Finite Group Databasebeta

← #6686265

Subgroups — #6686265

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3365C211yesmaximal, characteristic, complemented
2C2×#3338C211yesmaximal, characteristic, complemented
3#39173C211yesmaximal, characteristic, complemented
4#36496C211yesmaximal, characteristic
5#33202C211yesmaximal, characteristic, complemented
6#30643C214yesmaximal, complemented
7#30598C214yesmaximal, complemented
8#30278C211yesmaximal, characteristic, complemented
9#29827C211yesmaximal, characteristic, complemented
10#28676C211yesmaximal, characteristic, complemented
11#28597C211yesmaximal, characteristic
12#26555C214yesmaximal, complemented
13#26350C214yesmaximal, complemented
14#25836C211yesmaximal, characteristic, complemented
15#25702C211yesmaximal, characteristic, complemented
16#25648C211yesmaximal, characteristic, complemented
17#25569C211yesmaximal, characteristic
18#22208C211yesmaximal, characteristic, complemented
19#21119C211yesmaximal, characteristic, complemented
20C2×#559C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3365—28no—
25#3338—28no—
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#488C2212yes—
29C2×#486C2211yescharacteristic
30C2×#486C2211yescharacteristic
31C2×#484C2212yes—
32C2×#484C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#478C2212yes—
37C2×#467C2212yes—
38C2×#464C2212yes—
39C2×#460C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2211yescharacteristic
42C2×C4×SD16C2212yes—
43C2×C8×D4C2211yescharacteristic
44C22×#107C2212yes—
45#2862C2211yescharacteristic
46#2861—21no—
47#2858C2211yescharacteristic
48#2690—24no—
49#2642—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2566—21no—
53#2565—21no—
54#2547—21no—
55#2547C2211yescharacteristic
56#2544—21no—
57#2541C2211yescharacteristic
58#2461—24no—
59#2442C2214yes—
60#2426C2214yes—
61#2421C2214yes—
62#2361C2214yes—
63#2326—21no—
64C4×#127—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2092C2214yes—
69#2092C2214yes—
70#2087C2214yes—
71#2067C2214yes—
72#2059C2214yes—
73#2055C2214yes—
74#2042C2214yes—
75#2035C2214yes—
76#2033C2214yes—
77#2000C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1982C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1942C2214yes—
86#1928C2212yes—
87#1923C2214yes—
88#1921C2212yes—
89#1893C2214yes—
90#1878C2212yes—
91#1874C2214yes—
92#1865C2214yes—
93#1863C2212yes—
94#1862C2214yes—
95#1853C2214yes—
96#1850C2214yes—
97#1842C2211yescharacteristic
98#1805C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#103C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2311yescharacteristic
104C22×SD16C2312yes—
105C23×C8C2312yes—
106#559—24no—
107#559—42no—
108#557—24no—
109#557—42no—
110#554—24no—
111#554—24no—
112#554—42no—
113#554—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×#127—21no—
119C2×#127C2311yescharacteristic
120C2×#125—21no—
121C2×#125—21no—
122C2×#125—22no—
123C2×#125C2312yes—
124C2×#124—21no—
125C2×#124—21no—
126C2×#124—22no—
127C2×#124D411yescharacteristic
128C2×#124C2312yes—
129C2×#123—22no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8C2312yes—
133C2×C22⋊Q8C2312yes—
134C2×#121C2314yes—
135C2×C4×Q8—21no—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4—21no—
142C2×C4×D4—21no—
143C2×C4×D4—21no—
144C2×C4×D4—24no—
145C2×C4×D4C2312yes—
146C2×C4×D4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149#500—24no—
150#498—42no—
151#488—28no—
152#486—24no—
153#486—42no—
154#484—28no—
155#484—28no—
156#478—28no—
157#477—24no—
158#477—24no—
159#475—24no—
160#467—28no—
161#464—28no—
162#460—24no—
163C8×Q8—24no—
164C4×Q16D418yescomplemented
165C4×SD16—28no—
166C8×D4—24no—
167C2×#107—28no—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2311yescharacteristic
174C2×#106C2312yes—
175C2×#106C2314yes—
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2311yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2314yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2311yescharacteristic
186C2×#103C2311yescharacteristic
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2314yes—
191C2×#102C2314yes—
192C2×C22⋊C8C2312yes—
193C2×C22⋊C8C2314yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2312yes—
196C2×C4×C8C2311yescharacteristic
197#400—24no—
198#400—24no—
199#398—24no—
200#397—24no—
201#397—22no—
202#396—24no—
203#395—22no—
204#395—22no—
205#394—24no—
206#392—22no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—22no—
213#391D411yescharacteristic
214#391C2312yes—
215#390—21no—
216#390—24no—
217#390—21no—
218#390—21no—
219#390D411yescharacteristic
220#390C2314yes—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#388—24no—
225#388—24no—
226#388—24no—
227#388D414yes—
228#387—22no—
229#387—24no—
230#386—24no—
231#385—24no—
232#385—24no—
233#384—21no—
234#384—21no—
235#382D411yescharacteristic
236#382C2314yes—
