Finite Group Databasebeta

← #6805648

Subgroups — #6805648

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2Q8×#124C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#32457C214yesmaximal, complemented
5#32453C214yesmaximal, complemented
6#30213C212yesmaximal, complemented
7#29211C212yesmaximal, complemented
8#28726C211yesmaximal, characteristic, complemented
9#28621C212yesmaximal
10#28521C211yesmaximal, characteristic, complemented
11#24681C212yesmaximal, complemented
12#24677C212yesmaximal, complemented
13#24329C212yesmaximal, complemented
14#24326C212yesmaximal, complemented
15#22223C212yesmaximal, complemented
16C4×#496C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic, complemented
18C2×Q82C2211yescharacteristic
19C2×#556—21no—
20C2×D4×Q8C2211yescharacteristic
21C2×#496C2211yescharacteristic
22C2×#489C2212yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25C2×C4×SD16C2212yes—
26C2×C4×D8C2212yes—
27#2868C2211yescharacteristic
28#2862C2212yes—
29#2862C2211yescharacteristic
30#2858C2212yes—
31#2568C2211yescharacteristic
32#2568C2212yes—
33#2563C2211yescharacteristic
34#2563C2212yes—
35#2547—21no—
36#2545—22no—
37#2538—21no—
38#2538C2212yes—
39#2512—22no—
40#2512C2214yes—
41#2492—21no—
42#2492—22no—
43#2492—22no—
44#2492—21no—
45#2354—21no—
46#2354C2212yes—
47Q8×C22⋊C4—21no—
48Q8×C22⋊C4—21no—
49C4×C4⋊Q8C2212yes—
50C4×#127C2211yescharacteristic
51C4×#124C2211yescharacteristic
52C4×#124C2212yes—
53C42×Q8C2211yescharacteristic
54#2092C2214yes—
55#2073C2214yes—
56#2073C2214yes—
57#2072C2212yes—
58#2071C2212yes—
59#2058C2214yes—
60#2057C2214yes—
61#2055C2214yes—
62#2054C2214yes—
63#2023C2214yes—
64#2023C2214yes—
65#1993C2212yes—
66#1993C2212yes—
67#1993C2212yes—
68#1990C2212yes—
69#1990C2212yes—
70#1984C2212yes—
71#1973C2212yes—
72#1972C2212yes—
73#1961C2211yescharacteristic
74#1961C2212yes—
75#1959C2211yescharacteristic
76#1949C2214yes—
77#1949C2214yes—
78#1939C2212yes—
79#1928C2214yes—
80#1926C2214yes—
81#1917C2214yes—
82#1880C2214yes—
83#1879C2214yes—
84#1865C2214yes—
85#1862C2214yes—
86#1847C2211yescharacteristic
87#1847C2212yes—
88#1836C2214yes—
89#1803C2212yes—
90#1802C2212yes—
91#1767C2212yes—
92#1767C2211yescharacteristic
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C23×Q8—21no—
96C22×SD16C2312yes—
97C22×D8C2312yes—
98Q82—44no—
99#556—48no—
100D4×Q8—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#127C2311yescharacteristic
111C2×#126—24no—
112C2×#124—22no—
113C2×#124—22no—
114C2×#124C2311yescharacteristic
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—22no—
118C2×C4×Q8—21no—
119C2×C4×Q8—22no—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8D411yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2312yes—
130C2×#117—22no—
131#496—24no—
132#489—28no—
133#476—44no—
134#474—44no—
135C4×SD16—44no—
136C4×D8—44no—
137C2×#107C2312yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#97C2312yes—
152C2×#97C2312yes—
153C2×#97C2311yescharacteristic
154C2×#97C2311yescharacteristic
155C2×C4×C8C2312yes—
156#400—22no—
157#400—22no—
158#400—22no—
159#397—24no—
160#396—24no—
161#396—24no—
162#393—22no—
163#393—22no—
164#393—22no—
165#391—21no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391—22no—
170#391—21no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391—22no—
175#391—21no—
176#391—22no—
177#391—21no—
178#391C2312yes—
179#391C2311yescharacteristic
180#390—21no—
181#390—21no—
182#390—22no—
183#390—22no—
184#390C2312yes—
185#390C2312yes—
186#390C2311yescharacteristic
187#390C2314yes—
188#390C2314yes—
189#389—22no—
190#389C2314yes—
191#388—22no—
192#388—21no—
193#388—22no—
194#388—21no—
195#388—24no—
196#388—24no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2312yes—
202#382C2312yes—
203#382C2312yes—
204#382C2312yes—
205#382C2311yescharacteristic
206#382C2312yes—
207#380—21no—
208#380—22no—
209#380—21no—
210#380—22no—
211#380—22no—
212#380—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4—22no—
225C43C2311yescharacteristic
226C43C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236#336C2314yes—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—44no—
