Finite Group Databasebeta

← #6566089

Subgroups — #6566089

Order 512.

1867 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
1C2×#3338C212yesmaximal
2#36180C211yesmaximal, characteristic
3#34332C211yesmaximal, characteristic
4#29869C211yesmaximal, characteristic, complemented
5#29757C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28575C212yesmaximal, complemented
8#28206C211yesmaximal, characteristic, complemented
9#27648C212yesmaximal, complemented
10#25455C212yesmaximal, complemented
11#25206C214yesmaximal, complemented
12#24817C212yesmaximal, complemented
13#24582C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15C2×#2083C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic
17#22487C212yesmaximal
18C2×#554C2212yes—
19C2×#554C2211yescharacteristic
20#3338—216no—
21C2×#500C2212yes—
22C2×#484C2214yes—
23C2×#484C2214yes—
24C2×#477C2212yes—
25C2×#477C2212yes—
26C2×#467C2214yes—
27C2×C4×Q16C2212yes—
28C2×C8×D4C2212yes—
29C22×#107C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2567C2212yes—
33#2552—21no—
34#2552—21no—
35#2502C2214yes—
36#2440—22no—
37#2440—22no—
38#2370—21no—
39#2369C2211yescharacteristic
40#2360—22no—
41#2360C2212yes—
42Q8×C4⋊C4C2211yescharacteristic
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic
47#2345C2212yes—
48#2330—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#389C2212yes—
55C2×#384C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2083C2212yes—
59#2083C2212yes—
60#2083C2212yes—
61#2083C2214yes—
62#2083C2212yes—
63#2083C2212yes—
64#2083C2211yescharacteristic
65#2083C2214yes—
66#2072C2211yescharacteristic
67#2072C2212yes—
68#2072C2212yes—
69#2068C2212yes—
70#2068C2214yes—
71#2068C2214yes—
72#2056C2212yes—
73#2043C2212yes—
74#2043C2211yescharacteristic
75#2016C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1947C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1928C2214yes—
82#1928C2214yes—
83#1892C2214yes—
84#1878C2214yes—
85#1872C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1824C2214yes—
89#1803C2214yes—
90#1803C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94C2×#340C2214yes—
95C2×#336C2212yes—
96C22×Q16C2312yes—
97C23×C8C2314yes—
98#554—24no—
99#554—28no—
100#554—28no—
101#554D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—21no—
108C2×#125—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C42C2312yes—
130#500—28no—
131#484—216no—
132#484—216no—
133#477—28no—
134#477—28no—
135#467—216no—
136C4×Q16—28no—
137C8×D4—28no—
138C2×#107—216no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C4×C8C2312yes—
156#398—24no—
157#398—24no—
158#395—24no—
159#391—21no—
160#391—22no—
161#391C2312yes—
162#391C2314yes—
163#391C2311yescharacteristic
164#391C2311yescharacteristic
165#389—22no—
166#389—22no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2312yes—
182#389C2311yescharacteristic
183#387—21no—
184#387—22no—
185#387—21no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#385—24no—
193#385—24no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384—22no—
198#384—22no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2314yes—
202#384C2312yes—
203#384C2312yes—
204#380—22no—
205#380—22no—
206#380—22no—
207#380C2312yes—
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⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—21no—
228C2×#95—22no—
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×Q8C2411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C23×C4—22no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C2×Q16—24no—
254C2×Q16—24no—
255C2×Q16—42no—
256C22×C8—28no—
257C22×C8—28no—
258C22×C8—44no—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—42no—
266C4⋊Q8—82no—
267C4⋊Q8—82no—
268C4⋊Q8C2×D414yes—
269#125—28no—
270#125—48no—
271C22⋊Q8—24no—
272C22⋊Q8—216no—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C22⋊Q8—416no—
276C22⋊Q8C2×D418yes—
277C4×Q8—22no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—82no—
282C4×Q8C2×D414yes—
283C4×D4—28no—
284C4×D4—24no—
285C4×D4—48no—
286C4×D4—44no—
287C4×D4Q1618yescomplemented
288C4×D4C2×D414yes—
289C4×D4#4114yes—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—28no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4—84no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D418yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—24no—
359#107—28no—
360#107—28no—
361#107—28no—
362#107—28no—
363C4⋊C8—24no—
364C4⋊C8—24no—
365C4⋊C8—44no—
366#103—24no—
367#103—28no—
368#103—24no—
369#103—24no—
370#103—28no—
371#103—42no—
372#103—44no—
373C22⋊C8—28no—
374C4×C8—24no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—24no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24C22×D412yes—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—44no—
407C2×Q8—81no—
408C2×Q8—81no—
409C2×Q8—82no—
410C2×Q8—82no—
411C2×Q8C22×D412yes—
412C2×D4#12114yes—
413C2×D4C2×Q1614yes—
414C2×D4#13714yes—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—84no—
438C22×C4—82no—
439C22×C4—84no—
440C22×C4C2×Q1618yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4114yes—
452C22×C4C2511yescharacteristic
453Q16—48no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—416no—
463C2×C82−1+414yes—
464C4⋊C4—28no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—81no—
492C4⋊C4—82no—
493C4⋊C4—84no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4#12114yes—
502C4⋊C4C2×Q1614yes—
503C4⋊C4C22×D412yes—
504C4⋊C4C22×D414yes—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4#12118yes—
510C22⋊C4C2×Q1618yes—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—42no—
518C42—42no—
519C42—41no—
520C42—44no—
521C42—42no—
522C42—82no—
523C42—82no—
524C42—82no—
525C42C2×Q1614yes—
526C42C22×D412yes—
527C42C2×#4112yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23C2×#12114yes—
533C23C22×Q1614yes—
534C23C2×#13714yes—
535C23C23×D411yescharacteristic
536Q8—24no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—84no—
543Q8—164no—
544D4#462116yes—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—88no—
595C2×C4—168no—
596C2×C4—164no—
597C2×C4C2×#12112yes—
598C2×C4C2×#12114yes—
599C2×C4C2×#12112yes—
600C2×C4C22×Q1612yes—
601C2×C4C22×Q1612yes—
602C2×C4C22×Q1614yes—
603C2×C4C22×Q1612yes—
604C2×C4C2×#13712yes—
605C2×C4#57612yes—
606C2×C4C23×D411yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×2−1+412yes—
609C2×C4#58512yes—
610C8—24no—
611C8—42no—
612C8—44no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22C2×#46218yes—
620C22C22×#12111yescharacteristic
621C22#348511yescharacteristic
622C22#349911yescharacteristic
623C22C23×Q1611yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57611yescharacteristic
626C22#358511yescharacteristic
627C4—22no—
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—42no—
634C4—44no—
635C4—42no—
636C4—41no—
637C4—42no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4C2×#46214yes—
645C4#348912yes—
646C4#351512yes—
647C2—28no—
648C2#3602011yescharacteristic
649C2C22×#46211yescharacteristic
650C2#6042811yescharacteristic
651C2#6169811yescharacteristic
652C2#6172611yescharacteristic
653C2#6247211yescharacteristic
654C2#6250411yescharacteristic
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. #29757C2semidirect
6. #28689C2semidirect
7. #28575C2semidirect
8. #28206C2semidirect
9. #27648C2semidirect
10. #25455C2semidirect
11. #25206C2semidirect
12. #24817C2semidirect
13. #24582C2semidirect
14. #23679C2semidirect
15. C2×#2083C2semidirect
287. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#33382
#361802
#343322
#298692
#297572
#286892
#285752
#282062
#276482
#254552
#252062
#248172
#245822
#236792
C2×#20832
#226962
#224872