Finite Group Databasebeta

← #6461493

Subgroups — #6461493

Order 512.

1968 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34254C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#29383C214yesmaximal, complemented
4#29382C214yesmaximal, complemented
5#29375C214yesmaximal, complemented
6#29374C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23732C211yesmaximal, characteristic, complemented
9#23506C211yesmaximal, characteristic
10C2×#2015C211yesmaximal, characteristic
11#21064C211yesmaximal, characteristic, complemented
12#21063C211yesmaximal, characteristic
13#20936C211yesmaximal, characteristic
14#20935C211yesmaximal, characteristic, complemented
15#20935C211yesmaximal, characteristic, complemented
16#20933C211yesmaximal, characteristic
17#20877C211yesmaximal, characteristic, complemented
18#20876C211yesmaximal, characteristic
19#20774C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#428C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#102C2211yescharacteristic
36#2367—22no—
37#2367—22no—
38#2356—22no—
39#2355—22no—
40#2343—24no—
41#2342—24no—
42#2326C2212yes—
43#2326C2212yes—
44#2320C2214yescomplemented
45#2317C2211yescharacteristic
46#2310C2211yescharacteristic
47#2310C2211yescharacteristic
48#2309C2211yescharacteristic
49C4×#124C2212yes—
50C4×#124C2212yes—
51#2296C2211yescharacteristic
52#2295C2214yescomplemented
53C2×#391C2211yescharacteristic
54#2285C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2101C2212yescomplemented
58#2101C2212yescomplemented
59#2100C2212yescomplemented
60#2099C2212yescomplemented
61#2097C2212yescomplemented
62#2097C2212yescomplemented
63#2096C2212yescomplemented
64#2095C2212yescomplemented
65#2042C2214yes—
66#2042C2214yes—
67#2015C2214yes—
68#2015C2214yes—
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1926C2214yes—
72#1925C2214yes—
73#1880C2214yes—
74#1880C2214yes—
75#1878C2214yescomplemented
76#1878C2214yescomplemented
77#1878C2214yescomplemented
78#1878C2214yescomplemented
79#1877C2214yes—
80#1877C2214yes—
81#1865C2214yescomplemented
82#1864C2214yes—
83#1863C2214yescomplemented
84#1862C2214yes—
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×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—24no—
102C22×Q16—42no—
103C22×Q16C2314yes—
104C22×SD16—24no—
105C22×SD16—24no—
106C22×SD16—42no—
107C22×SD16C2314yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C2×#124—24no—
113C2×#124—24no—
114C2×#124—22no—
115C2×#124C2312yes—
116C2×#124C2312yes—
117#519—24no—
118#519—24no—
119#519—42no—
120#519—42no—
121#518—42no—
122#518—42no—
123C2×C4×Q8—22no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2314yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×#117—22no—
134C2×#117—22no—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—22no—
143C22×C22⋊C4—22no—
144C22×C42C2311yescharacteristic
145#428—42no—
146#428—42no—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151#419—24no—
152#419—24no—
153#419—42no—
154#419—42no—
155#418—24no—
156#418—24no—
157C2×#103—42no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#102—42no—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×#102C2314yes—
169#391—21no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391C2314yes—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#390—22no—
179#390C2312yes—
180#390C2312yes—
181#389C2314yes—
182#389C2312yes—
183#389C2312yes—
184#388—24no—
185#388—24no—
186#388—24no—
187#388—24no—
188#387—24no—
189#387—24no—
190#387—24no—
191#387—24no—
192#385—22no—
193#385—22no—
194#385—24no—
195#385—24no—
196#384—21no—
197#384C2311yescharacteristic
198#384C2311yescharacteristic
199#383—24no—
200#382—21no—
201#382—24no—
202#382C2311yescharacteristic
203#382C2311yescharacteristic
204#381—22no—
205#381—22no—
206#380—22no—
207#379—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2314yes—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—24no—
217C4×C22⋊C4—24no—
218C4×C22⋊C4D414yes—
219#376—24no—
220#376C2312yes—
221C2×#95—22no—
222C43C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8#4114yes—
253C22×Q8#4114yes—
254C22×Q8C2412yes—
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4#4114yes—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×Q16—28no—
272C2×Q16—28no—
273C2×Q16—44no—
274C2×Q16—416no—
275C2×SD16—28no—
276C2×SD16—28no—
277C2×SD16—44no—
278C2×SD16—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#124—216no—
296#124—24no—
297#124—24no—
298#124—24no—
299#124—24no—
300C4×Q8—216no—
301C4×Q8—28no—
302C4×Q8—28no—
303C4×Q8—42no—
304C4×Q8—42no—
