Finite Group Databasebeta

← #6434230

Subgroups — #6434230

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35313C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#30144C214yesmaximal, complemented
4#30143C214yesmaximal, complemented
5#30136C214yesmaximal, complemented
6#30135C214yesmaximal, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23732C211yesmaximal, characteristic, complemented
9#23615C211yesmaximal, characteristic, complemented
10#23614C211yesmaximal, characteristic
11#23542C211yesmaximal, characteristic, complemented
12#23541C211yesmaximal, characteristic
13#23506C211yesmaximal, characteristic
14C2×#2015C211yesmaximal, characteristic
15#20935C211yesmaximal, characteristic, complemented
16#20933C211yesmaximal, characteristic
17#20877C211yesmaximal, characteristic, complemented
18#20876C211yesmaximal, characteristic
19#20754C211yesmaximal, characteristic, complemented
20C23×Q16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#419C2211yescharacteristic
32C2×#419C2211yescharacteristic
33C2×#418C2211yescharacteristic
34C2×#418C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37#2863C2214yescomplemented
38#2645C2214yescomplemented
39#2514—22no—
40#2511—22no—
41#2493—24no—
42#2367—22no—
43#2367—22no—
44#2342—24no—
45#2326C2212yes—
46#2326C2212yes—
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2310C2211yescharacteristic
50#2310C2211yescharacteristic
51#2310C2211yescharacteristic
52#2309C2211yescharacteristic
53C4×#124C2212yes—
54C4×#124C2212yes—
55#2296C2211yescharacteristic
56C2×#391C2211yescharacteristic
57#2285C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2101C2212yescomplemented
60#2101C2212yescomplemented
61#2100C2212yescomplemented
62#2099C2212yescomplemented
63#2097C2212yescomplemented
64#2097C2212yescomplemented
65#2096C2212yescomplemented
66#2095C2212yescomplemented
67#2068C2214yescomplemented
68#2067C2214yes—
69#2066C2214yescomplemented
70#2065C2214yes—
71#2056C2214yescomplemented
72#2056C2214yescomplemented
73#2055C2214yes—
74#2055C2214yes—
75#2042C2214yes—
76#2042C2214yes—
77#2015C2214yes—
78#2015C2214yes—
79#1878C2214yescomplemented
80#1878C2214yescomplemented
81#1877C2214yes—
82#1877C2214yes—
83#1865C2214yescomplemented
84#1864C2214yes—
85#1863C2214yescomplemented
86#1862C2214yes—
87#1832C2214yescomplemented
88#1831C2214yescomplemented
89#1735C2211yescharacteristic
90#1735C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×Q16—24no—
101C22×Q16—42no—
102C22×Q16—42no—
103C22×Q16C2314yes—
104C22×SD16—24no—
105C22×SD16—42no—
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—42no—
118#519—42no—
119#518—42no—
120#518—42no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×#117—22no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C42C2311yescharacteristic
140#428—42no—
141#428—42no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146#424—42no—
147#424—42no—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2314yes—
150#419—24no—
151#419—42no—
152#419—42no—
153#419—42no—
154#418—24no—
155#418—42no—
156C2×#103—42no—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#102—42no—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168#401—24no—
169#400—24no—
170#400—24no—
171#400—24no—
172#391—22no—
173#391—21no—
174#391—22no—
175#391—21no—
176#391—22no—
177#391C2314yes—
178#391C2312yes—
179#391C2312yes—
180#391C2314yes—
181#391C2312yes—
182#391C2312yes—
183#390—22no—
184#390C2312yes—
185#390C2312yes—
186#389—22no—
187#389C2314yes—
188#389C2314yes—
189#389C2312yes—
190#389C2312yes—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#387—24no—
196#387—24no—
197#387—24no—
198#385—22no—
199#385—22no—
200#385—24no—
201#385—24no—
202#384—21no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#383—24no—
206#382—21no—
207#382C2311yescharacteristic
208#382C2311yescharacteristic
209#381—22no—
210#381—22no—
211#381—22no—
212#380—22no—
213#380—22no—
214#379—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4D412yes—
219C4×C22⋊C4—24no—
220C4×C22⋊C4—24no—
221C4×C22⋊C4D414yes—
222#376C2312yes—
223C2×#95—22no—
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—
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—22no—
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—44no—
273C2×Q16—44no—
274C2×Q16—416no—
275C2×SD16—28no—
