Finite Group Databasebeta

← #6606602

Subgroups — #6606602

Order 512.

1943 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3357C212yesmaximal, complemented
2#36187C211yesmaximal, characteristic
3C4×#554C211yesmaximal, characteristic, complemented
4#33198C211yesmaximal, characteristic, complemented
5#33106C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28280C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25450C212yesmaximal, complemented
10#25222C214yesmaximal, complemented
11#24761C212yesmaximal, complemented
12#24679C212yesmaximal, complemented
13#24591C214yesmaximal, complemented
14#22650C211yesmaximal, characteristic, complemented
15#22503C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17#21170C212yesmaximal, complemented
18C2×#555C2212yes—
19C2×#554C2211yescharacteristic
20#3357—216no—
21C2×#500C2212yes—
22C2×#485C2214yes—
23C2×#480C2214yes—
24C2×#477C2212yes—
25C2×#476C2212yes—
26C2×#468C2214yes—
27C2×#460C2211yescharacteristic
28C2×C4×SD16C2212yes—
29C2×#436C2212yes—
30C2×#428C2214yes—
31C2×#423C2212yes—
32#2868C2211yescharacteristic
33#2856C2212yes—
34#2563C2212yes—
35#2537—21no—
36#2537—21no—
37#2500C2214yes—
38#2491—22no—
39#2491—22no—
40#2370—21no—
41#2369C2211yescharacteristic
42#2368—22no—
43#2360C2212yes—
44#2355C2211yescharacteristic
45#2354—21no—
46#2354—21no—
47#2349—21no—
48D4×C4⋊C4C2211yescharacteristic
49#2345C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#125—21no—
53C4×C22⋊Q8—22no—
54C4×C22⋊Q8C2212yes—
55#2285C2212yes—
56C42×Q8C2211yescharacteristic
57C42×D4—21no—
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4C2212yes—
60#2085C2214yes—
61#2083C2214yes—
62#2074C2214yes—
63#2067C2214yes—
64#2065C2214yes—
65#2055C2214yes—
66#1990C2212yes—
67#1987C2212yes—
68#1982C2211yescharacteristic
69#1982C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1947C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1947C2212yes—
78#1939C2212yes—
79#1926C2212yes—
80#1925C2214yes—
81#1925C2214yes—
82#1895C2214yes—
83#1880C2212yes—
84#1874C2214yes—
85#1851C2214yes—
86#1842C2211yescharacteristic
87#1825C2214yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#1736C2214yes—
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×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C42—22no—
132C22×C42C2312yes—
133#500—28no—
134#485—216no—
135#480—216no—
136#477—28no—
137#476—28no—
138#468—216no—
139#460—24no—
140C4×SD16—28no—
141#436—28no—
142#428—216no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2312yes—
148#423—28no—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2311yescharacteristic
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C4×C8C2312yes—
164#394—24no—
165#394—24no—
166#393—24no—
167#391—21no—
168#391—22no—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390C2312yes—
175#390C2314yes—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389C2312yes—
181#389C2312yes—
182#388—24no—
183#388—24no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#382—21no—
199#382C2314yes—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4C2312yes—
232C2×#95—22no—
233C43—21no—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2411yescharacteristic
247C22×D4—24no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4C2×D411yescharacteristic
251C22×D4C2412yes—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—42no—
261C2×M16—28no—
262C2×M16—28no—
263C2×M16—44no—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—22no—
269C4⋊Q8—24no—
270C4⋊Q8—82no—
271C4⋊Q8—82no—
272C4⋊Q8C2×D414yes—
273#127—22no—
274#127—22no—
275#125—28no—
276#124—416no—
277C22⋊Q8—24no—
278C22⋊Q8—216no—
279C22⋊Q8C2×D418yes—
280#121—28no—
281#121—28no—
282#121—48no—
283#121—84no—
284#121—84no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—41no—
290C4×Q8—82no—
291C4×Q8C2×D414yes—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—42no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4C2×D414yes—
301C4×D4#4114yes—
302#117—28no—
303#117—28no—
304#117—48no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D418yes—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—21no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—24no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—84no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372#107—28no—
