Finite Group Databasebeta

← #6621050

Subgroups — #6621050

Order 512.

1819 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3364C212yesmaximal
2#36560C211yesmaximal, characteristic, complemented
3C4×#556C211yesmaximal, characteristic
4#33200C212yesmaximal, complemented
5#33195C211yesmaximal, characteristic
6#28721C212yesmaximal
7#28717C211yesmaximal, characteristic, complemented
8#28255C211yesmaximal, characteristic, complemented
9#28254C212yesmaximal
10#25329C214yesmaximal
11#25323C212yesmaximal, complemented
12#24687C214yesmaximal, complemented
13#24681C212yesmaximal
14Q8×C4⋊C8C211yesmaximal, characteristic, complemented
15#22521C212yesmaximal, complemented
16C4×#462C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic
18C2×#557C2212yes—
19C2×#556C2211yescharacteristic
20#3364—216no—
21C2×#488C2214yes—
22C2×#487C2212yes—
23C2×#479C2214yes—
24C2×#474C2212yes—
25C2×#464C2214yes—
26C2×#462C2212yes—
27C2×#462C2212yes—
28C2×#461C2211yescharacteristic
29C2×C8×Q8C2212yes—
30C2×C4×Q16C2214yes—
31C2×C4×SD16C2212yes—
32#2868C2211yescharacteristic
33#2859C2212yes—
34#2568C2212yes—
35#2565C2214yes—
36#2546—21no—
37#2546—21no—
38#2506—22no—
39#2506—22no—
40#2373—22no—
41#2367C2212yes—
42#2365—21no—
43#2365—21no—
44#2365C2211yescharacteristic
45#2362—21no—
46#2356C2212yes—
47#2355C2211yescharacteristic
48#2354C2211yescharacteristic
49#2347—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×#126—22no—
54C4×#124C2212yes—
55C4×C22⋊Q8—21no—
56C42×Q8C2212yes—
57C42×Q8C2211yescharacteristic
58C42×D4C2211yescharacteristic
59C4×#117—21no—
60#2074C2214yes—
61#2073C2214yes—
62#2072C2214yes—
63#2056C2214yes—
64#2056C2214yes—
65#2055C2214yes—
66#1991C2214yes—
67#1990C2212yes—
68#1989C2214yes—
69#1988C2212yes—
70#1981C2211yescharacteristic
71#1949C2214yes—
72#1948C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1939C2214yes—
77#1928C2214yes—
78#1927C2212yes—
79#1878C2214yes—
80#1877C2212yes—
81#1842C2212yes—
82#1842C2211yescharacteristic
83#1805C2214yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1802C2211yescharacteristic
90#1802C2212yes—
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C4×#103C2214yes—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98#557—28no—
99#557—28no—
100#556—24no—
101#556D418yes—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#126—22no—
109C2×#125—24no—
110C2×#125C2314yes—
111C2×#124C2312yes—
112C2×C22⋊Q8—21no—
113C2×C4×Q8—22no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—21no—
126C2×#117—21no—
127C22×C42—21no—
128#488—216no—
129#487—28no—
130#479—216no—
131#474—28no—
132#464—216no—
133#462—28no—
134#462—28no—
135#461—24no—
136C8×Q8—28no—
137C4×Q16—216no—
138C4×SD16—28no—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2311yescharacteristic
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2314yes—
159#400—24no—
160#398—24no—
161#398—24no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391C2312yes—
166#391C2314yes—
167#391C2312yes—
168#391C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—24no—
191#387—24no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2311yescharacteristic
198#384C2312yes—
199#383—22no—
200#382C2314yes—
201#382C2312yes—
202#380—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—21no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229C4×C22⋊C4—21no—
230C4×C22⋊C4—22no—
231#376—22no—
232C43—22no—
233C43C2311yescharacteristic
234C43C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244C22×Q8—22no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C2×Q16—28no—
253C2×Q16—28no—
254C2×Q16—44no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×SD16—42no—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C22×C8C2412yes—
261C4⋊Q8—24no—
262C4⋊Q8—22no—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—44no—
266C4⋊Q8—48no—
267C4⋊Q8—44no—
268C4⋊Q8C2×D414yes—
269C4⋊Q8#4114yes—
270#126—216no—
271#125—28no—
272#125—28no—
273#125—416no—
274#124—24no—
