—
40#2365C2211yescharacteristic
41#2360C2212yes—
42#2355C2211yescharacteristic
43#2349—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic
47Q8×C22⋊C4C2212yes—
48#2331—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#391C2212yes—
55C2×#384C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2083C2211yescharacteristic
59#2083C2212yes—
60#2083C2212yes—
61#2072C2212yes—
62#2072C2212yes—
63#2072C2212yes—
64#2072C2212yes—
65#2072C2211yescharacteristic
66#2072C2214yes—
67#2072C2214yes—
68#2072C2212yes—
69#2068C2212yes—
70#2056C2214yes—
71#2056C2214yes—
72#2056C2212yes—
73#2043C2211yescharacteristic
74#2016C2212yes—
75#2016C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1947C2214yes—
79#1939C2212yes—
80#1928C2214yes—
81#1892C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1872C2214yes—
85#1850C2214yes—
86#1831C2211yescharacteristic
87#1824C2214yes—
88#1803C2214yes—
89#1803C2212yes—
90#1803C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94C2×#340C2214yes—
95C2×#336C2212yes—
96C23×Q8C2312yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C23×C8C2314yes—
100#554—28no—
101#554—28no—
102D4×Q8—24no—
103D4×Q8D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×#125—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2314yes—
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C42C2312yes—
130#487—28no—
131#484—216no—
132#477—28no—
133#467—216no—
134#462—28no—
135#451—216no—
136C4×Q16—28no—
137C8×D4—28no—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#103—216no—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×C22⋊C8C2314yes—
154C2×C4×C8C2312yes—
155#398—24no—
156#398—24no—
157#395—24no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391C2312yes—
163#391C2312yes—
164#391C2314yes—
165#391C2311yescharacteristic
166#391C2312yes—
167#391C2312yes—
168#391C2311yescharacteristic
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2311yescharacteristic
181#389C2311yescharacteristic
182#387—21no—
183#387—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—24no—
192#385—24no—
193#384—21no—
194#384—21no—
195#384—22no—
196#384—22no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2314yes—
200#384C2312yes—
201#384C2312yes—
202#382—21no—
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—21no—
213C4×C4⋊C4—21no—
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#340C2312yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2314yes—
241#336C2312yes—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—44no—
246C22×Q8—44no—
247C22×Q8C2×D418yes—
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C22×Q8C2411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C23×C4—22no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×Q16—24no—
259C2×Q16—28no—
260C2×Q16—42no—
261C2×Q16—42no—
262C2×Q16—44no—
263C2×Q16—44no—
264C22×C8—28no—
265C22×C8—44no—
266C22×C8—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—28no—
271C4⋊Q8—24no—
272C4⋊Q8—22no—
273C4⋊Q8—24no—
274C4⋊Q8—42no—
275C4⋊Q8—82no—
276C4⋊Q8—82no—
277C4⋊Q8C2×D414yes—
278#125—48no—
279C22⋊Q8—24no—
280C22⋊Q8—216no—
281C22⋊Q8—28no—
282C22⋊Q8—28no—
283C22⋊Q8—416no—
284C22⋊Q8C2×D418yes—
285C4×Q8—22no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×Q8C2×D414yes—
291C4×D4—28no—
292C4×D4—24no—
293C4×D4—48no—
294C4×D4—44no—
295C4×D4Q1618yescomplemented
296C4×D4C2×D414yes—
297C4×D4#4114yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—28no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
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⋊C4—84no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2414yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C22⋊C4C2×D412yes—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—28no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364C4⋊C8—42no—
365#103—24no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—28no—
370#103—24no—
371#103—42no—
372#103—44no—
373#103—42no—
374C22⋊C8—28no—
375C4×C8—24no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—24no—
394#95—24no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24C22×D412yes—
399C2×Q8—28no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—416no—
413C2×Q8—44no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8—88no—
419C2×Q8C22×D412yes—
420C2×Q8C22×D414yes—
421C2×D4#12114yes—
422C2×D4C2×Q1614yes—
423C2×D4#13714yes—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—82no—
449C22×C4C2×Q1618yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4114yes—
461C22×C4C2511yescharacteristic
462Q16—48no—
463Q16—88no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8—88no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4—82no—
508C4⋊C4#12114yes—
509C4⋊C4C2×Q1614yes—
510C4⋊C4C22×D412yes—
511C22⋊C4—28no—
512C22⋊C4—24no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4#12118yes—
516C22⋊C4C2×Q1618yes—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—24no—
521C42—22no—
522C42—22no—
523C42—42no—
524C42—42no—
525C42—41no—
526C42—44no—
527C42—42no—
528C42—82no—
529C42—82no—
530C42—82no—
531C42C2×Q1614yes—
532C42C22×D412yes—
533C42C2×#4112yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23C2×#12114yes—
539C23C22×Q1614yes—
540C23C2×#13714yes—
541C23C23×D411yescharacteristic
542Q8—24no—
543Q8—28no—
544Q8—44no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—42no—
550Q8—88no—
551Q8—164no—
552D4#462116yes—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—88no—
603C2×C4—168no—
604C2×C4—164no—
605C2×C4C2×#12112yes—
606C2×C4C2×#12114yes—
607C2×C4C2×#12112yes—
608C2×C4C22×Q1612yes—
609C2×C4C22×Q1612yes—
610C2×C4C22×Q1614yes—
611C2×C4C22×Q1612yes—
612C2×C4C2×#13712yes—
613C2×C4#57712yes—
614C2×C4C23×D411yescharacteristic
615C2×C4C22×#4111yescharacteristic
616C2×C4#58512yes—
617C2×C4#58512yes—
618C8—42no—
619C8—44no—
620C8—42no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22C2×#46218yes—
628C22C22×#12111yescharacteristic
629C22#348511yescharacteristic
630C22#350011yescharacteristic
631C22C23×Q1611yescharacteristic
632C22C22×#13711yescharacteristic
633C22C2×#57711yescharacteristic
634C22#358511yescharacteristic
635C4—22no—
636C4—24no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—24no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—42no—
645C4—44no—
646C4—82no—
647C4—84no—
648C4—81no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4C2×#46214yes—
653C4#348512yes—
654C4#352112yes—
655C2—28no—
656C2#3600911yescharacteristic
657C2C22×#46211yescharacteristic
658C2#6043111yescharacteristic
659C2#6169811yescharacteristic
660C2#6173511yescharacteristic
661C2#6253011yescharacteristic
662C2#6258211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
4. #29869C2semidirect
5. #29715C2semidirect
6. #28717C2semidirect
7. #28575C2semidirect
8. #28161C2semidirect
9. #27648C2semidirect
10. #25324C2semidirect
11. #25206C2semidirect
12. #24682C2semidirect
13. #24532C2semidirect
14. #23679C2semidirect
15. C2×#2072C2semidirect
295. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32992
#361802
#343342
#298692
#297152
#287172
#285752
#281612
#276482
#253242
#252062
#246822
#245322
#236792
C2×#20722
#226962
#224872