Finite Group Databasebeta

← #6566101

Subgroups — #6566101

Order 512.

1873 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3357C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic
3#34332C211yesmaximal, characteristic, complemented
4#29869C211yesmaximal, characteristic, complemented
5#29754C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28580C212yesmaximal, complemented
8#28202C211yesmaximal, characteristic, complemented
9#27652C212yesmaximal, complemented
10#25450C212yesmaximal, complemented
11#25224C214yesmaximal, complemented
12#24814C212yesmaximal, complemented
13#24591C214yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23657C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22503C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×#554C2211yescharacteristic
20#3357—216no—
21C2×#500C2212yes—
22C2×#485C2214yes—
23C2×#480C2214yes—
24C2×#477C2212yes—
25C2×#476C2212yes—
26C2×#468C2214yes—
27C2×C4×SD16C2212yes—
28C2×#436C2212yes—
29C2×#428C2214yes—
30#2868C2211yescharacteristic
31#2856C2212yes—
32#2563C2212yes—
33#2537—21no—
34#2537—21no—
35#2500C2214yes—
36#2491—22no—
37#2491—22no—
38#2370—21no—
39#2369C2211yescharacteristic
40#2360—22no—
41#2360C2212yes—
42#2354—21no—
43#2354—21no—
44Q8×C4⋊C4C2211yescharacteristic
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2211yescharacteristic
47#2345C2212yes—
48#2330—21no—
49#2329—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2295C2211yescharacteristic
54C2×#389C2212yes—
55#2285C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2085C2214yes—
59#2085C2214yes—
60#2084C2212yes—
61#2084C2211yescharacteristic
62#2083C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2212yes—
66#2072C2211yescharacteristic
67#2072C2212yes—
68#2072C2212yes—
69#2067C2214yes—
70#2067C2214yes—
71#2065C2212yes—
72#2055C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2014C2211yescharacteristic
76#1990C2212yes—
77#1987C2212yes—
78#1949C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1926C2214yes—
82#1925C2214yes—
83#1895C2214yes—
84#1880C2214yes—
85#1874C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1825C2214yes—
89#1805C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×SD16C2312yes—
97C22×M16C2314yes—
98#555—28no—
99#555—28no—
100#554—24no—
101#554D418yescomplemented
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#124—24no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×#121—22no—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×C4×Q8—21no—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
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—
128C22×C4⋊C4—22no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C42C2312yes—
132#500—28no—
133#485—216no—
134#480—216no—
135#477—28no—
136#476—28no—
137#468—216no—
138C4×SD16—28no—
139#436—28no—
140#428—216no—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158#394—24no—
159#394—24no—
160#393—24no—
161#391—21no—
162#391—22no—
163#391C2311yescharacteristic
164#391C2311yescharacteristic
165#390—21no—
166#390—21no—
167#390—22no—
168#390—22no—
169#390C2312yes—
170#390C2314yes—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2311yescharacteristic
180#389C2312yes—
181#389C2312yes—
182#389C2311yescharacteristic
183#388—24no—
184#388—24no—
185#387—21no—
186#387—22no—
187#387—21no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—22no—
192#385—22no—
193#385—22no—
194#384—21no—
195#384—21no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#384C2312yes—
200#384C2312yes—
201#382C2314yes—
202#380—22no—
203#380—22no—
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—22no—
213C4×C4⋊C4—22no—
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×Q8C2411yescharacteristic
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2×D411yescharacteristic
249C22×D4C2412yes—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C2×SD16—24no—
257C2×SD16—24no—
258C2×SD16—42no—
259C2×M16—28no—
260C2×M16—28no—
261C2×M16—44no—
262C22×C8C2414yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—24no—
268C4⋊Q8—82no—
269C4⋊Q8—82no—
270C4⋊Q8C2×D414yes—
271#127—42no—
272#125—28no—
273#124—416no—
274C22⋊Q8—24no—
275C22⋊Q8—216no—
276C22⋊Q8C2×D418yes—
277#121—28no—
278#121—28no—
279#121—48no—
280#121—84no—
281#121—84no—
282C4×Q8—22no—
283C4×Q8—44no—
284C4×Q8—82no—
285C4×Q8C2×D414yes—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—42no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—44no—
294C4×D4C2×D414yes—
295C4×D4#4114yes—
296#117—28no—
297#117—28no—
298#117—48no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
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⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D418yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—44no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2×D411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362#107—28no—
363#107—28no—
364#107—28no—
365#106—24no—
366#106—28no—
367C4⋊C8—24no—
368C4⋊C8—24no—
369C4⋊C8—44no—
370#103—24no—
371#103—28no—
372#102—24no—
373#102—24no—
374#102—28no—
375#102—42no—
376#102—44no—
377C22⋊C8—28no—
378C4×C8—24no—
379#95—21no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95—21no—
391#95—22no—
392#95—22no—
393#95—21no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C24C22×D412yes—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—81no—
405C2×Q8—81no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8C22×D412yes—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4#12114yes—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D414yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2×#4114yes—
463C22×C4C2511yescharacteristic
464SD16—48no—
465M16—416no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C82−1+414yes—
475C4⋊C4—28no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—42no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—81no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—81no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4#12114yes—
507C4⋊C4C22×D412yes—
508C4⋊C4C22×D414yes—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—42no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4#12118yes—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—24no—
526C42—22no—
527C42—42no—
528C42—42no—
529C42—41no—
530C42—48no—
531C42—42no—
532C42—82no—
533C42—82no—
534C42—82no—
535C42#13714yes—
536C42C22×D412yes—
537C42C2×#4112yes—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23C2×#12114yes—
552C23C23×D411yescharacteristic
553Q8—24no—
554Q8—42no—
555Q8—84no—
556Q8—164no—
557D4—28no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—84no—
563D4—84no—
564D4—168no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
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—84no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—164no—
615C2×C4C2×#12114yes—
616C2×C4C2×#12112yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4#57512yes—
622C2×C4#57612yes—
623C2×C4C23×D411yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×2−1+412yes—
626C2×C4#58512yes—
627C8—24no—
628C8—42no—
629C8—44no—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22C22×#12111yescharacteristic
645C22#348511yescharacteristic
646C22#349911yescharacteristic
647C22C22×#13711yescharacteristic
648C22C2×#57511yescharacteristic
649C22C2×#57611yescharacteristic
650C22#358511yescharacteristic
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—24no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—42no—
662C4—84no—
663C4—82no—
664C4—81no—
665C4—81no—
666C4—82no—
667C4#303714yes—
668C4#348912yes—
669C4#351512yes—
670C2—44no—
671C2—162no—
672C2#3602011yescharacteristic
673C2C2×#303711yescharacteristic
674C2#6042811yescharacteristic
675C2#6172111yescharacteristic
676C2#6174011yescharacteristic
677C2#6248011yescharacteristic
678C2#6251211yescharacteristic
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×#3357C2semidirect
3. #34332C2semidirect
4. #29869C2semidirect
5. #29754C2semidirect
6. #28689C2semidirect
7. #28580C2semidirect
8. #28202C2semidirect
9. #27652C2semidirect
10. #25450C2semidirect
11. #25224C2semidirect
12. #24814C2semidirect
13. #24591C2semidirect
14. #23678C2semidirect
15. #23657C2semidirect
16. #22700C2semidirect
17. #22503C2semidirect
101. #554D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33572
#361872
#343322
#298692
#297542
#286892
#285802
#282022
#276522
#254502
#252242
#248142
#245912
#236782
#236572
#227002
#225032