Finite Group Databasebeta

← #6493612

Subgroups — #6493612

Order 512.

2588 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#33106C214yesmaximal, complemented
4#29920C214yesmaximal, complemented
5#27973C212yesmaximal, complemented
6#27909C212yesmaximal, complemented
7#27891C214yesmaximal, complemented
8#27582C214yesmaximal, complemented
9#24002C214yesmaximal, complemented
10#21982C212yesmaximal, complemented
11#21803C212yesmaximal, complemented
12#20714C211yesmaximal, characteristic, complemented
13C22×#127C2211yescharacteristic
14C22×#121—21no—
15C22×#121—21no—
16C22×#121C2212yes—
17C22×#117—21no—
18C23×C42C2211yescharacteristic
19C2×#480C2218yescomplemented
20C2×#431C2212yes—
21#2856—21no—
22#2856—21no—
23#2856—21no—
24#2856—21no—
25#2856C2212yes—
26#2856C2212yes—
27#2856C2212yes—
28#2856C2212yes—
29C2×#390—21no—
30C2×#390—21no—
31C2×#390C2212yes—
32#2285—21no—
33#2285—21no—
34#2285—21no—
35#2285—21no—
36#2285C2212yes—
37#2285C2212yes—
38#2285C2212yescomplemented
39#2285C2212yescomplemented
40C2×#384C2211yescharacteristic
41C2×#384C2211yescharacteristic, complemented
42C2×#384C2211yescharacteristic, complemented
43C2×#382—21no—
44C2×#380—21no—
45C2×#380C2212yes—
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4C2212yescomplemented
48#2093C2218yescomplemented
49#2071C2218yescomplemented
50#2033C22116yescomplemented
51#1982C2218yescomplemented
52#1957C2214yescomplemented
53#1957C2214yescomplemented
54#1947C2218yescomplemented
55#1925C22116yescomplemented
56#1874C2218yescomplemented
57#1852C2218yes—
58#1845C2214yescomplemented
59#1834C2214yescomplemented
60#1817C2212yes—
61#1797C2212yescomplemented
62#1797C2212yescomplemented
63#1786C2218yes—
64#1779C2212yes—
65C4×C22⋊C8C2212yescomplemented
66C4×C22⋊C8C2212yescomplemented
67C23×D4—22no—
68C23×D4—21no—
69C23×D4—21no—
70C23×D4C2312yes—
71C24×C4D411yescharacteristic
72C24×C4D411yescharacteristic
73C24×C4C2311yescharacteristic
74C2×#127—21no—
75C2×#127—21no—
76C2×#127—28no—
77C2×#127C2312yes—
78C2×#127C2312yes—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—22no—
85C2×#121—22no—
86C2×#121—22no—
87C2×#121—22no—
88C2×#121—22no—
89C2×#121—48no—
90C2×#121—48no—
91C2×#121—48no—
92C2×#121C2314yes—
93C2×#121C2314yes—
94C2×#117—22no—
95C2×#117—22no—
96C2×#117—22no—
97C2×#117—22no—
98C2×#117—48no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—22no—
109C22×C22⋊C4—22no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—22no—
112C22×C42—28no—
113C22×C42—22no—
114C22×C42D411yescharacteristic
115C22×C42D411yescharacteristic
116C22×C42D411yescharacteristic
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#480—416no—
127#431—28no—
128C2×#107C23116yes—
129C2×C4⋊C8C2314yes—
130C2×#102C23116yes—
131C2×#102C23116yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C4×C8C2314yes—
136#390—22no—
137#390—22no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390C2314yes—
147#390C2314yes—
148#384—21no—
149#384—21no—
150#384—21no—
151#384—21no—
152#384—21no—
153#384—21no—
154#384C2312yes—
155#384C2312yes—
156#384C2312yes—
157#384C2312yes—
158#384C2312yes—
159#384C2312yes—
160#382—22no—
161#382—22no—
162#382—22no—
163#382—22no—
164#380—22no—
165#380—22no—
166#380—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380C2314yes—
171#380C2314yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4C2314yes—
179C4×C4⋊C4C2314yes—
180C2×#95—22no—
181C2×#95—22no—
182#340C23116yes—
183#336C2314yes—
184#336C2314yes—
185C25—22no—
186C25C2×D411yescharacteristic
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—28no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—48no—
200C22×D4—48no—
201C22×D4—48no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4C2414yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4D818yes—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C22×C8C2418yes—
238#127—28no—
239#127—28no—
240#127—44no—
241#127—44no—
242#121—216no—
243#121—48no—
244#121—48no—
245#121—48no—
246#121—48no—
247#121—48no—
248#121—48no—
249#121—48no—
250#121—48no—
251#121—48no—
252#121—48no—
253#121—48no—
254#117—48no—
255#117—48no—
256#117—48no—
