Finite Group Databasebeta

← #6565752

Subgroups — #6565752

Order 512.

1833 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3360C212yesmaximal, complemented
2#36594C211yesmaximal, characteristic, complemented
3#34317C211yesmaximal, characteristic, complemented
4#29800C212yesmaximal
5#29798C211yesmaximal, characteristic, complemented
6#28751C212yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28209C212yesmaximal, complemented
9#28202C211yesmaximal, characteristic, complemented
10#25282C214yesmaximal, complemented
11#25273C212yesmaximal, complemented
12#24636C214yesmaximal, complemented
13#24627C212yesmaximal, complemented
14#23677C212yesmaximal, complemented
15#23676C211yesmaximal, characteristic
16#22734C211yesmaximal, characteristic, complemented
17#22541C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#554C2211yescharacteristic
20#3360—216no—
21C2×#488C2214yes—
22C2×#487C2212yes—
23C2×#479C2214yes—
24C2×#476C2212yes—
25C2×#463C2214yes—
26C2×#461C2212yes—
27C2×C8×Q8C2212yes—
28C2×C4×SD16C2214yes—
29C2×C4×D8C2212yes—
30#2868C2211yescharacteristic
31#2858C2212yes—
32#2547C2212yes—
33#2545C2214yes—
34#2538—21no—
35#2538—21no—
36#2512—22no—
37#2512—22no—
38#2370C2212yes—
39#2369C2211yescharacteristic
40#2362—22no—
41#2362—21no—
42#2357C2212yes—
43#2355C2211yescharacteristic
44#2354—21no—
45#2354—21no—
46#2354C2211yescharacteristic
47#2344—21no—
48#2332—22no—
49#2323—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2212yes—
52C4×#127C2211yescharacteristic
53C4×#125C2212yes—
54#2293C2211yescharacteristic
55#2293C2212yes—
56#2274—21no—
57C42×Q8C2211yescharacteristic
58#2084C2212yes—
59#2083C2211yescharacteristic
60#2071C2211yescharacteristic
61#2067C2212yes—
62#2066C2214yes—
63#2055C2214yes—
64#2054C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2043C2212yes—
68#2043C2212yes—
69#2042C2214yes—
70#2041C2212yes—
71#2041C2211yescharacteristic
72#2013C2212yes—
73#2013C2212yes—
74#2012C2214yes—
75#2011C2211yescharacteristic
76#1983C2214yes—
77#1983C2214yes—
78#1983C2214yes—
79#1981C2212yes—
80#1981C2214yes—
81#1958C2214yes—
82#1956C2212yes—
83#1939C2212yes—
84#1939C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1805C2214yes—
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2214yes—
95C4×#102C2212yes—
96C22×SD16C2314yes—
97C22×D8C2312yes—
98#556—28no—
99#556—28no—
100#554—24no—
101#554D418yescomplemented
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#126—24no—
108C2×#125C2312yes—
109C2×#124C2314yes—
110C2×#124C2314yes—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—22no—
128C22×C4⋊C4—21no—
129#488—216no—
130#487—28no—
131#479—216no—
132#476—28no—
133#463—216no—
134#461—28no—
135C8×Q8—28no—
136C4×SD16—216no—
137C4×D8—28no—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#397—24no—
160#396—24no—
161#396—24no—
162#391—21no—
163#391—21no—
164#391—22no—
165#391C2314yes—
166#390—21no—
167#390—21no—
168#390—22no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390C2312yes—
173#390C2311yescharacteristic
174#390C2312yes—
175#390C2311yescharacteristic
176#389—21no—
177#389—22no—
178#389—21no—
179#389—22no—
180#389—22no—
181#389C2311yescharacteristic
182#389C2312yes—
183#389C2312yes—
184#389C2311yescharacteristic
185#388—22no—
186#388—22no—
187#388—22no—
188#388—21no—
189#388—21no—
190#388—22no—
191#388—22no—
192#388—24no—
193#388—24no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382C2312yes—
206#381—22no—
207#380—21no—
208#380—22no—
209#380—22no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—21no—
231C4×C22⋊C4—22no—
232C43C2312yes—
233C43C2311yescharacteristic
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2312yes—
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2414yes—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2412yes—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×D8—24no—
256C2×D8—42no—
