Finite Group Databasebeta

← #6856738

Subgroups — #6856738

Order 512.

1565 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic, complemented
3#33163C211yesmaximal, characteristic, complemented
4#32106C214yesmaximal, complemented
5#30904C214yesmaximal, complemented
6#29797C212yesmaximal
7#29255C212yesmaximal, complemented
8#28754C212yesmaximal
9#28748C211yesmaximal, characteristic, complemented
10#28419C212yesmaximal, complemented
11#28280C211yesmaximal, characteristic
12#27776C212yesmaximal, complemented
13#25646C212yesmaximal, complemented
14#24189C212yesmaximal, complemented
15#22825C211yesmaximal, characteristic, complemented
16#22154C212yesmaximal, complemented
17C4×#460C211yesmaximal, characteristic
18C2×#556C2212yes—
19C2×#489C2214yes—
20C2×#487C2212yes—
21C2×#476C2212yes—
22C2×#462C2212yes—
23C2×#462C2212yes—
24C2×#462C2212yes—
25C2×#460C2211yescharacteristic
26C2×C4×SD16C2212yes—
27#2868C2211yescharacteristic
28#2858C2212yes—
29#2586—22no—
30#2582C2211yescharacteristic
31#2567C2211yescharacteristic
32#2567C2211yescharacteristic
33#2551C2211yescharacteristic
34#2547C2212yes—
35#2545C2214yes—
36#2538—21no—
37#2538—21no—
38#2512—22no—
39#2512—22no—
40#2507—21no—
41#2507C2212yes—
42#2451—21no—
43#2451C2212yes—
44#2399—22no—
45#2354—21no—
46#2354—21no—
47C4×C4⋊Q8—21no—
48C4×C4⋊Q8C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51C4×#127C2211yescharacteristic
52#2295—21no—
53#2295C2212yes—
54C42×Q8C2211yescharacteristic
55C42×Q8C2211yescharacteristic
56#2096C2214yes—
57#2065C2214yes—
58#2043C2214yes—
59#2042C2214yes—
60#2042C2214yes—
61#2042C2214yes—
62#2042C2214yes—
63#2024C2214yes—
64#2015C2214yes—
65#2015C2214yes—
66#1984C2212yes—
67#1982C2211yescharacteristic
68#1982C2212yes—
69#1981C2212yes—
70#1981C2212yes—
71#1981C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1971C2214yes—
75#1959C2214yes—
76#1956C2212yes—
77#1948C2211yescharacteristic
78#1948C2212yes—
79#1935C2214yes—
80#1926C2212yes—
81#1917C2214yes—
82#1880C2212yes—
83#1878C2214yes—
84#1865C2214yes—
85#1865C2214yes—
86#1865C2214yes—
87#1842C2211yescharacteristic
88#1831C2212yes—
89#1803C2211yescharacteristic
90#1802C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97#556—44no—
98#556—44no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—24no—
107C2×#124C2314yes—
108C2×#124C2314yes—
109C2×C22⋊Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2312yes—
122C2×#117—22no—
123#489—48no—
124#487—44no—
125#476—44no—
126#462—28no—
127#462—44no—
128#462—44no—
129#460—24no—
130C4×SD16—44no—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2312yes—
147C2×#102C2312yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#97C2314yes—
152C2×C4×C8C2312yes—
153#398—22no—
154#398—22no—
155#398—22no—
156#398—22no—
157#397—24no—
158#396—24no—
159#396—24no—
160#395—22no—
161#395—22no—
162#395—22no—
163#395—22no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391C2314yes—
170#391C2312yes—
171#391C2312yes—
172#390—21no—
173#390—21no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—21no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—24no—
194#388—24no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384—21no—
203#384—21no—
204#384C2311yescharacteristic
205#384C2312yes—
206#384C2311yescharacteristic
207#380—22no—
208#380—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227#376—22no—
228C43—21no—
229C43C2311yescharacteristic
230C43C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#336C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2312yes—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8C2×D411yescharacteristic
241C22×Q8C2414yes—
242C22×Q8C2411yescharacteristic
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2412yes—
246C23×C4—22no—
247C2×Q16—28no—
248C2×Q16—44no—
249C2×Q16—44no—
250C2×SD16—24no—
251C2×SD16—42no—
