Finite Group Databasebeta

← #6815474

Subgroups — #6815474

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3374C211yesmaximal, characteristic, complemented
2C2×#3299C211yesmaximal, characteristic
3#39616C211yesmaximal, characteristic
4#39173C211yesmaximal, characteristic
5#33236C211yesmaximal, characteristic, complemented
6#31146C214yesmaximal, complemented
7#31006C214yesmaximal, complemented
8#29867C211yesmaximal, characteristic, complemented
9#29799C211yesmaximal, characteristic, complemented
10#28647C211yesmaximal, characteristic, complemented
11#28600C211yesmaximal, characteristic
12#27268C214yesmaximal, complemented
13#26981C214yesmaximal, complemented
14#25931C211yesmaximal, characteristic, complemented
15#25884C211yesmaximal, characteristic
16#25651C211yesmaximal, characteristic, complemented
17#25573C211yesmaximal, characteristic
18#21633C211yesmaximal, characteristic
19#21278C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3374—28no—
25#3299—28no—
26C2×#491C2212yes—
27C2×#487C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#484C2212yes—
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#475C2211yescharacteristic
35C2×#467C2212yes—
36C2×#462C2212yes—
37C2×#462C2211yescharacteristic
38C2×#462C2211yescharacteristic
39C2×#451C2212yes—
40C2×#446C2211yescharacteristic
41C2×#441C2212yes—
42C2×C4×Q16C2211yescharacteristic
43C2×C8×D4C2211yescharacteristic
44C22×#103C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2720—24no—
48#2690—24no—
49#2660—21no—
50#2642—21no—
51#2568C2211yescharacteristic
52#2567—21no—
53#2567C2211yescharacteristic
54#2566—21no—
55#2565—21no—
56#2547—21no—
57#2547C2211yescharacteristic
58#2542C2211yescharacteristic
59#2442C2214yes—
60#2434C2214yes—
61#2429C2214yes—
62#2421C2214yes—
63#2326—21no—
64#2326—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2295C2211yescharacteristic
68#2085C2214yes—
69#2074C2214yes—
70#2066C2214yes—
71#2043C2214yes—
72#2043C2214yes—
73#2035C2214yes—
74#2035C2214yes—
75#2026C2214yes—
76#2016C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1984C2211yescharacteristic
81#1981C2211yescharacteristic
82#1981C2211yescharacteristic
83#1971C2212yes—
84#1960C2214yes—
85#1960C2214yes—
86#1954C2211yescharacteristic
87#1943C2212yes—
88#1935C2212yes—
89#1935C2212yes—
90#1935C2214yes—
91#1924C2214yes—
92#1892C2214yes—
93#1878C2214yes—
94#1872C2214yes—
95#1863C2214yes—
96#1831C2211yescharacteristic
97#1787C2214yes—
98C4×#106C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#103C2211yescharacteristic
101#1752C2214yes—
102C4×#97C2211yescharacteristic
103C23×Q8C2312yes—
104C22×Q16C2312yes—
105C22×Q16C2312yes—
106C22×Q16C2311yescharacteristic
107C23×C8C2312yes—
108#557—24no—
109#557—24no—
110#557—42no—
111#557—42no—
112#554—24no—
113#554—42no—
114D4×Q8—24no—
115D4×Q8—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2314yes—
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×#125—21no—
125C2×#125—21no—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#124—21no—
130C2×#124D411yescharacteristic
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8—22no—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×C22⋊Q8C2314yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140C2×C4×Q8C2311yescharacteristic
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×D4—21no—
144C2×C4×D4—21no—
145C2×C4×D4C2311yescharacteristic
146C22×C4⋊C4C2312yes—
147#491—28no—
148#487—24no—
149#487—24no—
150#487—42no—
151#487—42no—
152#484—28no—
153#479—28no—
154#477—24no—
155#475—24no—
156#467—28no—
157#462—28no—
158#462—24no—
159#462—24no—
160#451—28no—
161#446—24no—
162#441—28no—
163C4×Q16D418yes—
164C8×D4—24no—
165C2×#107C2312yes—
166C2×#107C2311yescharacteristic
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2312yes—
172C2×#103—28no—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2311yescharacteristic
188C2×#103C2311yescharacteristic
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2314yes—
191C2×#97C2312yes—
192C2×#97C2312yes—
193C2×#97C2312yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2311yescharacteristic
196#400—22no—
197#400—24no—
198#400—24no—
199#399—22no—
200#398—24no—
201#398—24no—
202#397—24no—
203#397—22no—
204#396—24no—
205#395—22no—
206#395—24no—
207#395—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391D411yescharacteristic
215#391C2314yes—
216#391C2312yes—
217#391C2312yes—
218#390—21no—
219#390D411yescharacteristic
220#389—24no—
221#389C2314yes—
222#389C2312yes—
223#389C2312yes—
224#388—24no—
225#388—24no—
226#388D414yes—
227#387—22no—
228#387—22no—
229#387—24no—
230#385—24no—
231#385—24no—
232#385—24no—
233#384—21no—
234#384—21no—
235#384—21no—
236#384—21no—