305C4×Q8—42no—
306C4×Q8—42no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×D4—42no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4—42no—
313C4×D4—44no—
314C4×D4—44no—
315#117—216no—
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#4114yes—
329C2×C4⋊C4—24no—
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—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
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—21no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—48no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42C2×D412yes—
375C2×C42C2×D414yes—
376C2×C42C2×D414yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—42no—
382#107—42no—
383#106—42no—
384#106—42no—
385#103—28no—
386#103—28no—
387#103—44no—
388#103—42no—
389#103—42no—
390#103—42no—
391#103—42no—
392#102—28no—
393#102—28no—
394#102—44no—
395#95—24no—
396#95—24no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—22no—
407#95C2×D414yes—
408#95C2×D414yes—
409C24—22no—
410C24—24no—
411C24—42no—
412C24—42no—
413C24—82no—
414C24—82no—
415C24C22×D411yescharacteristic
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—28no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8C2×#4114yes—
441C2×Q8C2×#4114yes—
442C2×D4—24no—
443C2×D4—28no—
444C2×D4—28no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4C2×#4114yes—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—82no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4C2511yescharacteristic
492Q16—416no—
493SD16—416no—
494M16—44no—
495M16—44no—
496C2×C8—22no—
497C2×C8—22no—
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—44no—
509C2×C8—44no—
510C4⋊C4—216no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
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—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—84no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4C2×#4114yes—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C42—216no—
549C42—24no—
550C42—24no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—42no—
564C42—42no—
565C42—42no—
566C42—84no—
567C23—22no—
568C23—22no—
569C23—24no—
570C23—44no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—48no—
575C23—88no—
576C23—162no—
577C23—162no—
578C23—164no—
579C23#52512yes—
580C23#52512yes—
581C23#53512yes—
582C23C23×D411yescharacteristic
583Q8—24no—
584Q8—24no—
585Q8—24no—
586Q8—24no—
587Q8—48no—
588Q8—48no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593D4—24no—
594D4—24no—
595D4—48no—
596D4—42no—
597D4—42no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—164no—
637C2×C4—164no—
638C2×C4—162no—
639C2×C4#52512yes—
640C2×C4#52512yes—
641C2×C4#53512yes—
642C2×C4#55114yes—
643C2×C4#55114yes—
644C2×C4#55314yes—
645C2×C4#55314yes—
646C2×C4C22×#4111yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C8—42no—
653C8—42no—
654C8—42no—
655C8—42no—
656C22—22no—
657C22—22no—
658C22—24no—
659C22—24no—
660C22—44no—
661C22—48no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22—168no—
666C22#259814yes—
667C22C2×#52511yescharacteristic
668C22C2×#52511yescharacteristic
669C22C2×#53511yescharacteristic
670C22#358412yes—
671C22#358612yes—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—81no—
681C4—82no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—82no—
686C4#259814yes—
687C4C2×#55111yescharacteristic
688C4C2×#55111yescharacteristic
689C4C2×#55311yescharacteristic
690C4C2×#55311yescharacteristic
691C2—24no—
692C2—84no—
693C2C2×#259811yescharacteristic
694C2#6058112yes—
695C2#6059512yes—
696C2#6081911yescharacteristic
697C2#6082611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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. #34254C2semidirect
3. #29383C2semidirect
4. #29382C2semidirect
5. #29375C2semidirect
6. #29374C2semidirect
7. #23744C2semidirect
8. #23732C2semidirect
11. #21064C2semidirect
14. #20935C2semidirect
15. #20935C2semidirect
17. #20877C2semidirect
19. #20774C2semidirect
44. #2320C22semidirect
52. #2295C22semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
61. #2097C22semidirect
62. #2097C22semidirect
63. #2096C22semidirect
64. #2095C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
75. #1878C22semidirect
76. #1878C22semidirect
77. #1878C22semidirect
78. #1878C22semidirect
81. #1865C22semidirect
83. #1863C22semidirect
85. #1840C22semidirect
86. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#342542
#342112
#293832
#293822
#293752
#293742
#237442
#237322
#235062
C2×#20152
#210642
#210632
#209362
#209352
#209332
#208772
#208762
#207742