252C2×SD16—42no—
253C22×C8C2412yes—
254C22×C8C2414yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262C4⋊Q8—24no—
263C4⋊Q8—42no—
264C4⋊Q8—42no—
265C4⋊Q8—42no—
266C4⋊Q8—44no—
267#127—22no—
268#127—22no—
269#126—416no—
270#124—44no—
271#124—44no—
272#124—44no—
273#124—44no—
274C22⋊Q8—48no—
275C4×Q8—24no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—22no—
282C4×Q8—42no—
283C4×Q8—41no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8—84no—
287C4×Q8—82no—
288C4×Q8#4114yes—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293#117—48no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
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⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C42—21no—
337C2×C42—24no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#106—42no—
358#106—44no—
359C4⋊C8—22no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—22no—
363C4⋊C8—42no—
364#103—28no—
365#103—28no—
366#103—44no—
367#103—42no—
368#103—44no—
369#103—44no—
370#103—44no—
371#103—44no—
372#102—24no—
373#102—24no—
374#102—42no—
375#102—44no—
376#102—44no—
377#97—44no—
378C4×C8—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—24no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—24no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—81no—
416C2×Q8—82no—
417C2×Q8—81no—
418C2×Q8C2×#4112yes—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2511yescharacteristic
453Q16—88no—
454SD16—48no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—81no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C2×#4112yes—
504C4⋊C4H(5,𝔽2)12yes—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—24no—
517C42—22no—
518C42—21no—
519C42—22no—
520C42—22no—
521C42—21no—
522C42—22no—
523C42—41no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—48no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—44no—
532C42—81no—
533C42#13714yes—
534C42C2×#4112yes—
535C42H(5,𝔽2)12yes—
536C42H(5,𝔽2)12yes—
537C422−1+412yes—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—44no—
549Q8—41no—
550Q8—41no—
551Q8—44no—
552Q8—44no—
553Q8—44no—
554Q8—44no—
555Q8—84no—
556Q8—88no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—42no—
561D4—84no—
562D4—84no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—88no—
609C2×C4#53512yes—
610C2×C4#53512yes—
611C2×C4#54812yes—
612C2×C4#54812yes—
613C2×C4C2×#13712yes—
614C2×C4C2×#13712yes—
615C2×C4C2×#13712yes—
616C2×C4#57512yes—
617C2×C4#57712yes—
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×H(5,𝔽2)11yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×2−1+411yescharacteristic
623C2×C4#58511yescharacteristic
624C8—42no—
625C8—44no—
626C8—42no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C2×#53511yescharacteristic
635C22C2×#54811yescharacteristic
636C22#348811yescharacteristic
637C22C22×#13711yescharacteristic
638C22C2×#57511yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—84no—
654C4—82no—
655C4—81no—
656C4—82no—
657C4#321614yes—
658C4#336914yes—
659C4#350611yescharacteristic
660C4#350811yescharacteristic
661C4#354811yescharacteristic
662C4#355411yescharacteristic
663C2—162no—
664C2#3972511yescharacteristic
665C2C2×#321611yescharacteristic
666C2#6083611yescharacteristic
667C2C2×#336911yescharacteristic
668C2#6132311yescharacteristic
669C2#6196611yescharacteristic
670C2#6199611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34950C2semidirect
3. #33163C2semidirect
4. #32106C2semidirect
5. #30904C2semidirect
7. #29255C2semidirect
9. #28748C2semidirect
10. #28419C2semidirect
12. #27776C2semidirect
13. #25646C2semidirect
14. #24189C2semidirect
15. #22825C2semidirect
16. #22154C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#349502
#331632
#321062
#309042
#297972
#292552
#287542
#287482
#284192
#282802
#277762
#256462
#241892
#228252
#221542
C4×#4602