Finite Group Databasebeta

← #6621053

Subgroups — #6621053

Order 512.

1975 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3361C212yesmaximal, complemented
2#36559C211yesmaximal, characteristic, complemented
3C4×#559C211yesmaximal, characteristic, complemented
4#33199C212yesmaximal, complemented
5#33198C211yesmaximal, characteristic, complemented
6#28721C212yesmaximal, complemented
7#28716C211yesmaximal, characteristic, complemented
8#28253C212yesmaximal, complemented
9#28251C211yesmaximal, characteristic
10#25326C214yesmaximal, complemented
11#25321C212yesmaximal, complemented
12#24684C214yesmaximal, complemented
13#24679C212yesmaximal, complemented
14Q8×C4⋊C8C211yesmaximal, characteristic, complemented
15#22519C212yesmaximal, complemented
16C4×#460C211yesmaximal, characteristic, complemented
17C4×#459C212yesmaximal, complemented
18C2×#559C2211yescharacteristic
19C2×#554C2212yes—
20#3361—216no—
21C2×#488C2214yes—
22C2×#486C2212yes—
23C2×#477C2212yes—
24C2×#478C2214yes—
25C2×#460C2211yescharacteristic
26C2×#463C2214yes—
27C2×#459C2212yes—
28C2×#459C2212yes—
29C2×C8×Q8C2212yes—
30C2×C4×SD16C2212yes—
31C2×C4×D8C2214yes—
32#2868C2211yescharacteristic
33#2859C2212yes—
34#2566C2214yes—
35#2563C2212yes—
36#2538—21no—
37#2538—21no—
38#2452—22no—
39#2452—22no—
40#2374—22no—
41#2367—21no—
42#2367C2212yes—
43#2366C2211yescharacteristic
44#2357—21no—
45#2357C2212yes—
46#2355C2211yescharacteristic
47#2355C2211yescharacteristic
48D4×C4⋊C4—21no—
49D4×C4⋊C4—21no—
50C4×#127C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#127C2212yes—
53C4×#124—21no—
54C4×#124C2212yes—
55C4×#123—22no—
56C42×Q8C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58C42×D4—21no—
59C42×D4C2212yes—
60#2074C2214yes—
61#2073C2214yes—
62#2071C2214yes—
63#2055C2214yes—
64#2054C2214yes—
65#2054C2214yes—
66#1991C2214yes—
67#1990C2212yes—
68#1989C2214yes—
69#1987C2212yes—
70#1982C2211yescharacteristic
71#1949C2214yes—
72#1948C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1939C2212yes—
76#1939C2214yes—
77#1926C2212yes—
78#1925C2214yes—
79#1880C2212yes—
80#1879C2214yes—
81#1842C2211yescharacteristic
82#1842C2212yes—
83#1804C2214yes—
84#1804C2214yes—
85#1803C2211yescharacteristic
86#1803C2212yes—
87#1803C2212yes—
88#1802C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#102C2212yes—
95C4×#102C2214yes—
96C22×SD16C2312yes—
97C22×D8C2314yes—
98#559—24no—
99#559D418yescomplemented
100#554—28no—
101#554—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125C2314yes—
108C2×#124—21no—
109C2×#124C2312yes—
110C2×#123—22no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
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×D4—21no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2314yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C22×C4⋊C4—22no—
133C22×C42—22no—
134#488—216no—
135#486—28no—
136#478—216no—
137#477—28no—
138#463—216no—
139#460—24no—
140#459—28no—
141#459—28no—
142C8×Q8—28no—
143C4×SD16—28no—
144C4×D8—216no—
145C2×#107C2314yes—
146C2×#106C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2311yescharacteristic
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×#103C2314yes—
155C2×#103C2312yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×C4×C8C2312yes—
164C2×C4×C8C2314yes—
165#396—24no—
166#396—24no—
167#393—24no—
168#391—21no—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2312yes—
179#390C2312yes—
180#390C2312yes—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#385—22no—
195#385—24no—
196#385—24no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2311yescharacteristic
202#384C2312yes—
203#382—21no—
204#382C2314yes—
205#382C2312yes—
206#381—22no—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C43—21no—
233C43C2311yescharacteristic
234C43C2312yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244C22×Q8—22no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4C2412yes—
252C22×D4C2412yes—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—24no—
258C2×SD16—24no—
259C2×SD16—24no—
260C2×SD16—42no—
261C2×D8—28no—
262C2×D8—28no—
263C2×D8—44no—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—82no—
270C4⋊Q8—82no—
271#127—22no—
272#127—22no—
273#127C2×D414yes—
274#127#4114yes—
275#125—28no—
276#125—28no—
277#124—28no—
278#124—24no—
279#124C2×D418yes—
280#123—216no—
281C22⋊Q8—48no—
282C22⋊Q8—416no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8—22no—
288C4×Q8—42no—
