Finite Group Databasebeta

← #6793189

Subgroups — #6793189

Order 512.

1595 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic, complemented
4#31492C214yesmaximal, complemented
5#30600C214yesmaximal, complemented
6#29923C212yesmaximal, complemented
7#29290C211yesmaximal, characteristic, complemented
8#28748C211yesmaximal, characteristic, complemented
9#28598C212yesmaximal
10#26532C212yesmaximal, complemented
11#25386C212yesmaximal
12#25324C212yesmaximal, complemented
13#24752C212yesmaximal, complemented
14#24155C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic
16#21885C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#487C2212yes—
20C2×#477C2211yescharacteristic
21C2×#467C2214yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×C8×D4C2212yes—
26C22×C4⋊C8C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2574—22no—
30#2567C2211yescharacteristic
31#2563C2211yescharacteristic
32#2552—21no—
33#2552C2211yescharacteristic
34#2542C2212yes—
35#2538—21no—
36#2538C2211yescharacteristic
37#2508—21no—
38#2497C2212yes—
39#2454—21no—
40#2441—22no—
41#2429C2214yes—
42#2423—22no—
43#2423C2212yes—
44#2396—22no—
45#2365C2212yes—
46#2354—21no—
47Q8×C4⋊C4—21no—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#125—21no—
51C4×C22⋊Q8C2212yes—
52#2293—21no—
53#2285C2212yes—
54C42×Q8C2211yescharacteristic
55C42×D4C2211yescharacteristic
56#2095C2214yes—
57#2087C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2056C2214yes—
61#2043C2214yes—
62#2036C2214yes—
63#2036C2214yes—
64#2036C2214yes—
65#2001C2214yes—
66#1987C2212yes—
67#1987C2211yescharacteristic
68#1987C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1956C2212yes—
75#1947C2212yes—
76#1947C2211yescharacteristic
77#1947C2212yes—
78#1939C2212yes—
79#1921C2212yes—
80#1892C2214yes—
81#1892C2214yes—
82#1892C2214yes—
83#1878C2214yes—
84#1863C2214yes—
85#1863C2212yes—
86#1850C2214yes—
87#1842C2211yescharacteristic
88#1803C2214yes—
89#1803C2211yescharacteristic
90#1803C2214yes—
91#1797C2212yes—
92C4×#107C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×Q16C2314yes—
96C23×C8C2314yes—
97#533—28no—
98#533—44no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—21no—
105C2×#125—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2314yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2314yes—
110C2×C4×Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2314yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4C2311yescharacteristic
120C2×#117—22no—
121C2×#117—22no—
122C22×C42C2312yes—
123#487—44no—
124#477—24no—
125#467—48no—
126#462—44no—
127#462—44no—
128#462—44no—
129C8×D4—44no—
130C2×#107C2312yes—
131C2×#107C2314yes—
132C2×#107C2312yes—
133C2×C4⋊C8—44no—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2314yes—
150C2×C4×C8C2312yes—
151#400—22no—
152#399—22no—
153#399—24no—
154#398—22no—
155#398—22no—
156#398—24no—
157#396—22no—
158#396—22no—
159#396—24no—
160#395—22no—
161#393—22no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2314yes—
166#390—21no—
167#390—22no—
168#390—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—24no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#388—24no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—24no—
189#385—22no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—21no—
200#382—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227C43—21no—
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8C2414yes—
241C22×Q8C2411yescharacteristic
242C22×D4C2×D411yescharacteristic
243C23×C4C2×D412yes—
244C23×C4C2×D412yes—
245C23×C4C2412yes—
246C2×Q16—28no—
247C2×Q16—44no—
248C2×Q16—44no—
249C22×C8—44no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264C4⋊Q8#4114yes—
265#125—48no—
266#125—44no—
267C22⋊Q8—24no—
268C22⋊Q8—28no—
269C22⋊Q8—42no—
270C22⋊Q8—44no—
271C22⋊Q8—48no—
272C22⋊Q8—44no—
273C22⋊Q8—44no—
274C4×Q8—22no—
275C4×Q8—28no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—41no—
281C4×Q8—44no—
282C4×Q8—48no—
283C4×Q8—44no—
284C4×Q8—82no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—42no—
288C4×D4Q1618yescomplemented
289#117—48no—
290#117—84no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
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—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—42no—
359#107—44no—
360C4⋊C8—22no—
361C4⋊C8—22no—
362C4⋊C8—24no—
363C4⋊C8—24no—
364C4⋊C8—44no—
365C4⋊C8—42no—
366#103—24no—
367#103—24no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375C22⋊C8—44no—
376C4×C8—24no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24C22×D412yes—
396C2×Q8—24no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—24no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—21no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—81no—
412C2×Q8—81no—
413C2×Q8H(5,𝔽2)12yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4C2×Q1614yes—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4C2×Q1618yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448Q16—88no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—88no—
459C4⋊C4—22no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—42no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4C2×Q1614yes—
501C4⋊C4C2×#4114yes—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C22⋊C4C2×Q1618yes—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—21no—
518C42—21no—
519C42—41no—
520C42—48no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—44no—
528C42—81no—
529C42C2×Q1614yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C42H(5,𝔽2)12yes—
533C422−1+412yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23C22×Q1614yes—
541C23C23×D411yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—48no—
545Q8—41no—
546Q8—41no—
547Q8—44no—
548Q8—44no—
549Q8—44no—
550Q8—44no—
551Q8—84no—
552D4—28no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—216no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4#53712yes—
604C2×C4#53712yes—
605C2×C4#54612yes—
606C2×C4#54612yes—
607C2×C4C22×Q1612yes—
608C2×C4C22×Q1614yes—
609C2×C4C22×Q1612yes—
610C2×C4C22×Q1612yes—
611C2×C4#57512yes—
612C2×C4#57712yes—
613C2×C4C22×#4111yescharacteristic
614C2×C4C2×H(5,𝔽2)11yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×2−1+411yescharacteristic
618C2×C4#58511yescharacteristic
619C8—42no—
620C8—42no—
621C8—44no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22C2×#53711yescharacteristic
630C22C2×#54611yescharacteristic
631C22#348811yescharacteristic
632C22C23×Q1611yescharacteristic
633C22C2×#57511yescharacteristic
634C22C2×#57711yescharacteristic
635C22#358611yescharacteristic
636C4—22no—
637C4—24no—
638C4—22no—
639C4—22no—
640C4—41no—
641C4—42no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4—81no—
650C4—82no—
651C4—84no—
652C4#324914yes—
653C4#329914yes—
654C4#351711yescharacteristic
655C4#352211yescharacteristic
656C4#354311yescharacteristic
657C4#355411yescharacteristic
658C2—44no—
659C2#3890911yescharacteristic
660C2C2×#324911yescharacteristic
661C2#6092311yescharacteristic
662C2C2×#329911yescharacteristic
663C2#6115911yescharacteristic
664C2#6195211yescharacteristic
665C2#6198711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33163C2semidirect
4. #31492C2semidirect
5. #30600C2semidirect
6. #29923C2semidirect
7. #29290C2semidirect
8. #28748C2semidirect
10. #26532C2semidirect
12. #25324C2semidirect
13. #24752C2semidirect
14. #24155C2semidirect
16. #21885C2semidirect
17. C4×#477C2semidirect
288. C4×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#361342
#349412
#331632
#314922
#306002
#299232
#292902
#287482
#285982
#265322
#253862
#253242
#247522
#241552
D4×C4⋊C82
#218852
C4×#4772