Finite Group Databasebeta

← #6539671

Subgroups — #6539671

Order 512.

2047 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3328C212yesmaximal
2D4×C4⋊Q8C211yesmaximal, characteristic
3#34324C211yesmaximal, characteristic
4#29799C211yesmaximal, characteristic, complemented
5#29759C212yesmaximal, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28581C212yesmaximal, complemented
8#28205C211yesmaximal, characteristic, complemented
9#25276C212yesmaximal, complemented
10#25095C214yesmaximal, complemented
11#24732C212yesmaximal, complemented
12#24498C214yesmaximal, complemented
13#24212C212yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23507C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic
17#22430C212yesmaximal
18C2×#556C2211yescharacteristic
19C2×D4×Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21#3328—216no—
22C2×#487C2212yes—
23C2×#485C2214yes—
24C2×#475C2212yes—
25C2×#470C2214yes—
26C2×#462C2212yes—
27C2×#462C2214yes—
28C2×#451C2214yes—
29C2×C4×Q16C2212yes—
30C2×#436C2212yes—
31C2×#419C2214yes—
32C2×#419C2214yes—
33#2868C2211yescharacteristic
34#2857C2212yes—
35#2551—21no—
36#2551—21no—
37#2546C2212yes—
38#2440—22no—
39#2440—22no—
40#2434C2214yes—
41#2373—22no—
42#2367—21no—
43#2365C2211yescharacteristic
44#2362C2212yes—
45#2355—21no—
46#2354C2211yescharacteristic
47Q8×C4⋊C4C2211yescharacteristic
48D4×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50#2346C2212yes—
51#2334—22no—
52#2326—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C22⋊Q8C2212yes—
56#2296C2212yes—
57#2295C2211yescharacteristic
58#2276—21no—
59C42×D4C2211yescharacteristic
60#2084C2212yes—
61#2083C2211yescharacteristic
62#2072C2211yescharacteristic
63#2068C2214yes—
64#2068C2212yes—
65#2066C2214yes—
66#2056C2212yes—
67#2043C2214yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2043C2212yes—
73#2043C2211yescharacteristic
74#2043C2214yes—
75#2016C2212yes—
76#2016C2212yes—
77#2016C2211yescharacteristic
78#1981C2214yes—
79#1981C2212yes—
80#1981C2214yes—
81#1956C2212yes—
82#1939C2212yes—
83#1927C2214yes—
84#1927C2214yes—
85#1894C2214yes—
86#1878C2214yes—
87#1873C2214yes—
88#1851C2214yes—
89#1831C2211yescharacteristic
90#1803C2212yes—
91#1797C2212yes—
92#1787C2214yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1728C2212yes—
96C23×Q8C2314yes—
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C22×M16C2314yes—
100#556—24no—
101#556D418yes—
102D4×Q8—28no—
103D4×Q8—28no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—48no—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2314yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C2×C4⋊Q8C2311yescharacteristic
114C2×#126—22no—
115C2×#124—21no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×D4—22no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117C2312yes—
130C2×#117C2312yes—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C42C2312yes—
134#487—28no—
135#485—216no—
136#475—28no—
137#470—216no—
138#462—28no—
139#462—48no—
140#451—216no—
141C4×Q16—28no—
142#436—28no—
143C2×#107C2312yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148#419—216no—
149#419—48no—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8C2312yes—
165#398—24no—
166#395—24no—
167#395—24no—
168#391—21no—
169#391—21no—
170#391C2312yes—
171#391C2311yescharacteristic
172#391C2314yes—
173#391C2311yescharacteristic
174#390—21no—
175#389—22no—
176#389—22no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389C2312yes—
184#389C2312yes—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2311yescharacteristic
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—21no—
194#387—22no—
195#387—21no—
196#387—22no—
197#387—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#385—24no—
202#385—24no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#382—22no—
211#380—22no—
212#380—22no—
213#380C2312yes—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234#376—21no—
235#376—22no—
236C43C2311yescharacteristic
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2312yes—
244#336C2314yes—
245C22×Q8—21no—
246C22×Q8—24no—
247C22×Q8—28no—
248C22×Q8—28no—
249C22×Q8—28no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8—44no—
253C22×Q8C2414yes—
254C22×Q8C2411yescharacteristic
255C22×Q8C2412yes—
256C22×D4C2×D411yescharacteristic
257C23×C4—24no—
258C23×C4C2×D412yes—
259C23×C4C2×D412yes—
260C23×C4C2412yes—