275#124C2×D418yes—
276C22⋊Q8—28no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×Q8—42no—
288C4×Q8—48no—
289C4×Q8C2×D418yes—
290C4×Q8C2×D414yes—
291C4×Q8#4114yes—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—41no—
297C4×D4C2×D414yes—
298#117—28no—
299#117—44no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—22no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363#107—28no—
364#106—24no—
365C4⋊C8—24no—
366C4⋊C8—28no—
367C4⋊C8—22no—
368C4⋊C8—24no—
369C4⋊C8—22no—
370C4⋊C8—42no—
371C4⋊C8—44no—
372#103—28no—
373#103—28no—
374#103—28no—
375#103—24no—
376#103—28no—
377#103—24no—
378#103—24no—
379#102—24no—
380#102—28no—
381C4×C8—24no—
382C4×C8—28no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8#12114yes—
420C2×Q8C22×D414yes—
421C2×Q8C2×#4114yes—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4C22×D412yes—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
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
456Q16—416no—
457SD16—48no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C4⋊C4—24no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4#12114yes—
502C4⋊C4#12118yes—
503C4⋊C4C22×D412yes—
504C4⋊C42−1+412yes—
505C22⋊C4—28no—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—42no—
509C22⋊C4—44no—
510C22⋊C4—42no—
511C22⋊C4—82no—
512C42—28no—
513C42—24no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—21no—
518C42—22no—
519C42—21no—
520C42—22no—
521C42—42no—
522C42—41no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—48no—
527C42—44no—
528C42—44no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—82no—
533C42#13514yes—
534C42C22×D412yes—
535C42C22×D414yes—
536C42C2×#4112yes—
537C42C2×#4114yes—
538C42C2×#4112yes—
539C422−1+412yes—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C23×D411yescharacteristic
548Q8—28no—
549Q8—24no—
550Q8—24no—
551Q8—24no—
552Q8—24no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—88no—
558D4—22no—
559D4—22no—
560D4—41no—
561D4—41no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—48no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4C2×#12114yes—
609C2×C4C2×#12112yes—
610C2×C4C2×#12114yes—
611C2×C4C2×#12112yes—
612C2×C4#54812yes—
613C2×C4#54812yes—
614C2×C4C2×#13512yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4#57512yes—
618C2×C4#57812yes—
619C2×C4C23×D411yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C22×#4111yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4C2×2−1+411yescharacteristic
624C2×C4#58511yescharacteristic
625C8—42no—
626C8—44no—
627C8—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22C22×#12111yescharacteristic
636C22C2×#54811yescharacteristic
637C22#348911yescharacteristic
638C22C22×#13511yescharacteristic
639C22C2×#57511yescharacteristic
640C22C2×#57811yescharacteristic
641C22#358611yescharacteristic
642C4—24no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—24no—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—42no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4—84no—
658C4#303514yes—
659C4#336414yescomplemented
660C4D4×#4111yescharacteristic
661C4#348211yescharacteristic
662C4#349911yescharacteristic
663C4#351811yescharacteristic
664C2—82no—
665C2#3644411yescharacteristic
666C2C2×#303511yescharacteristic
667C2#6043211yescharacteristic
668C2C2×#336411yescharacteristic
669C2#6132011yescharacteristic
670C2#6209711yescharacteristic
671C2#6214611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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
2. #36560C2semidirect
4. #33200C2semidirect
7. #28717C2semidirect
8. #28255C2semidirect
11. #25323C2semidirect
12. #24687C2semidirect
14. Q8×C4⋊C8C2semidirect
15. #22521C2semidirect
16. C4×#462C2semidirect
659. C4#3364semidirect

Maximal subgroups

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

subgroupindex
C2×#33642
#365602
C4×#5562
#332002
#331952
#287212
#287172
#282552
#282542
#253292
#253232
#246872
#246812
Q8×C4⋊C82
#225212
C4×#4622
C4×#4612