237#380—24no—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C22⋊C4C2314yes—
246#376—21no—
247C43—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2411yescharacteristic
263C22×Q8C2411yescharacteristic
264C22×D4—24no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2×D411yescharacteristic
268C22×D4C2412yes—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4C2412yes—
273C2×Q16—42no—
274C2×Q16C2×D414yes—
275C2×SD16—24no—
276C2×SD16—44no—
277C22×C8—24no—
278C22×C8—44no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C22×C8C2411yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—44no—
291#127—22no—
292#127—22no—
293#127—44no—
294#125—24no—
295#125—42no—
296#125—44no—
297#125—44no—
298#125—48no—
299#124—24no—
300#124—24no—
301#124—42no—
302#124—44no—
303#124—44no—
304#124—84no—
305#123—48no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—48no—
309C22⋊Q8—48no—
310C22⋊Q8—42no—
311C22⋊Q8—42no—
312#121—28no—
313#121—44no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—22no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—44no—
323C4×Q8C2×D414yes—
324C4×Q8C2×D414yes—
325C4×D4—24no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—42no—
329C4×D4—44no—
330C4×D4—44no—
331C4×D4—44no—
332C4×D4—88no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D414yes—
374C2×C22⋊C4C2×D414yes—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388#107—24no—
389#107—24no—
390#107—24no—
391#107—24no—
392#107—24no—
393#107C2×D414yes—
394#106—24no—
395#106—44no—
396C4⋊C8—24no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—44no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103—24no—
404#103—42no—
405#103C2×D414yes—
406#103C2×D414yes—
407#102—24no—
408#102—24no—
409#102—24no—
410#102—44no—
411#102—44no—
412C22⋊C8—24no—
413C22⋊C8—44no—
414C4×C8—24no—
415C4×C8—22no—
416C4×C8C2×D414yes—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—21no—
421#95—22no—
422#95—21no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95C2×D414yes—
429#95C2×D414yes—
430C24—22no—
431C24C22×D412yes—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8C22×D412yes—
449C2×Q8C22×D412yes—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—48no—
457C2×D4—48no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—82no—
463C2×D4—84no—
464C2×D4#13514yes—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C2511yescharacteristic
492Q16—44no—
493SD16—48no—
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—48no—
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—22no—
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—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4C22×D412yes—
537C4⋊C4C22×D412yes—
538C4⋊C42−1+414yes—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—48no—
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—44no—
563C42—42no—
564C42—42no—
565C42C22×D412yes—
566C42H(5,𝔽2)12yes—
567C23—24no—
568C23—22no—
569C23—24no—
570C23—22no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23C2×#13514yes—
577C23C23×D411yescharacteristic
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—44no—
583Q8—42no—
584Q8—42no—
585Q8—84no—
586Q8—162no—
587Q8—162no—
588D4—28no—
589D4—42no—
590D4—42no—
591D4—44no—
592D4—816no—
593D4—84no—
594D4—162no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4—164no—
636C2×C4D4212yes—
637C2×C4D4212yes—
638C2×C4C2×#13512yes—
639C2×C4#57812yes—
640C2×C4C23×D411yescharacteristic
641C2×C4C2×H(5,𝔽2)11yescharacteristic
642C2×C4C2×2−1+412yes—
643C2×C4#58512yes—
644C8—22no—
645C8—22no—
646C8—42no—
647C8—42no—
648C8—42no—
649C22—24no—
650C22—28no—
651C22—42no—
652C22—42no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—82no—
658C22—84no—
659C22C2×D4211yescharacteristic
660C22#348912yes—
661C22C22×#13511yescharacteristic
662C22C2×#57811yescharacteristic
663C22#358512yes—
664C4—22no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4#328514yes—
680C4#344911yescharacteristic
681C4#349911yescharacteristic
682C4#355412yes—
683C2—44no—
684C2—162no—
685C2#3768011yescharacteristic
686C2C2×#328511yescharacteristic
687C2#6108211yescharacteristic
688C2#6210312yes—
689C2#6219312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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×#3365C2semidirect
2. C2×#3338C2semidirect
3. #39173C2semidirect
5. #33202C2semidirect
6. #30643C2semidirect
7. #30598C2semidirect
8. #30278C2semidirect
9. #29827C2semidirect
10. #28676C2semidirect
12. #26555C2semidirect
13. #26350C2semidirect
14. #25836C2semidirect
15. #25702C2semidirect
16. #25648C2semidirect
18. #22208C2semidirect
19. #21119C2semidirect
164. C4×Q16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33652
C2×#33382
#391732
#364962
#332022
#306432
#305982
#302782
#298272
#286762
#285972
#265552
#263502
#258362
#257022
#256482
#255692
#222082
#211192