261C2×Q16—24no—
262C2×Q16—24no—
263C2×Q16—28no—
264C2×Q16—42no—
265C2×Q16—44no—
266C2×Q16—44no—
267C2×M16—28no—
268C2×M16—44no—
269C2×M16—44no—
270C22×C8C2412yes—
271C22×C8C2414yes—
272C22×C8C2412yes—
273C4⋊Q8—24no—
274C4⋊Q8—28no—
275C4⋊Q8—24no—
276C4⋊Q8—22no—
277C4⋊Q8—24no—
278C4⋊Q8—44no—
279C4⋊Q8—44no—
280C4⋊Q8—44no—
281C4⋊Q8—48no—
282C4⋊Q8—48no—
283C4⋊Q8—48no—
284C4⋊Q8C2×D414yes—
285#126—216no—
286#124—28no—
287C22⋊Q8—24no—
288C22⋊Q8—28no—
289C22⋊Q8—28no—
290C22⋊Q8—416no—
291C22⋊Q8C2×D418yes—
292C4×Q8—28no—
293C4×Q8—22no—
294C4×Q8—24no—
295C4×Q8—24no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8C2×D414yes—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—48no—
303C4×D4C2×D414yes—
304C4×D4#4114yes—
305#117—24no—
306#117—44no—
307#117C2×D418yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—48no—
334C2×C4⋊C4—48no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371#107—24no—
372#106—28no—
373C4⋊C8—24no—
374C4⋊C8—42no—
375C4⋊C8—44no—
376#103—24no—
377#103—24no—
378#103—28no—
379#103—24no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—42no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389C22⋊C8—28no—
390C4×C8—24no—
391#95—21no—
392#95—22no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—24no—
407#95—24no—
408#95C2×D412yes—
409#95C2×D412yes—
410#95C2×D412yes—
411C24C22×D412yes—
412C2×Q8—24no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—24no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—48no—
428C2×Q8—416no—
429C2×Q8—416no—
430C2×Q8—48no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—84no—
434C2×Q8—88no—
435C2×Q8—88no—
436C2×Q8C22×D412yes—
437C2×Q8H(5,𝔽2)14yes—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4#12114yes—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—84no—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4114yes—
472C22×C4C2511yescharacteristic
473Q16—48no—
474Q16—88no—
475M16—88no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—28no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4#12114yes—
519C4⋊C4#13714yes—
520C4⋊C4C22×D414yes—
521C4⋊C4C22×D412yes—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—48no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4#12118yes—
530C42—28no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—42no—
538C42—42no—
539C42—41no—
540C42—44no—
541C42—44no—
542C42—44no—
543C42—42no—
544C42—82no—
545C42—82no—
546C42#13714yes—
547C42C22×D412yes—
548C42C2×#4112yes—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23C2×#12114yes—
556C23C23×D411yescharacteristic
557Q8—24no—
558Q8—24no—
559Q8—24no—
560Q8—42no—
561Q8—42no—
562Q8—42no—
563Q8—48no—
564Q8—44no—
565Q8—44no—
566Q8—44no—
567Q8—44no—
568Q8—816no—
569D4—28no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—816no—
620C2×C4C2×#12114yes—
621C2×C4C2×#12112yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13714yes—
625C2×C4C2×#13712yes—
626C2×C4C2×#13712yes—
627C2×C4C2×#13712yes—
628C2×C4#57812yes—
629C2×C4C23×D411yescharacteristic
630C2×C4C22×#4111yescharacteristic
631C2×C4C2×H(5,𝔽2)12yes—
632C2×C4#58512yes—
633C8—42no—
634C8—44no—
635C8—42no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22C22×#12111yescharacteristic
644C22#348411yescharacteristic
645C22#350011yescharacteristic
646C22C22×#13711yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358611yescharacteristic
650C4—22no—
651C4—22no—
652C4—24no—
653C4—22no—
654C4—22no—
655C4—24no—
656C4—42no—
657C4—41no—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—82no—
663C4—81no—
664C4—84no—
665C4—82no—
666C4—81no—
667C4#303914yes—
668C4#348812yes—
669C4#351312yes—
670C2—44no—
671C2#3572811yescharacteristic
672C2C2×#303911yescharacteristic
673C2#6043211yescharacteristic
674C2#6158411yescharacteristic
675C2#6160211yescharacteristic
676C2#6253611yescharacteristic
677C2#6258911yescharacteristic
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. #29799C2semidirect
5. #29759C2semidirect
6. #28748C2semidirect
7. #28581C2semidirect
8. #28205C2semidirect
9. #25276C2semidirect
10. #25095C2semidirect
11. #24732C2semidirect
12. #24498C2semidirect
13. #24212C2semidirect
14. #23679C2semidirect
15. #23507C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33282
D4×C4⋊Q82
#343242
#297992
#297592
#287482
#285812
#282052
#252762
#250952
#247322
#244982
#242122
#236792
#235072
#227002
#224302