Finite Group Databasebeta

← #6461492

Subgroups — #6461492

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic, complemented
3#29381C214yesmaximal, complemented
4#29380C214yesmaximal, complemented
5#29373C214yesmaximal, complemented
6#29372C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic, complemented
9#23506C211yesmaximal, characteristic, complemented
10C2×#2012C211yesmaximal, characteristic
11#21064C211yesmaximal, characteristic, complemented
12#21063C211yesmaximal, characteristic, complemented
13#20936C211yesmaximal, characteristic, complemented
14#20934C211yesmaximal, characteristic
15#20934C211yesmaximal, characteristic
16#20933C211yesmaximal, characteristic, complemented
17#20877C211yesmaximal, characteristic, complemented
18#20876C211yesmaximal, characteristic, complemented
19#20774C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#428C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36#2367—22no—
37#2367—22no—
38#2357—22no—
39#2354—22no—
40#2342—24no—
41#2342—24no—
42#2326C2212yescomplemented
43#2326C2212yescomplemented
44#2319C2214yescomplemented
45#2317C2211yescharacteristic
46#2310C2211yescharacteristic
47#2309C2211yescharacteristic
48#2309C2211yescharacteristic
49C4×#124C2212yescomplemented
50C4×#124C2212yescomplemented
51#2296C2211yescharacteristic
52#2293C2214yescomplemented
53C2×#390C2211yescharacteristic
54#2285C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2101C2212yescomplemented
58#2101C2212yescomplemented
59#2100C2212yescomplemented
60#2099C2212yescomplemented
61#2094C2212yescomplemented
62#2093C2212yescomplemented
63#2092C2212yescomplemented
64#2092C2212yescomplemented
65#2042C2214yescomplemented
66#2042C2214yescomplemented
67#2012C2214yes—
68#2012C2214yes—
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1926C2214yescomplemented
72#1925C2214yescomplemented
73#1880C2214yescomplemented
74#1880C2214yescomplemented
75#1879C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1879C2214yes—
79#1877C2214yescomplemented
80#1877C2214yescomplemented
81#1865C2214yescomplemented
82#1864C2214yescomplemented
83#1863C2214yescomplemented
84#1862C2214yescomplemented
85#1840C2214yescomplemented
86#1839C2214yescomplemented
87#1735C2211yescharacteristic
88#1735C2211yescharacteristic
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1729C2211yescharacteristic
92#1729C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16C2314yes—
104C22×D8—24no—
105C22×D8—24no—
106C22×D8—42no—
107C22×D8C2314yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C2×#124—24no—
113C2×#124—24no—
114C2×#124—22no—
115C2×#124C2312yes—
116C2×#124C2312yes—
117#519—42no—
118#519—42no—
119#518—24no—
120#518—24no—
121#518—42no—
122#518—42no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2314yes—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—22no—
134C2×#117—24no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C22⋊C4—22no—
145C22×C22⋊C4—22no—
146C22×C42C2311yescharacteristic
147#428—42no—
148#428—42no—
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#106C2312yes—
152C2×#106C2312yes—
153#419—24no—
154#419—24no—
155#418—24no—
156#418—24no—
157#418—42no—
158#418—42no—
159C2×#103—42no—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102—42no—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171#391—22no—
172#391C2312yes—
173#391C2312yes—
174#390—21no—
175#390—21no—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#390C2312yes—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#389C2314yes—
184#389C2312yes—
185#389C2312yes—
186#388—24no—
187#388—24no—
188#388—24no—
189#388—24no—
190#388—24no—
191#388—24no—
192#388—24no—
193#388—24no—
194#385—22no—
195#385—22no—
196#385—24no—
197#385—24no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#382—21no—
202#382C2311yescharacteristic
203#382C2311yescharacteristic
204#381—22no—
205#381—24no—
206#380—22no—
207#380—24no—
208#379—22no—
209#379—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2314yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—24no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4D414yes—
221#376C2312yes—
222C43C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C25—21no—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8#4114yes—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4#4114yes—
262C22×D4#4114yes—
263C22×D4C2412yes—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—24no—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272C2×SD16—28no—
273C2×SD16—28no—
274C2×SD16—44no—
275C2×SD16—416no—
276C2×D8—28no—
277C2×D8—28no—
278C2×D8—44no—
279C2×D8—416no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296#124—216no—
297#124—24no—
298#124—24no—
299#124—24no—
300#124—24no—
301C4×Q8—42no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×D4—216no—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4—42no—
313C4×D4—42no—
314C4×D4—44no—
315C4×D4—44no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—48no—