237#384—22no—
238#382—21no—
239#382D411yescharacteristic
240#382C2314yes—
241#380—24no—
242#380C2314yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C22⋊C4C2314yes—
248#376—21no—
249#376—21no—
250C43C2311yescharacteristic
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—24no—
262C22×Q8—42no—
263C22×Q8—44no—
264C22×Q8—44no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2412yes—
267C22×Q8C2412yes—
268C22×Q8C2411yescharacteristic
269C22×Q8C2411yescharacteristic
270C22×D4C2×D411yescharacteristic
271C23×C4—22no—
272C23×C4—22no—
273C23×C4C2412yes—
274C2×Q16—24no—
275C2×Q16—24no—
276C2×Q16—42no—
277C2×Q16—44no—
278C2×Q16—44no—
279C2×Q16C2×D414yes—
280C22×C8—24no—
281C22×C8—44no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C22×C8C2411yescharacteristic
287C4⋊Q8—24no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—44no—
297C4⋊Q8—42no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301#125—24no—
302#125—42no—
303#125—44no—
304#125—44no—
305#125—48no—
306#125—48no—
307#124—24no—
308#124—44no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—42no—
314C22⋊Q8—42no—
315C22⋊Q8—44no—
316C22⋊Q8—44no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—24no—
322C4×Q8—24no—
323C4×Q8—42no—
324C4×Q8—42no—
325C4×Q8—44no—
326C4×Q8—44no—
327C4×Q8C2×D414yes—
328C4×Q8C2×D414yes—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—44no—
332C4×D4—44no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—42no—
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D412yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386#107—24no—
387#107C2×D414yes—
388#106—24no—
389#106—42no—
390C4⋊C8—24no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393#103—24no—
394#103—24no—
395#103—24no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—42no—
402#103—44no—
403#103—44no—
404#103—42no—
405#103—44no—
406#103—44no—
407#103C2×D414yes—
408#103C2×D414yes—
409C22⋊C8—24no—
410C22⋊C8—44no—
411#97—24no—
412#97—42no—
413#97—42no—
414C4×C8—22no—
415C4×C8C2×D414yes—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—21no—
423#95—21no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95C2×D414yes—
431#95C2×D414yes—
432C24C22×D412yes—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—24no—
449C2×Q8—24no—
450C2×Q8—44no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—48no—
456C2×Q8—84no—
457C2×Q8—82no—
458C2×Q8—84no—
459C2×Q8—88no—
460C2×Q8C22×D412yes—
461C2×Q8C22×D412yes—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4#13714yes—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C2511yescharacteristic
491Q16—44no—
492Q16—48no—
493Q16—84no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—84no—
504C2×C8C22×D412yes—
505C2×C8C22×D412yes—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—82no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4C22×D412yes—
543C4⋊C4C22×D412yes—
544C22⋊C4—28no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C42—24no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—42no—
560C42—44no—
561C42—44no—
562C42—42no—
563C42—42no—
564C42C22×D412yes—
565C42H(5,𝔽2)12yes—
566C23—24no—
567C23—22no—
568C23—24no—
569C23—22no—
570C23C2×#13714yes—
571C23C23×D411yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—22no—
576Q8—44no—
577Q8—42no—
578Q8—42no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—44no—
584Q8—44no—
585Q8—88no—
586Q8—84no—
587Q8—162no—
588D4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—164no—
631C2×C4D4212yes—
632C2×C4D4212yes—
633C2×C4C2×#13712yes—
634C2×C4#57812yes—
635C2×C4C23×D411yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4#58512yes—
638C2×C4#58512yes—
639C8—22no—
640C8—22no—
641C8—42no—
642C8—42no—
643C8—42no—
644C22—24no—
645C22—28no—
646C22—44no—
647C22—42no—
648C22—42no—
649C22C2×D4211yescharacteristic
650C22#350012yes—
651C22C22×#13711yescharacteristic
652C22C2×#57811yescharacteristic
653C22#358612yes—
654C4—22no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4#327014yes—
671C4#344911yescharacteristic
672C4#350011yescharacteristic
673C4#356212yes—
674C2—44no—
675C2#3935711yescharacteristic
676C2C2×#327011yescharacteristic
677C2#6108311yescharacteristic
678C2#6257712yes—
679C2#6261112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3374C2semidirect
5. #33236C2semidirect
6. #31146C2semidirect
7. #31006C2semidirect
8. #29867C2semidirect
9. #29799C2semidirect
10. #28647C2semidirect
12. #27268C2semidirect
13. #26981C2semidirect
14. #25931C2semidirect
16. #25651C2semidirect
19. #21278C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33742
C2×#32992
#396162
#391732
#332362
#311462
#310062
#298672
#297992
#286472
#286002
#272682
#269812
#259312
#258842
#256512
#255732
#216332
#212782