257#117—48no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2414yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2414yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—48no—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42#4118yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—416no—
329C4⋊C8—28no—
330#102—416no—
331#102—416no—
332C22⋊C8—28no—
333C22⋊C8—28no—
334C22⋊C8—44no—
335C4×C8—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—41no—
346C24—41no—
347C24—44no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—44no—
354C24C2×D814yes—
355C24C22×D411yescharacteristic
356C24C22×D411yescharacteristic
357C24C22×D411yescharacteristic
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—816no—
383C2×D4—816no—
384C2×D4—832no—
385C2×D4—816no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—232no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4#12314yes—
444C22×C4#12314yes—
445C22×C4C2×D814yes—
446C22×C4C2×D814yes—
447C22×C4C2×D814yes—
448C22×C4C2×D814yes—
449C22×C4C2×D814yes—
450C22×C4C2×D814yes—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4#13714yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C4C2511yescharacteristic
467C2×C8—28no—
468C2×C8—28no—
469C2×C8—28no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—48no—
509C42—48no—
510C42#12314yes—
511C42#12314yes—
512C42C2×#4114yes—
513C42C2×#4114yes—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—28no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—28no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—88no—
543C23—88no—
544C23—88no—
545C23—88no—
546C23—88no—
547C23—88no—
548C23—88no—
549C23C2×#12312yes—
550C23#53512yes—
551C23C22×D812yes—
552C23C22×D812yes—
553C23C22×D812yes—
554C23C2×AGL(1,ℤ/8)12yes—
555C23C2×#13712yes—
556C23C23×D411yescharacteristic
557D4—416no—
558D4—88no—
559D4—88no—
560D4—816no—
561D4—816no—
562D4—88no—
563D4—88no—
564D4—88no—
565C2×C4—28no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—88no—
610C2×C4—88no—
611C2×C4—88no—
612C2×C4#480116yescomplemented
613C2×C4C2×#12312yes—
614C2×C4C2×#12312yes—
615C2×C4C2×#12312yes—
616C2×C4C2×#12312yes—
617C2×C4C2×#12312yes—
618C2×C4#53512yes—
619C2×C4C22×D812yes—
620C2×C4C22×D812yes—
621C2×C4C22×D812yes—
622C2×C4C22×D812yes—
623C2×C4C2×AGL(1,ℤ/8)12yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×#13712yes—
626C2×C4C2×#13712yes—
627C2×C4#57512yes—
628C2×C4C22×#4112yes—
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4#58512yes—
631C8—48no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—416no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—1616no—
660C22C2×#48014yes—
661C22#320514yes—
662C22C22×#12311yescharacteristic
663C22C2×#53511yescharacteristic
664C22#344911yescharacteristic
665C22C23×D811yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C22×#13711yescharacteristic
668C22C2×#57511yescharacteristic
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—22no—
675C4—41no—
676C4—41no—
677C4—84no—
678C4—84no—
679C4—84no—
680C4C2×#48014yes—
681C4C2×#48014yes—
682C4#309414yes—
683C4#349414yes—
684C2—24no—
685C2—24no—
686C2—42no—
687C2—164no—
688C2#3526211yescharacteristic
689C2C22×#48011yescharacteristic
690C2C2×#309411yescharacteristic
691C2C2×#320511yescharacteristic
692C2#6080411yescharacteristic
693C2#6094011yescharacteristic
694C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2856C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2285C2semidirect
3. #33106C2semidirect
4. #29920C2semidirect
5. #27973C2semidirect
6. #27909C2semidirect
7. #27891C2semidirect
8. #27582C2semidirect
9. #24002C2semidirect
10. #21982C2semidirect
11. #21803C2semidirect
12. #20714C2semidirect
19. C2×#480C22semidirect
38. #2285C22semidirect
39. #2285C22semidirect
41. C2×#384C22semidirect
42. C2×#384C22semidirect
47. C2×C4×C4⋊C4C22semidirect
48. #2093C22semidirect
49. #2071C22semidirect
50. #2033C22semidirect
51. #1982C22semidirect
52. #1957C22semidirect
53. #1957C22semidirect
54. #1947C22semidirect
55. #1925C22semidirect
56. #1874C22semidirect
58. #1845C22semidirect
59. #1834C22semidirect
61. #1797C22semidirect
62. #1797C22semidirect
65. C4×C22⋊C8C22semidirect
66. C4×C22⋊C8C22semidirect
612. C2×C4#480semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#22852
#331062
#299202
#279732
#279092
#278912
#275822
#240022
#219822
#218032
#207142