328#117—88no—
329#117#4114yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—216no—
351C2×C22⋊C4—216no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—48no—
368C2×C22⋊C4—84no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D412yes—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—24no—
380C2×C42C2×D412yes—
381C2×C42C2×D414yes—
382C2×C42C2×D414yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—42no—
388#107—42no—
389#106—42no—
390#106—42no—
391#103—28no—
392#103—28no—
393#103—44no—
394#102—28no—
395#102—28no—
396#102—44no—
397#102—42no—
398#102—42no—
399#102—42no—
400#102—42no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95C2×D414yes—
410#95C2×D414yes—
411C24—22no—
412C24—22no—
413C24—24no—
414C24—24no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24—82no—
420C24—82no—
421C24—82no—
422C24—82no—
423C24C22×D411yescharacteristic
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—28no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8C2×#4114yes—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—28no—
440C2×D4—28no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4C2×#4114yes—
462C2×D4C2×#4114yes—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—216no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—82no—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)12yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4C2511yescharacteristic
499SD16—416no—
500D8—416no—
501M16—44no—
502M16—44no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—44no—
516C2×C8—44no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—84no—
541C4⋊C4C2×#4114yes—
542C22⋊C4—216no—
543C22⋊C4—216no—
544C22⋊C4—216no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—48no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C42—24no—
560C42—24no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—44no—
570C42—44no—
571C42—44no—
572C42—44no—
573C42—42no—
574C42—42no—
575C42—42no—
576C42—84no—
577C23—22no—
578C23—22no—
579C23—24no—
580C23—24no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—48no—
590C23—48no—
591C23—88no—
592C23—88no—
593C23—162no—
594C23—162no—
595C23—162no—
596C23—162no—
597C23—164no—
598C23—164no—
599C23#52512yes—
600C23#52512yes—
601C23#53512yes—
602C23C23×D411yescharacteristic
603Q8—24no—
604Q8—24no—
605Q8—48no—
606Q8—42no—
607Q8—42no—
608D4—24no—
609D4—24no—
610D4—24no—
611D4—24no—
612D4—48no—
613D4—48no—
614D4—42no—
615D4—42no—
616D4—42no—
617D4—42no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—48no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—816no—
653C2×C4—164no—
654C2×C4—164no—
655C2×C4—162no—
656C2×C4#52512yes—
657C2×C4#52512yes—
658C2×C4#53512yes—
659C2×C4#55114yes—
660C2×C4#55114yes—
661C2×C4#55314yes—
662C2×C4#55314yes—
663C2×C4C22×#4111yescharacteristic
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C8—42no—
670C8—42no—
671C8—42no—
672C8—42no—
673C22—22no—
674C22—22no—
675C22—24no—
676C22—24no—
677C22—44no—
678C22—44no—
679C22—48no—
680C22—48no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—84no—
686C22—84no—
687C22—168no—
688C22—168no—
689C22#259814yes—
690C22C2×#52511yescharacteristic
691C22C2×#52511yescharacteristic
692C22C2×#53511yescharacteristic
693C22#358412yes—
694C22#358512yes—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—44no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—82no—
706C4—81no—
707C4—82no—
708C4#259814yes—
709C4C2×#55111yescharacteristic
710C4C2×#55111yescharacteristic
711C4C2×#55311yescharacteristic
712C4C2×#55311yescharacteristic
713C2—24no—
714C2—84no—
715C2—84no—
716C2C2×#259811yescharacteristic
717C2#6058012yes—
718C2#6059212yes—
719C2#6081911yescharacteristic
720C2#6082511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #34255C2semidirect
2. #34211C2semidirect
3. #29381C2semidirect
4. #29380C2semidirect
5. #29373C2semidirect
6. #29372C2semidirect
7. #23744C2semidirect
8. #23731C2semidirect
9. #23506C2semidirect
11. #21064C2semidirect
12. #21063C2semidirect
13. #20936C2semidirect
16. #20933C2semidirect
17. #20877C2semidirect
18. #20876C2semidirect
19. #20774C2semidirect
42. #2326C22semidirect
43. #2326C22semidirect
44. #2319C22semidirect
49. C4×#124C22semidirect
50. C4×#124C22semidirect
52. #2293C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
61. #2094C22semidirect
62. #2093C22semidirect
63. #2092C22semidirect
64. #2092C22semidirect
65. #2042C22semidirect
66. #2042C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
71. #1926C22semidirect
72. #1925C22semidirect
73. #1880C22semidirect
74. #1880C22semidirect
79. #1877C22semidirect
80. #1877C22semidirect
81. #1865C22semidirect
82. #1864C22semidirect
83. #1863C22semidirect
84. #1862C22semidirect
85. #1840C22semidirect
86. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#342112
#293812
#293802
#293732
#293722
#237442
#237312
#235062
C2×#20122
#210642
#210632
#209362
#209342
#209332
#208772
#208762
#207742