276C2×SD16—44no—
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—42no—
301C4×Q8—42no—
302C4×Q8—42no—
303C4×Q8—42no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×D4—42no—
307C4×D4—42no—
308C4×D4—42no—
309C4×D4—42no—
310C4×D4—44no—
311C4×D4—44no—
312#117—22no—
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—48no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—216no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42C2×D412yes—
369C2×C42C2×D414yes—
370C2×C42C2×D414yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—42no—
376#107—42no—
377#106—42no—
378#106—42no—
379C4⋊C8—44no—
380C4⋊C8—44no—
381#103—28no—
382#103—44no—
383#103—44no—
384#103—42no—
385#103—42no—
386#103—42no—
387#103—42no—
388#102—28no—
389#102—44no—
390#102—44no—
391#95—24no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—24no—
396#95—24no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95C2×D414yes—
406#95C2×D414yes—
407C24—22no—
408C24—24no—
409C24—42no—
410C24—42no—
411C24—82no—
412C24—82no—
413C24C22×D411yescharacteristic
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8C2×#4114yes—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4C2×#4114yes—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4—82no—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+412yes—
489C22×C4C2511yescharacteristic
490Q16—416no—
491SD16—416no—
492M16—44no—
493M16—44no—
494C2×C8—22no—
495C2×C8—22no—
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—44no—
507C2×C8—44no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C22⋊C4—216no—
535C22⋊C4—216no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C42—24no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—44no—
556C42—44no—
557C42—44no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—42no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—48no—
571C23—88no—
572C23—162no—
573C23—162no—
574C23—164no—
575C23#52512yes—
576C23#53512yes—
577C23#53712yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—24no—
581Q8—48no—
582Q8—42no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—84no—
589D4—24no—
590D4—24no—
591D4—48no—
592D4—42no—
593D4—42no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—164no—
633C2×C4—164no—
634C2×C4—162no—
635C2×C4#52512yes—
636C2×C4#53512yes—
637C2×C4#53712yes—
638C2×C4#53814yes—
639C2×C4#53814yes—
640C2×C4#53914yes—
641C2×C4#53914yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×H(5,𝔽2)11yescharacteristic
645C2×C4C2×H(5,𝔽2)11yescharacteristic
646C2×C4C2×H(5,𝔽2)11yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C8—42no—
649C8—42no—
650C8—42no—
651C8—42no—
652C22—22no—
653C22—22no—
654C22—24no—
655C22—24no—
656C22—44no—
657C22—48no—
658C22—82no—
659C22—82no—
660C22—84no—
661C22—168no—
662C22#263614yes—
663C22C2×#52511yescharacteristic
664C22C2×#53511yescharacteristic
665C22C2×#53711yescharacteristic
666C22#358412yes—
667C22#358612yes—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—82no—
681C4—82no—
682C4#263614yes—
683C4C2×#53811yescharacteristic
684C4C2×#53811yescharacteristic
685C4C2×#53911yescharacteristic
686C4C2×#53911yescharacteristic
687C2—24no—
688C2—84no—
689C2C2×#263611yescharacteristic
690C2#6058112yes—
691C2#6081911yescharacteristic
692C2#6082611yescharacteristic
693C2#6092712yes—
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. #35313C2semidirect
3. #30144C2semidirect
4. #30143C2semidirect
5. #30136C2semidirect
6. #30135C2semidirect
7. #23744C2semidirect
8. #23732C2semidirect
9. #23615C2semidirect
11. #23542C2semidirect
15. #20935C2semidirect
17. #20877C2semidirect
19. #20754C2semidirect
37. #2863C22semidirect
38. #2645C22semidirect
59. #2101C22semidirect
60. #2101C22semidirect
61. #2100C22semidirect
62. #2099C22semidirect
63. #2097C22semidirect
64. #2097C22semidirect
65. #2096C22semidirect
66. #2095C22semidirect
67. #2068C22semidirect
69. #2066C22semidirect
71. #2056C22semidirect
72. #2056C22semidirect
79. #1878C22semidirect
80. #1878C22semidirect
83. #1865C22semidirect
85. #1863C22semidirect
87. #1832C22semidirect
88. #1831C22semidirect

Maximal subgroups

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

subgroupindex
#353132
#342112
#301442
#301432
#301362
#301352
#237442
#237322
#236152
#236142
#235422
#235412
#235062
C2×#20152
#209352
#209332
#208772
#208762
#207542