Finite Group Databasebeta

← #6795013

Subgroups — #6795013

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32430C214yesmaximal, complemented
5#32346C214yesmaximal, complemented
6#29907C212yesmaximal
7#28530C212yesmaximal, complemented
8#27729C212yesmaximal, complemented
9#27583C212yesmaximal, complemented
10#25811C211yesmaximal, characteristic, complemented
11#25453C212yesmaximal, complemented
12#24680C212yesmaximal, complemented
13#24503C212yesmaximal
14#24299C211yesmaximal, characteristic
15#22599C211yesmaximal, characteristic, complemented
16#21981C212yesmaximal, complemented
17C4×#448C211yesmaximal, characteristic
18C2×#552—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#498C2212yes—
22C2×#475C2212yes—
23C2×#476C2212yes—
24C2×#448C2211yescharacteristic
25C2×C4×SD16C2212yes—
26C2×#431C2212yes—
27#2857C2211yescharacteristic
28#2854C2212yes—
29#2567—21no—
30#2567—21no—
31#2567—22no—
32#2567C2212yes—
33#2567C2214yes—
34#2567C2212yes—
35#2552C2211yescharacteristic
36#2542C2211yescharacteristic
37#2519—22no—
38#2512C2212yes—
39#2506C2212yes—
40#2498—22no—
41#2489—21no—
42#2417—21no—
43C2×#395—22no—
44#2365C2211yescharacteristic
45#2362—21no—
46#2354C2211yescharacteristic
47C4×#121C2211yescharacteristic
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51#2285C2212yes—
52C2×#384C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54#2100C2214yes—
55#2096C2214yes—
56#2094C2214yes—
57#2084C2214yes—
58#2072C2214yes—
59#2071C2214yes—
60#2066C2214yes—
61#2058C2212yes—
62#2058C2214yes—
63#2055C2214yes—
64#2023C2212yes—
65#1990C2212yes—
66#1987C2212yes—
67#1968C2214yes—
68#1948C2214yes—
69#1948C2214yes—
70#1939C2212yes—
71#1933C2214yes—
72#1927C2214yes—
73#1927C2214yes—
74#1916C2214yes—
75#1880C2212yes—
76#1874C2212yes—
77#1874C2211yescharacteristic
78#1865C2212yes—
79#1865C2214yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2212yes—
83#1853C2212yes—
84#1853C2211yescharacteristic
85#1844C2214yes—
86#1834C2214yes—
87#1817C2212yes—
88#1803C2212yes—
89#1797C2211yescharacteristic
90#1787C2212yes—
91#1787C2211yescharacteristic
92C8×C4⋊C4C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98#552—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—44no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#126—22no—
106C2×#125—21no—
107C2×#125—22no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×#121C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—44no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—21no—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#498—44no—
135#476—44no—
136#475—44no—
137#448—24no—
138C4×SD16—44no—
139#431—28no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2311yescharacteristic
157C2×C4×C8C2314yes—
158C2×C4×C8C2312yes—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#397—22no—
164#396—22no—
165#395—24no—
166#395—24no—
167#395—24no—
168#395—24no—
169#391—21no—
170#391—22no—
171#391—22no—
172#391—21no—
173#391C2314yes—
174#391C2314yes—
175#390C2312yes—
176#390C2312yes—
177#390C2311yescharacteristic
178#389—22no—
179#389—24no—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—22no—
184#389—21no—
185#389—21no—
186#389C2312yes—
187#389C2311yescharacteristic
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—21no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#386—22no—
198#386—21no—
199#384—22no—
200#384—22no—
201#384C2314yes—
202#384C2314yes—
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2312yes—
206#382—22no—
207#380—21no—
208#380—22no—
209#380—21no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C2×#95—22no—
224C2×#95—22no—
225C2×#95D412yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#336C2312yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—42no—
245C22×Q8—44no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—44no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272#126—48no—
273#125—48no—
274#125—44no—
275#123—48no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279#121—22no—
280#121—22no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×Q8—42no—
285C4×Q8—44no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—44no—
290#117—48no—
291#117—44no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4SD1618yescomplemented
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#106—42no—
362#106—44no—
363#106—44no—
364#106—44no—
365#106—44no—
366C4⋊C8—28no—
367C4⋊C8—24no—
368#103—28no—
369#103—24no—
370#103—24no—
371#103—42no—
372#103—44no—
373#102—42no—
374#102—44no—
375C22⋊C8—24no—
376C22⋊C8—22no—
377C22⋊C8#4114yes—
378C4×C8—28no—
379C4×C8—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24C22×D411yescharacteristic
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—42no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8—816no—
419C2×Q82−1+414yes—
420C2×D4—21no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4C2×SD1618yes—
460C22×C4C2×SD1614yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C42−1+412yes—
472C22×C4C2511yescharacteristic
473SD16—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8C2×#4114yes—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4—81no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4C2×SD1614yes—
517C4⋊C4C2×SD1614yes—
518C4⋊C4C2×SD1618yes—
519C4⋊C4#13714yes—
520C22⋊C4—42no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—81no—
530C42—22no—
531C42—22no—
532C42—24no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—44no—
540C42—44no—
541C42—44no—
542C42—44no—
543C42—42no—
544C42H(5,𝔽2)14yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23#54712yes—
557C23C22×SD1612yes—
558C23#57712yes—
559C23C23×D411yescharacteristic
560Q8—24no—
561Q8—24no—
562Q8—48no—
563Q8—42no—
564Q8—42no—
565Q8—88no—
566Q8—88no—
567Q8—84no—
568D4—42no—
569D4—42no—
570D4—84no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—216no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—84no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4#53512yes—
624C2×C4#53512yes—
625C2×C4#54712yes—
626C2×C4C22×SD1614yes—
627C2×C4C22×SD1614yes—
628C2×C4C22×SD1612yes—
629C2×C4C22×SD1612yes—
630C2×C4C2×#13714yes—
631C2×C4C2×#13712yes—
632C2×C4C22×#4111yescharacteristic
633C2×C4C2×H(5,𝔽2)12yes—
634C2×C4C2×2−1+412yes—
635C2×C4C2×2−1+411yescharacteristic
636C8—24no—
637C8—44no—
638C8—42no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22#333514yes—
653C22C2×#53511yescharacteristic
654C22C2×#54711yescharacteristic
655C22#348611yescharacteristic
656C22C23×SD1611yescharacteristic
657C22C22×#13711yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—24no—
662C4—24no—
663C4—24no—
664C4—22no—
665C4—22no—
666C4—44no—
667C4—42no—
668C4—41no—
669C4—81no—
670C4—81no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—82no—
675C4—84no—
676C4#320614yes—
677C4#352012yes—
678C4#353812yes—
679C2—24no—
680C2—161no—
681C2#3891711yescharacteristic
682C2C2×#320611yescharacteristic
683C2#6081211yescharacteristic
684C2C2×#333511yescharacteristic
685C2#6125411yescharacteristic
686C2#6182011yescharacteristic
687C2#6183811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2567C2semidirect
3. #32954C2semidirect
4. #32430C2semidirect
5. #32346C2semidirect
7. #28530C2semidirect
8. #27729C2semidirect
9. #27583C2semidirect
10. #25811C2semidirect
11. #25453C2semidirect
12. #24680C2semidirect
15. #22599C2semidirect
16. #21981C2semidirect
320. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#361862
C2×#25672
#329542
#324302
#323462
#299072
#285302
#277292
#275832
#258112
#254532
#246802
#245032
#242992
#225992
#219812
C4×#4482