Finite Group Databasebeta

← #6434103

Subgroups — #6434103

Order 512.

1956 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35262C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic, complemented
3#29917C214yesmaximal, complemented
4#29916C214yesmaximal, complemented
5#29912C212yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#29907C212yesmaximal, complemented
8#29906C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23735C212yesmaximal, complemented
11#23662C212yesmaximal, complemented
12#23632C212yesmaximal, complemented
13#20912C211yesmaximal, characteristic
14C2×#1874C211yesmaximal, characteristic
15#20844C211yesmaximal, characteristic, complemented
16#20843C211yesmaximal, characteristic, complemented
17#20685C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#418C2212yes—
25C22×#102C2212yes—
26C2×#407C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28#2854C2212yes—
29#2854C2212yes—
30#2624C2212yes—
31#2624C2212yes—
32#2504—21no—
33#2501—21no—
34#2500—21no—
35#2500—21no—
36#2482—22no—
37#2405—22no—
38#2317—22no—
39#2317—21no—
40#2317—21no—
41#2317C2211yescharacteristic
42#2317C2211yescharacteristic
43#2317C2212yes—
44#2317C2212yes—
45#2317C2214yescomplemented
46#2309C2211yescharacteristic
47#2309C2211yescharacteristic
48C2×#389—21no—
49C2×#389—21no—
50C2×#385—22no—
51C2×#384C2212yescomplemented
52C2×#382C2212yescomplemented
53#2280C2211yescharacteristic
54#2101C2214yescomplemented
55#2101C2214yescomplemented
56#2100C2214yescomplemented
57#2099C2214yescomplemented
58#2094C2214yescomplemented
59#2093C2214yescomplemented
60#2092C2214yescomplemented
61#2092C2214yescomplemented
62#2085C2214yescomplemented
63#2085C2214yescomplemented
64#2084C2214yescomplemented
65#2083C2214yescomplemented
66#2073C2214yescomplemented
67#2073C2214yescomplemented
68#2071C2214yescomplemented
69#2071C2214yescomplemented
70#1875C2214yes—
71#1874C2212yes—
72#1874C2212yes—
73#1874C2214yes—
74#1874C2212yes—
75#1874C2212yes—
76#1854C2212yescomplemented
77#1854C2212yescomplemented
78#1853C2211yescharacteristic, complemented
79#1853C2211yescharacteristic, complemented
80#1853C2212yescomplemented
81#1853C2212yescomplemented
82#1852C2211yescharacteristic, complemented
83#1852C2211yescharacteristic, complemented
84#1852C2212yescomplemented
85#1852C2212yescomplemented
86#1818C2214yescomplemented
87#1817C2212yescomplemented
88#1817C2212yescomplemented
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C23×D4—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C24×C4C2311yescharacteristic
99C22×M16C2312yes—
100C23×C8C2312yes—
101C2×#123—22no—
102C2×#123—22no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121—24no—
106C2×#121—24no—
107C2×#121C2312yes—
108C2×#121C2312yes—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117—21no—
112C2×#117—21no—
113C2×#117—44no—
114C2×#117D412yes—
115C2×#117D411yescharacteristic
116C2×#117D411yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2312yes—
120C2×#117C2312yes—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—44no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—21no—
139C22×C22⋊C4—21no—
140C22×C22⋊C4D412yes—
141C22×C42—21no—
142C22×C42C2312yes—
143#428—44no—
144#427—44no—
145C2×#107C2312yes—
146C2×#107C2314yes—
147C2×#107C2312yes—
148C2×#106C2312yes—
149C2×#106C2314yes—
150C2×#106C2312yes—
151#418—44no—
152C2×#102—44no—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157#407—24no—
158C2×C22⋊C8—24no—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164#399—22no—
165#399—22no—
166#397—22no—
167#397—22no—
168#394—22no—
169#394—22no—
170#394—22no—
171#394—22no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389—24no—
185#389—24no—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#386—22no—
195#386—22no—
196#385—24no—
197#385—24no—
198#385—24no—
199#385—24no—
200#385—22no—
201#385—22no—
202#384—21no—
203#384—21no—
204#384—22no—
205#384—22no—
206#384C2314yes—
207#384C2314yes—
208#382—21no—
209#382—21no—
210#382—22no—
211#382—22no—
212#382C2314yes—
213#382C2314yes—
214#381—21no—
215#380C2311yescharacteristic
216#380C2311yescharacteristic
217#380C2311yescharacteristic
218#380C2311yescharacteristic
219#379—21no—
220#379—22no—
221#379—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C2×#95—22no—
225C2×#95—22no—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240C25—21no—
241C25C2×D411yescharacteristic
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—24no—
248C22×D4—24no—
249C22×D4—44no—
250C22×D4—82no—
251C22×D4—82no—
252C22×D4C2412yes—
253C23×C4—24no—
254C23×C4—24no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—44no—
262C23×C4—82no—
263C23×C4—82no—
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4#4114yes—
270C23×C4#4114yes—
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×M16—24no—
275C2×M16—24no—
276C2×M16—24no—
277C22×C8—24no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2414yes—
283#123—84no—
284#123—84no—
285#121—24no—
286#121—24no—
287#121—24no—
288#121—24no—
289#121—44no—
290#121—44no—
291#117—28no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—24no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—42no—
305#117—42no—
306#117—44no—
