Finite Group Databasebeta

← #6434231

Subgroups — #6434231

Order 512.

1914 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35266C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic, complemented
3#30142C214yesmaximal, complemented
4#30141C214yesmaximal, complemented
5#30134C214yesmaximal, complemented
6#30133C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic, complemented
9#23615C211yesmaximal, characteristic, complemented
10#23614C211yesmaximal, characteristic, complemented
11#23541C211yesmaximal, characteristic, complemented
12#23540C211yesmaximal, characteristic
13#23506C211yesmaximal, characteristic, complemented
14C2×#2012C211yesmaximal, characteristic
15#20936C211yesmaximal, characteristic, complemented
16#20934C211yesmaximal, characteristic
17#20877C211yesmaximal, characteristic, complemented
18#20876C211yesmaximal, characteristic, complemented
19#20754C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#418C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37#2860C2214yescomplemented
38#2644C2214yescomplemented
39#2513—22no—
40#2512—22no—
41#2480—24no—
42#2367—22no—
43#2367—22no—
44#2342—24no—
45#2326C2212yescomplemented
46#2326C2212yescomplemented
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2310C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53C4×#124C2212yescomplemented
54C4×#124C2212yescomplemented
55#2296C2211yescharacteristic
56C2×#390C2211yescharacteristic
57#2285C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2212yescomplemented
60#2101C2212yescomplemented
61#2100C2212yescomplemented
62#2099C2212yescomplemented
63#2094C2212yescomplemented
64#2093C2212yescomplemented
65#2092C2212yescomplemented
66#2092C2212yescomplemented
67#2068C2214yescomplemented
68#2067C2214yescomplemented
69#2066C2214yescomplemented
70#2065C2214yescomplemented
71#2055C2214yescomplemented
72#2055C2214yescomplemented
73#2054C2214yes—
74#2054C2214yes—
75#2042C2214yescomplemented
76#2042C2214yescomplemented
77#2012C2214yes—
78#2012C2214yes—
79#1880C2214yescomplemented
80#1880C2214yescomplemented
81#1879C2214yes—
82#1879C2214yes—
83#1865C2214yescomplemented
84#1864C2214yescomplemented
85#1863C2214yescomplemented
86#1862C2214yescomplemented
87#1832C2214yescomplemented
88#1831C2214yescomplemented
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—42no—
102C22×SD16—42no—
103C22×SD16C2314yes—
104C22×D8—24no—
105C22×D8—42no—
106C22×D8—42no—
107C22×D8C2314yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C2×#126—24no—
113C2×#124—24no—
114C2×#124—24no—
115C2×#124—22no—
116C2×#124C2312yes—
117C2×#124C2312yes—
118#519—42no—
119#519—42no—
120#518—42no—
121#518—42no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117—22no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C22⋊C4—22no—
141C22×C42C2311yescharacteristic
142#428—42no—
143#428—42no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106C2312yes—
147C2×#106C2312yes—
148#424—42no—
149#424—42no—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152#419—24no—
153#419—42no—
154#418—24no—
155#418—42no—
156#418—42no—
157#418—42no—
158C2×#103—42no—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102—42no—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2314yes—
170#397—24no—
171#397—24no—
172#397—24no—
173#396—24no—
174#391—22no—
175#391C2312yes—
176#391C2312yes—
177#390—22no—
178#390—21no—
179#390—22no—
180#390—21no—
181#390—22no—
182#390C2314yes—
183#390C2312yes—
184#390C2312yes—
185#390C2314yes—
186#390C2312yes—
187#390C2312yes—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#389C2312yes—
192#389C2312yes—
193#388—24no—
194#388—24no—
195#388—24no—
196#388—24no—
197#388—24no—
198#388—24no—
199#388—24no—
200#385—22no—
201#385—22no—
202#385—24no—
203#385—24no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382—21no—
208#382C2311yescharacteristic
209#382C2311yescharacteristic
210#381—22no—
211#380—22no—
212#380—22no—
213#379—22no—
214#379—22no—
215#379—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D412yes—
220C4×C22⋊C4—24no—
221C4×C22⋊C4—24no—
222C4×C22⋊C4D414yes—
223#376C2312yes—
224C43C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
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—22no—
262C22×D4#4114yes—
263C22×D4C2412yes—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×SD16—28no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×SD16—416no—
275C2×D8—28no—
276C2×D8—44no—
277C2×D8—44no—
278C2×D8—416no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295#126—88no—
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—42no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—42no—
311C4×D4—44no—
312C4×D4—44no—
