Finite Group Databasebeta

← #6573179

Subgroups — #6573179

Order 512.

1852 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36473C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic, complemented
3#30145C212yesmaximal, complemented
4#29687C211yesmaximal, characteristic, complemented
5#29493C212yesmaximal, complemented
6#28201C212yesmaximal, complemented
7#27950C211yesmaximal, characteristic, complemented
8#27644C211yesmaximal, characteristic, complemented
9#27094C214yesmaximal, complemented
10#26784C214yesmaximal, complemented
11#26040C212yesmaximal, complemented
12#24811C212yesmaximal, complemented
13#23835C212yesmaximal, complemented
14#23567C211yesmaximal, characteristic
15#23060C212yesmaximal, complemented
16#22807C212yesmaximal, complemented
17#22632C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#532C2211yescharacteristic
20C2×#520—21no—
21C2×#480C2212yes—
22C2×#474C2212yes—
23C2×#474C2214yes—
24C2×C4×D8C2212yes—
25#2860C2212yes—
26#2857C2211yescharacteristic
27#2604—22no—
28#2551C2211yescharacteristic
29#2542C2211yescharacteristic
30#2521—22no—
31#2496C2212yes—
32#2492—21no—
33#2491C2212yes—
34#2415—21no—
35#2365C2211yescharacteristic
36#2362—21no—
37#2361—22no—
38#2361C2214yes—
39#2354—21no—
40#2354C2212yes—
41#2354C2211yescharacteristic
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44#2326—22no—
45#2311—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#127C2212yes—
49C4×#121C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2086C2214yes—
53#2083C2212yes—
54#2083C2212yes—
55#2083C2214yes—
56#2083C2212yes—
57#2071C2212yes—
58#2067C2214yes—
59#2057C2211yescharacteristic
60#2057C2214yes—
61#2057C2212yes—
62#2041C2211yescharacteristic
63#2041C2212yes—
64#2033C2212yes—
65#2033C2214yes—
66#2023C2211yescharacteristic
67#2011C2211yescharacteristic
68#1999C2212yes—
69#1990C2212yes—
70#1970C2214yes—
71#1947C2212yes—
72#1939C2212yes—
73#1934C2214yes—
74#1925C2214yes—
75#1925C2214yes—
76#1925C2214yes—
77#1923C2214yes—
78#1918C2214yes—
79#1879C2214yes—
80#1862C2214yes—
81#1852C2214yes—
82#1845C2212yes—
83#1831C2212yes—
84#1823C2212yes—
85#1802C2212yes—
86#1802C2212yes—
87#1802C2214yes—
88#1797C2211yescharacteristic
89C4×#107C2214yes—
90C4×#107C2214yes—
91C8×C4⋊C4C2212yes—
92C4×#102C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1744C2212yes—
95C22×#41—21no—
96C22×D8C2312yes—
97#556—48no—
98#532—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#126—22no—
105C2×#126—22no—
106C2×#124—22no—
107C2×#124—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×#121—21no—
112C2×#121C2312yes—
113C2×#121C2311yescharacteristic
114#520—48no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—22no—
118C2×C4×D4—21no—
119C2×C4×D4—24no—
120C2×C4×D4—21no—
121C2×C4×D4—22no—
122C2×C4×D4—22no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2314yes—
126C2×#117—22no—
127C2×#117—21no—
128C2×#117D411yescharacteristic
129C2×#117D411yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135#480—44no—
136#474—28no—
137#474—48no—
138C4×D8—28no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#107C2312yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2314yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#400—22no—
158#399—22no—
159#397—22no—
160#395—22no—
161#394—22no—
162#393—22no—
163#391—21no—
164#391—22no—
165#391—22no—
166#391—21no—
167#390—22no—
168#390—22no—
169#390C2314yes—
170#390C2312yes—
171#390C2311yescharacteristic
172#390C2312yes—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389C2311yescharacteristic
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—21no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—24no—
187#388—24no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—22no—
192#386—21no—
193#386—24no—
194#385—24no—
195#385—24no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2312yes—
200#384C2311yescharacteristic
201#384C2312yes—
202#380—21no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380C2311yescharacteristic
207#380C2312yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376D412yes—
222C43C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2311yescharacteristic
234#336C2314yes—
235C2×#41—28no—