289C4×Q8—41no—
290C4×Q8C2×D414yes—
291C4×Q8C2×D414yes—
292C4×Q8#4114yes—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—28no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—28no—
299C4×D4—24no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—42no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—84no—
308C4×D4—82no—
309C4×D4—84no—
310C4×D4C2×D418yes—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C42—21no—
354C2×C42—21no—
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×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—28no—
376#106—24no—
377C4⋊C8—24no—
378C4⋊C8—28no—
379C4⋊C8—24no—
380C4⋊C8—22no—
381C4⋊C8—22no—
382C4⋊C8—44no—
383C4⋊C8—42no—
384#103—24no—
385#103—28no—
386#102—28no—
387#102—28no—
388#102—28no—
389#102—28no—
390#102—24no—
391#102—24no—
392#102—24no—
393C4×C8—28no—
394C4×C8—24no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—24no—
405#95—24no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—22no—
410C24—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—44no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8#12114yes—
422C2×Q8C22×D412yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—81no—
444C2×D4—81no—
445C2×D4C22×D414yes—
446C2×D4C2×#4114yes—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—28no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480SD16—48no—
481D8—416no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—81no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4#12114yes—
520C4⋊C4#12118yes—
521C4⋊C4C22×D412yes—
522C4⋊C42−1+412yes—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—82no—
538C42—24no—
539C42—24no—
540C42—22no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—22no—
546C42—21no—
547C42—42no—
548C42—42no—
549C42—41no—
550C42—42no—
551C42—42no—
552C42—42no—
553C42—42no—
554C42—81no—
555C42—82no—
556C42—82no—
557C42#13514yes—
558C42C22×D414yes—
559C42C22×D412yes—
560C42C2×#4114yes—
561C42C2×#4112yes—
562C42C2×#4112yes—
563C422−1+412yes—
564C23—24no—
565C23—24no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23C23×D411yescharacteristic
579Q8—28no—
580Q8—22no—
581Q8—22no—
582Q8—41no—
583Q8—41no—
584Q8—164no—
585D4—24no—
586D4—24no—
587D4—24no—
588D4—24no—
589D4—48no—
590D4—48no—
591D4—42no—
592D4—42no—
593D4—42no—
594D4—42no—
595D4—48no—
596D4—84no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—88no—
638C2×C4—88no—
639C2×C4—88no—
640C2×C4—88no—
641C2×C4—168no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12114yes—
644C2×C4C2×#12112yes—
645C2×C4C2×#12112yes—
646C2×C4#54812yes—
647C2×C4#54812yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13512yes—
650C2×C4C2×#13512yes—
651C2×C4#57512yes—
652C2×C4#57612yes—
653C2×C4C23×D411yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C2×C4#58511yescharacteristic
659C8—44no—
660C8—42no—
661C8—42no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C22×#12111yescharacteristic
677C22C2×#54811yescharacteristic
678C22#348911yescharacteristic
679C22C22×#13511yescharacteristic
680C22C2×#57511yescharacteristic
681C22C2×#57611yescharacteristic
682C22#358511yescharacteristic
683C4—24no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—41no—
692C4—42no—
693C4—42no—
694C4—84no—
695C4—82no—
696C4—81no—
697C4—82no—
698C4#303514yes—
699C4#336114yescomplemented
700C4D4×#4111yescharacteristic
701C4#348211yescharacteristic
702C4#349911yescharacteristic
703C4#351811yescharacteristic
704C2—84no—
705C2—84no—
706C2#3644411yescharacteristic
707C2C2×#303511yescharacteristic
708C2#6042811yescharacteristic
709C2C2×#336111yescharacteristic
710C2#6131811yescharacteristic
711C2#6210211yescharacteristic
712C2#6214311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28251C2
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×#3361C2semidirect
2. #36559C2semidirect
3. C4×#559C2semidirect
4. #33199C2semidirect
5. #33198C2semidirect
6. #28721C2semidirect
7. #28716C2semidirect
8. #28253C2semidirect
10. #25326C2semidirect
11. #25321C2semidirect
12. #24684C2semidirect
13. #24679C2semidirect
14. Q8×C4⋊C8C2semidirect
15. #22519C2semidirect
16. C4×#460C2semidirect
17. C4×#459C2semidirect
99. #559D4semidirect
699. C4#3361semidirect

Maximal subgroups

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

subgroupindex
C2×#33612
#365592
C4×#5592
#331992
#331982
#287212
#287162
#282532
#282512
#253262
#253212
#246842
#246792
Q8×C4⋊C82
#225192
C4×#4602
C4×#4592