313#117—22no—
314#117—22no—
315#117—22no—
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—48no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—216no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—48no—
365C2×C22⋊C4—84no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42C2×D412yes—
377C2×C42C2×D414yes—
378C2×C42C2×D414yes—
379C2×C42C2412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—42no—
384#107—42no—
385#106—42no—
386#106—42no—
387C4⋊C8—44no—
388C4⋊C8—44no—
389#103—28no—
390#103—44no—
391#103—44no—
392#102—28no—
393#102—44no—
394#102—44no—
395#102—42no—
396#102—42no—
397#102—42no—
398#102—42no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—24no—
406#95—24no—
407#95C2×D414yes—
408#95C2×D414yes—
409C24—22no—
410C24—24no—
411C24—22no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—82no—
418C24—82no—
419C24—82no—
420C24—82no—
421C24C22×D411yescharacteristic
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—28no—
425C2×Q8—28no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8C2×#4114yes—
435C2×D4—24no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4C2×#4114yes—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C42−1+412yes—
493C22×C4C2511yescharacteristic
494SD16—416no—
495D8—416no—
496M16—44no—
497M16—44no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—44no—
511C2×C8—44no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C22⋊C4—216no—
539C22⋊C4—216no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—44no—
566C42—44no—
567C42—44no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42—42no—
572C42—42no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—48no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—88no—
587C23—88no—
588C23—84no—
589C23—162no—
590C23—162no—
591C23—162no—
592C23—162no—
593C23—164no—
594C23—164no—
595C23#52512yes—
596C23#53512yes—
597C23#53712yes—
598C23C23×D411yescharacteristic
599Q8—24no—
600Q8—24no—
601Q8—48no—
602Q8—42no—
603Q8—42no—
604D4—24no—
605D4—24no—
606D4—48no—
607D4—42no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—84no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—48no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—84no—
648C2×C4—164no—
649C2×C4—164no—
650C2×C4—162no—
651C2×C4#52512yes—
652C2×C4#53512yes—
653C2×C4#53712yes—
654C2×C4#53814yes—
655C2×C4#53814yes—
656C2×C4#53914yes—
657C2×C4#53914yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×2−1+411yescharacteristic
664C8—42no—
665C8—42no—
666C8—42no—
667C8—42no—
668C22—22no—
669C22—22no—
670C22—24no—
671C22—24no—
672C22—44no—
673C22—48no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—84no—
681C22—84no—
682C22—168no—
683C22—168no—
684C22#263614yes—
685C22C2×#52511yescharacteristic
686C22C2×#53511yescharacteristic
687C22C2×#53711yescharacteristic
688C22#358412yes—
689C22#358512yes—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—44no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—82no—
700C4—81no—
701C4—82no—
702C4—82no—
703C4#263614yes—
704C4C2×#53811yescharacteristic
705C4C2×#53811yescharacteristic
706C4C2×#53911yescharacteristic
707C4C2×#53911yescharacteristic
708C2—24no—
709C2—84no—
710C2—162no—
711C2C2×#263611yescharacteristic
712C2#6058012yes—
713C2#6081911yescharacteristic
714C2#6082511yescharacteristic
715C2#6092612yes—
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. #35266C2semidirect
2. #34211C2semidirect
3. #30142C2semidirect
4. #30141C2semidirect
5. #30134C2semidirect
6. #30133C2semidirect
7. #23744C2semidirect
8. #23731C2semidirect
9. #23615C2semidirect
10. #23614C2semidirect
11. #23541C2semidirect
13. #23506C2semidirect
15. #20936C2semidirect
17. #20877C2semidirect
18. #20876C2semidirect
19. #20754C2semidirect
37. #2860C22semidirect
38. #2644C22semidirect
45. #2326C22semidirect
46. #2326C22semidirect
53. C4×#124C22semidirect
54. C4×#124C22semidirect
59. #2101C22semidirect
60. #2101C22semidirect
61. #2100C22semidirect
62. #2099C22semidirect
63. #2094C22semidirect
64. #2093C22semidirect
65. #2092C22semidirect
66. #2092C22semidirect
67. #2068C22semidirect
68. #2067C22semidirect
69. #2066C22semidirect
70. #2065C22semidirect
71. #2055C22semidirect
72. #2055C22semidirect
75. #2042C22semidirect
76. #2042C22semidirect
79. #1880C22semidirect
80. #1880C22semidirect
83. #1865C22semidirect
84. #1864C22semidirect
85. #1863C22semidirect
86. #1862C22semidirect
87. #1832C22semidirect
88. #1831C22semidirect

Maximal subgroups

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

subgroupindex
#352662
#342112
#301422
#301412
#301342
#301332
#237442
#237312
#236152
#236142
#235412
#235402
#235062
C2×#20122
#209362
#209342
#208772
#208762
#207542