Finite Group Databasebeta

← #6566085

Subgroups — #6566085

Order 512.

1859 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×#3340C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic
3#34324C211yesmaximal, characteristic, complemented
4#29869C211yesmaximal, characteristic, complemented
5#29758C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28574C212yesmaximal, complemented
8#28201C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25452C212yesmaximal, complemented
11#25213C214yesmaximal, complemented
12#24814C212yesmaximal, complemented
13#24587C214yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23656C212yesmaximal, complemented
16#22696C211yesmaximal, characteristic, complemented
17#22489C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#555C2212yes—
20#3340—216no—
21C2×#498C2212yes—
22C2×#485C2214yes—
23C2×#482C2214yes—
24C2×#477C2212yes—
25C2×#474C2212yes—
26C2×#465C2214yes—
27C2×C4×SD16C2212yes—
28C2×C8×D4C2212yes—
29C2×#427C2214yes—
30#2868C2211yescharacteristic
31#2856C2212yes—
32#2563C2212yes—
33#2537—21no—
34#2537—21no—
35#2500C2214yes—
36#2491—22no—
37#2491—22no—
38#2373—22no—
39#2367—21no—
40#2365C2211yescharacteristic
41#2362C2212yes—
42#2355—21no—
43#2354—21no—
44#2354—21no—
45#2354C2211yescharacteristic
46Q8×C4⋊C4C2211yescharacteristic
47#2346C2212yes—
48#2334—22no—
49#2326—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2296C2212yes—
54#2295C2211yescharacteristic
55#2285C2212yes—
56#2276—21no—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2212yes—
62#2084C2211yescharacteristic
63#2083C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2072C2211yescharacteristic
67#2072C2212yes—
68#2072C2212yes—
69#2067C2214yes—
70#2065C2212yes—
71#2065C2214yes—
72#2055C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2014C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1949C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1926C2214yes—
82#1925C2214yes—
83#1893C2214yes—
84#1880C2214yes—
85#1875C2214yes—
86#1850C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1805C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1735C2214yes—
95C2×#336C2212yes—
96C22×SD16C2312yes—
97C23×C8C2314yes—
98#556—24no—
99#556D418yescomplemented
100#555—28no—
101#555—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#126—22no—
108C2×#124—21no—
109C2×#124—24no—
110C2×C22⋊Q8C2312yes—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×#117—22no—
127C2×#117C2314yes—
128C2×#117C2312yes—
129C2×#117C2312yes—
130C22×C42C2312yes—
131#498—28no—
132#485—216no—
133#482—216no—
134#477—28no—
135#474—28no—
136#465—216no—
137C4×SD16—28no—
138C8×D4—28no—
139#427—216no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2312yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C4×C8C2312yes—
157#394—24no—
158#394—24no—
159#393—24no—
160#391—21no—
161#391—21no—
162#391C2311yescharacteristic
163#391C2311yescharacteristic
164#390—21no—
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#390C2314yes—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2312yes—
176#389C2312yes—
177#389C2311yescharacteristic
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—24no—
184#388—24no—
185#387—21no—
186#387—22no—
187#387—21no—
188#387—22no—
189#387—22no—
190#385—22no—
191#385—22no—
192#385—22no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#384C2312yes—
199#384C2312yes—
200#382—22no—
201#382C2314yes—
202#380—22no—
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—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—21no—
228#376—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×Q8C2411yescharacteristic
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C22×C8—28no—
258C22×C8—28no—
259C22×C8—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—82no—
267C4⋊Q8—82no—
268C4⋊Q8C2×D414yes—
269#127—42no—
270#126—216no—
271#124—28no—
272#124—416no—
273C22⋊Q8—24no—
274C22⋊Q8C2×D418yes—
275#121—28no—
276#121—28no—
277#121—48no—
278#121—84no—
279#121—84no—
280C4×Q8—28no—
281C4×Q8—22no—
282C4×Q8—44no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8C2×D414yes—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—42no—
290C4×D4—44no—
291C4×D4—44no—
292C4×D4C2×D414yes—
293C4×D4#4114yes—
294#117—24no—
295#117—28no—
296#117—28no—
297#117—48no—
298#117—44no—
299#117C2×D418yes—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—24no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—28no—
362#107—28no—
363#106—24no—
364#106—28no—
365#106—28no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368C4⋊C8—44no—
369#103—24no—
370#103—28no—
371#102—24no—
372#102—24no—
373#102—28no—
374#102—42no—
375#102—44no—
376C22⋊C8—28no—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—22no—
397C24C22×D412yes—
398C2×Q8—24no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8C22×D412yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4#12114yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4114yes—
456C22×C4C2511yescharacteristic
457SD16—48no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—416no—
467C2×C82−1+414yes—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4#12114yes—
498C4⋊C4C22×D414yes—
499C4⋊C4C22×D412yes—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4#12118yes—
514C42—28no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—24no—
520C42—22no—
521C42—42no—
522C42—42no—
523C42—41no—
524C42—44no—
525C42—48no—
526C42—44no—
527C42—42no—
528C42—82no—
529C42—82no—
530C42#13514yes—
531C42C22×D412yes—
532C42C2×#4112yes—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C2×#12114yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—42no—
550Q8—48no—
551Q8—164no—
552D4—28no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—84no—
558D4—84no—
559D4—168no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4C2×#12114yes—
609C2×C4C2×#12112yes—
610C2×C4C2×#12112yes—
611C2×C4C2×#13512yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4#57512yes—
615C2×C4#57712yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×2−1+412yes—
619C2×C4#58512yes—
620C8—24no—
621C8—42no—
622C8—44no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22C22×#12111yescharacteristic
638C22#348511yescharacteristic
639C22#349911yescharacteristic
640C22C22×#13511yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57711yescharacteristic
643C22#358511yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—24no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—41no—
653C4—42no—
654C4—44no—
655C4—82no—
656C4—84no—
657C4—81no—
658C4—81no—
659C4—82no—
660C4#303514yes—
661C4#348912yes—
662C4#351512yes—
663C2—44no—
664C2—162no—
665C2#3602011yescharacteristic
666C2C2×#303511yescharacteristic
667C2#6043111yescharacteristic
668C2#6170011yescharacteristic
669C2#6173111yescharacteristic
670C2#6247011yescharacteristic
671C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
1. C2×#3340C2semidirect
3. #34324C2semidirect
4. #29869C2semidirect
5. #29758C2semidirect
6. #28687C2semidirect
7. #28574C2semidirect
8. #28201C2semidirect
9. #27652C2semidirect
10. #25452C2semidirect
11. #25213C2semidirect
12. #24814C2semidirect
13. #24587C2semidirect
14. #23678C2semidirect
15. #23656C2semidirect
16. #22696C2semidirect
17. #22489C2semidirect
99. #556D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33402
#361872
#343242
#298692
#297582
#286872
#285742
#282012
#276522
#254522
#252132
#248142
#245872
#236782
#236562
#226962
#224892