307#117—44no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4—84no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—28no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C22⋊C4C2×D412yes—
376C2×C42—24no—
377C2×C42—24no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—44no—
383C2×C42C2×D412yes—
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387#107—42no—
388#107—42no—
389#107—44no—
390#106—42no—
391#106—42no—
392#106—44no—
393#102—44no—
394#102—44no—
395#102—44no—
396#102—44no—
397C22⋊C8—22no—
398C22⋊C8—22no—
399C22⋊C8—24no—
400C22⋊C8—24no—
401C22⋊C8—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414C24—22no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24—41no—
420C24—41no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—81no—
425C24—81no—
426C24—81no—
427C24—81no—
428C24#12314yes—
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C24C22×D411yescharacteristic
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—48no—
444C2×D4—48no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4—816no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—216no—
455C22×C4—216no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—82no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—816no—
493C22×C4#12314yes—
494C22×C4#12314yes—
495C22×C4#12314yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4H(5,𝔽2)14yes—
510C22×C42−1+414yes—
511C22×C4C2511yescharacteristic
512M16—48no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—24no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—24no—
522C2×C8—48no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C4⋊C4—84no—
550C4⋊C4—84no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C4⋊C4—82no—
554C4⋊C4—82no—
555C22⋊C4—24no—
556C22⋊C4—24no—
557C22⋊C4—24no—
558C22⋊C4—24no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—42no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—81no—
576C22⋊C4—81no—
577C42—24no—
578C42—24no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—44no—
584C42—44no—
585C42—42no—
586C42—42no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—28no—
598C23—28no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—84no—
616C23—84no—
617C23—84no—
618C23—162no—
619C23—162no—
620C23—162no—
621C23—162no—
622C23C2×#12312yes—
623C23C2×#12312yes—
624C23C2×#12312yes—
625C23#53512yes—
626C23#53612yes—
627C23#57612yes—
628C23#57712yes—
629C23C23×D411yescharacteristic
630D4—44no—
631D4—44no—
632D4—88no—
633D4—88no—
634D4—84no—
635C2×C4—28no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—48no—
659C2×C4—48no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—44no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—84no—
685C2×C4—84no—
686C2×C4—88no—
687C2×C4—88no—
688C2×C4—84no—
689C2×C4—88no—
690C2×C4C2×#12312yes—
691C2×C4C2×#12312yes—
692C2×C4C2×#12312yes—
693C2×C4#53512yes—
694C2×C4#53612yes—
695C2×C4#53918yes—
696C2×C4#53918yes—
697C2×C4C22×#4111yescharacteristic
698C2×C4C22×#4111yescharacteristic
699C2×C4C2×H(5,𝔽2)12yes—
700C2×C4C2×2−1+412yes—
701C8—42no—
702C8—42no—
703C8—44no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—416no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—168no—
732C22#262314yes—
733C22#309614yes—
734C22C22×#12311yescharacteristic
735C22C2×#53511yescharacteristic
736C22C2×#53611yescharacteristic
737C22C2×#57611yescharacteristic
738C22C2×#57711yescharacteristic
739C22#358511yescharacteristic
740C22#358611yescharacteristic
741C4—22no—
742C4—22no—
743C4—42no—
744C4—42no—
745C4—42no—
746C4—42no—
747C4—82no—
748C4—81no—
749C4—82no—
750C4—84no—
751C4—81no—
752C4—82no—
753C4—81no—
754C4—81no—
755C4#262314yes—
756C4C2×#53912yes—
757C4C2×#53912yes—
758C2—24no—
759C2—24no—
760C2—42no—
761C2—162no—
762C2C2×#262311yescharacteristic
763C2C2×#309611yescharacteristic
764C2#6051711yescharacteristic
765C2#6081111yescharacteristic
766C2#6081311yescharacteristic
767C2#6087111yescharacteristic
768C2#6087311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2317C2semidirect
3. #29917C2semidirect
4. #29916C2semidirect
5. #29912C2semidirect
6. #29912C2semidirect
7. #29907C2semidirect
8. #29906C2semidirect
9. #23748C2semidirect
10. #23735C2semidirect
11. #23662C2semidirect
12. #23632C2semidirect
15. #20844C2semidirect
16. #20843C2semidirect
17. #20685C2semidirect
45. #2317C22semidirect
51. C2×#384C22semidirect
52. C2×#382C22semidirect
54. #2101C22semidirect
55. #2101C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
58. #2094C22semidirect
59. #2093C22semidirect
60. #2092C22semidirect
61. #2092C22semidirect
62. #2085C22semidirect
63. #2085C22semidirect
64. #2084C22semidirect
65. #2083C22semidirect
66. #2073C22semidirect
67. #2073C22semidirect
68. #2071C22semidirect
69. #2071C22semidirect
76. #1854C22semidirect
77. #1854C22semidirect
78. #1853C22semidirect
79. #1853C22semidirect
80. #1853C22semidirect
81. #1853C22semidirect
82. #1852C22semidirect
83. #1852C22semidirect
84. #1852C22semidirect
85. #1852C22semidirect
86. #1818C22semidirect
87. #1817C22semidirect
88. #1817C22semidirect

Maximal subgroups

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

subgroupindex
#352622
C2×#23172
#299172
#299162
#299122
#299072
#299062
#237482
#237352
#236622
#236322
#209122
C2×#18742
#208442
#208432
#206852