236C2×#41—44no—
237C2×#41—44no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×D8—24no—
255C2×D8—24no—
256C2×D8—42no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C4⋊Q8—44no—
265#127—24no—
266#126—48no—
267#126—48no—
268#124—28no—
269#124—48no—
270C22⋊Q8—44no—
271C22⋊Q8—44no—
272C22⋊Q8—44no—
273#121—28no—
274#121—24no—
275#121—24no—
276#121—22no—
277#121—41no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—48no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—416no—
288C4×D4—48no—
289C4×D4—84no—
290C4×D4#4118yes—
291#117—42no—
292#117—48no—
293#117—44no—
294#117D818yescomplemented
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—82no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2412yes—
359C2×C42C2411yescharacteristic
360#107—24no—
361#107—28no—
362#107—28no—
363#107—44no—
364#107—44no—
365C4⋊C8—24no—
366C4⋊C8—44no—
367#102—24no—
368#102—28no—
369#102—24no—
370#102—24no—
371#102—24no—
372#102—44no—
373#102—44no—
374C22⋊C8—24no—
375C22⋊C8—42no—
376C4×C8—28no—
377C4×C8—24no—
378#95—21no—
379#95—21no—
380#95—24no—
381#95—22no—
382#95—24no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24—21no—
395C24C22×D411yescharacteristic
396#41—432no—
397C2×Q8—24no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—81no—
427C2×D4C2×#4114yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—48no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4C2×D814yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C2511yescharacteristic
467D8—48no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C82−1+414yes—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4C2×D814yes—
506C4⋊C4C2×D814yes—
507C4⋊C4C2×#4114yes—
508C4⋊C4H(5,𝔽2)12yes—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—44no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—82no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4C2×D818yes—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—24no—
532C42—22no—
533C42—42no—
534C42—42no—
535C42—48no—
536C42—44no—
537C42—42no—
538C42—44no—
539C42C2×D818yes—
540C42C2×#4114yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—24no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23C22×D812yes—
562C23C23×D411yescharacteristic
563Q8—48no—
564Q8—88no—
565Q8—84no—
566D4—24no—
567D4—24no—
568D4—48no—
569D4—48no—
570D4—416no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—84no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—84no—
622C2×C4—168no—
623C2×C4—164no—
624C2×C4#52512yes—
625C2×C4#52512yes—
626C2×C4C22×D814yes—
627C2×C4C22×D812yes—
628C2×C4C22×D814yes—
629C2×C4C22×D812yes—
630C2×C4#57512yes—
631C2×C4#57812yes—
632C2×C4C22×#4112yes—
633C2×C4C2×H(5,𝔽2)11yescharacteristic
634C2×C4C2×2−1+412yes—
635C2×C4#58511yescharacteristic
636C8—24no—
637C8—42no—
638C8—44no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22C2×#52511yescharacteristic
661C22#348811yescharacteristic
662C22#349911yescharacteristic
663C22C23×D811yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57811yescharacteristic
666C22#358511yescharacteristic
667C4—24no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—41no—
676C4—42no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4#310014yes—
683C4#349512yes—
684C4#351112yes—
685C2—42no—
686C2—84no—
687C2—161no—
688C2#3608911yescharacteristic
689C2C2×#310011yescharacteristic
690C2#6058911yescharacteristic
691C2#6194711yescharacteristic
692C2#6199811yescharacteristic
693C2#6247111yescharacteristic
694C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23567C2
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. #36473C2semidirect
2. #34207C2semidirect
3. #30145C2semidirect
4. #29687C2semidirect
5. #29493C2semidirect
6. #28201C2semidirect
7. #27950C2semidirect
8. #27644C2semidirect
9. #27094C2semidirect
10. #26784C2semidirect
11. #26040C2semidirect
12. #24811C2semidirect
13. #23835C2semidirect
15. #23060C2semidirect
16. #22807C2semidirect
17. #22632C2semidirect
294. #117D8semidirect

Maximal subgroups

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

subgroupindex
#364732
#342072
#301452
#296872
#294932
#282012
#279502
#276442
#270942
#267842
#260402
#248112
#238352
#235672
#230602
#228072
#226322