257C2×D8—42no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8C2×D414yes—
267#127—42no—
268#126—416no—
269#125—24no—
270#125C2×D418yes—
271#124—28no—
272#124—28no—
273#124—44no—
274#124—44no—
275C22⋊Q8—28no—
276C22⋊Q8—216no—
277C22⋊Q8—48no—
278C4×Q8—24no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—42no—
282C4×Q8—82no—
283C4×Q8C2×D414yes—
284C4×Q8#4114yes—
285C4×D4—24no—
286C4×D4—22no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—44no—
290C4×D4—44no—
291C4×D4C2×D414yes—
292C4×D4C2×D418yes—
293#117—44no—
294#117—48no—
295#117—44no—
296#117—84no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—28no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—21no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362#107—24no—
363#106—28no—
364C4⋊C8—24no—
365C4⋊C8—28no—
366C4⋊C8—44no—
367C4⋊C8—42no—
368#103—28no—
369#103—28no—
370#103—28no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—24no—
376#102—24no—
377#102—28no—
378#102—42no—
379#102—44no—
380#102—42no—
381C4×C8—24no—
382C4×C8—28no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95C2×D412yes—
397#95C2×D412yes—
398#95C2×D412yes—
399C24—21no—
400C24—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×Q8#12114yes—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4C22×D412yes—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—416no—
457D8—48no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—82no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4#12114yes—
497C4⋊C4#12118yes—
498C4⋊C4C22×D414yes—
499C4⋊C4C22×D412yes—
500C22⋊C4—24no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—81no—
512C22⋊C4—82no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C42—24no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—41no—
524C42—42no—
525C42—44no—
526C42—48no—
527C42—42no—
528C42—82no—
529C42—82no—
530C42—82no—
531C42#13514yes—
532C42C22×D412yes—
533C42C22×D414yes—
534C42C2×#4112yes—
535C42C2×#4114yes—
536C23—22no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C23×D411yescharacteristic
551Q8—28no—
552Q8—44no—
553Q8—44no—
554Q8—44no—
555Q8—44no—
556Q8—84no—
557D4—24no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—84no—
564D4—84no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4C2×#12114yes—
611C2×C4C2×#12114yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4#57512yes—
618C2×C4#57712yes—
619C2×C4C23×D411yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4#58512yes—
622C2×C4#58512yes—
623C8—42no—
624C8—42no—
625C8—44no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22C22×#12111yescharacteristic
641C22#348711yescharacteristic
642C22#350011yescharacteristic
643C22C22×#13511yescharacteristic
644C22C2×#57511yescharacteristic
645C22C2×#57711yescharacteristic
646C22#358611yescharacteristic
647C4—24no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—24no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—41no—
657C4—42no—
658C4—81no—
659C4—84no—
660C4—81no—
661C4—82no—
662C4—82no—
663C4#303514yes—
664C4#348712yes—
665C4#352412yes—
666C2—82no—
667C2—162no—
668C2#3601811yescharacteristic
669C2C2×#303511yescharacteristic
670C2#6042911yescharacteristic
671C2#6188211yescharacteristic
672C2#6189911yescharacteristic
673C2#6253211yescharacteristic
674C2#6259111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2011C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3360C2semidirect
2. #36594C2semidirect
3. #34317C2semidirect
5. #29798C2semidirect
6. #28751C2semidirect
7. #28748C2semidirect
8. #28209C2semidirect
9. #28202C2semidirect
10. #25282C2semidirect
11. #25273C2semidirect
12. #24636C2semidirect
13. #24627C2semidirect
14. #23677C2semidirect
16. #22734C2semidirect
17. #22541C2semidirect
101. #554D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33602
#365942
#343172
#298002
#297982
#287512
#287482
#282092
#282022
#252822
#252732
#246362
#246272
#236772
#236762
#227342
#225412