373#107—28no—
374#107—28no—
375#106—24no—
376#106—28no—
377C4⋊C8—28no—
378C4⋊C8—24no—
379C4⋊C8—24no—
380C4⋊C8—22no—
381C4⋊C8—42no—
382C4⋊C8#4114yes—
383#103—24no—
384#103—28no—
385#102—24no—
386#102—24no—
387#102—28no—
388#102—24no—
389#102—44no—
390C22⋊C8—28no—
391C4×C8—24no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—21no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—22no—
409C24C22×D412yes—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—44no—
417C2×Q8—81no—
418C2×Q8—81no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8C22×D412yes—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—48no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4#12114yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D414yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4114yes—
478C22×C4C2511yescharacteristic
479SD16—48no—
480M16—416no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8C2×#4114yes—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—28no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—82no—
517C4⋊C4—81no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4#12114yes—
522C4⋊C4C22×D412yes—
523C4⋊C4C22×D414yes—
524C4⋊C42−1+412yes—
525C22⋊C4—28no—
526C22⋊C4—24no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—84no—
535C22⋊C4—84no—
536C22⋊C4#12118yes—
537C42—24no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—21no—
546C42—41no—
547C42—42no—
548C42—48no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—81no—
554C42—82no—
555C42—82no—
556C42#13714yes—
557C42C22×D412yes—
558C42C2×#4112yes—
559C42C2×#4112yes—
560C422−1+412yes—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23C2×#12114yes—
575C23C23×D411yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—84no—
581Q8—164no—
582D4—28no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—84no—
588D4—84no—
589D4—168no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—48no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—84no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4C2×#12114yes—
641C2×C4C2×#12112yes—
642C2×C4C2×#12112yes—
643C2×C4#54712yes—
644C2×C4#54712yes—
645C2×C4C2×#13712yes—
646C2×C4C2×#13712yes—
647C2×C4C2×#13712yes—
648C2×C4#57512yes—
649C2×C4#57612yes—
650C2×C4C23×D411yescharacteristic
651C2×C4C22×#4111yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×2−1+411yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C2×C4#58511yescharacteristic
656C8—24no—
657C8—44no—
658C8—42no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22C22×#12111yescharacteristic
674C22C2×#54711yescharacteristic
675C22#348911yescharacteristic
676C22C22×#13711yescharacteristic
677C22C2×#57511yescharacteristic
678C22C2×#57611yescharacteristic
679C22#358511yescharacteristic
680C4—22no—
681C4—22no—
682C4—22no—
683C4—24no—
684C4—42no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—41no—
690C4—42no—
691C4—84no—
692C4—81no—
693C4—82no—
694C4—82no—
695C4#303714yes—
696C4#335714yescomplemented
697C4D4×#4111yescharacteristic
698C4#348211yescharacteristic
699C4#349911yescharacteristic
700C4#351511yescharacteristic
701C2—44no—
702C2—162no—
703C2#3625911yescharacteristic
704C2C2×#303711yescharacteristic
705C2#6042811yescharacteristic
706C2C2×#335711yescharacteristic
707C2#6125611yescharacteristic
708C2#6211411yescharacteristic
709C2#6214311yescharacteristic
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. C4×#554C2semidirect
4. #33198C2semidirect
5. #33106C2semidirect
6. #28689C2semidirect
7. #28280C2semidirect
8. #27649C2semidirect
9. #25450C2semidirect
10. #25222C2semidirect
11. #24761C2semidirect
12. #24679C2semidirect
13. #24591C2semidirect
14. #22650C2semidirect
15. #22503C2semidirect
16. C4×#460C2semidirect
17. #21170C2semidirect
101. #554D4semidirect
696. C4#3357semidirect

Maximal subgroups

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

subgroupindex
C2×#33572
#361872
C4×#5542
#331982
#331062
#286892
#282802
#276492
#254502
#252222
#247612
#246792
#245912
#226502
#225032
C4×#4602
#211702