243C22×Q8—44no—
244C22×Q8—44no—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C22×D4C2411yescharacteristic
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—21no—
254C2×SD16—24no—
255C2×SD16—24no—
256C2×SD16—42no—
257C2×D8—24no—
258C2×D8—24no—
259C2×D8—42no—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—44no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8—48no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273#127—22no—
274#127—22no—
275#126—416no—
276#124—22no—
277#124—48no—
278#124—84no—
279#124#4114yes—
280C22⋊Q8—44no—
281C22⋊Q8—48no—
282C22⋊Q8—48no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—42no—
287C4×Q8—44no—
288C4×Q8—48no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—44no—
295C4×D4—44no—
296#117—48no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2414yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—42no—
355#107—44no—
356#106—42no—
357#106—44no—
358C4⋊C8—28no—
359C4⋊C8—24no—
360#103—28no—
361#103—42no—
362#102—28no—
363#102—42no—
364#102—42no—
365#102—42no—
366#102—44no—
367#102—44no—
368#97—24no—
369#97—24no—
370#97—22no—
371#97—22no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—24no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—21no—
393C24—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×Q8—82no—
413C2×Q8—82no—
414C2×Q8—816no—
415C2×Q8AGL(1,ℤ/8)14yes—
416C2×Q8C2×#4112yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×#4112yes—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C2511yescharacteristic
460SD16—48no—
461D8—48no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C82−1+414yes—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—42no—
507C22⋊C4—48no—
508C22⋊C4—82no—
509C22⋊C4—81no—
510C22⋊C4—82no—
511C22⋊C4—81no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—22no—
515C42—24no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—24no—
520C42—41no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—41no—
525C42—48no—
526C42—44no—
527C42—42no—
528C42C2×#4112yes—
529C42H(5,𝔽2)14yes—
530C422−1+412yes—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546Q8—28no—
547Q8—24no—
548Q8—42no—
549Q8—88no—
550Q8—88no—
551Q8—84no—
552Q8—162no—
553Q8—164no—
554D4—24no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—42no—
560D4—84no—
561D4—84no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—88no—
607C2×C4—88no—
608C2×C4—84no—
609C2×C4#53512yes—
610C2×C4#53512yes—
611C2×C4#54812yes—
612C2×C4#54812yes—
613C2×C4C2×AGL(1,ℤ/8)14yes—
614C2×C4C2×AGL(1,ℤ/8)12yes—
615C2×C4#57512yes—
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)12yes—
618C2×C4C2×2−1+412yes—
619C2×C4C2×2−1+411yescharacteristic
620C8—24no—
621C8—44no—
622C8—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22C2×#53511yescharacteristic
638C22C2×#54811yescharacteristic
639C22#349911yescharacteristic
640C22C22×AGL(1,ℤ/8)11yescharacteristic
641C22C2×#57511yescharacteristic
642C22#358511yescharacteristic
643C22#358611yescharacteristic
644C4—24no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—84no—
658C4—82no—
659C4—81no—
660C4#321314yes—
661C4#352012yes—
662C4#355012yes—
663C2—82no—
664C2—162no—
665C2#3916711yescharacteristic
666C2C2×#321311yescharacteristic
667C2#6082911yescharacteristic
668C2#6132111yescharacteristic
669C2#6132311yescharacteristic
670C2#6250811yescharacteristic
671C2#6250911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#496C22
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
1. #36594C2semidirect
2. Q8×#124C2semidirect
3. #33293C2semidirect
4. #32457C2semidirect
5. #32453C2semidirect
6. #30213C2semidirect
7. #29211C2semidirect
8. #28726C2semidirect
10. #28521C2semidirect
11. #24681C2semidirect
12. #24677C2semidirect
13. #24329C2semidirect
14. #24326C2semidirect
15. #22223C2semidirect
17. #21345C2semidirect

Maximal subgroups

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

subgroupindex
#365942
Q8×#1242
#332932
#324572
#324532
#302132
#292112
#287262
#286212
#285212
#246812
#246772
#243292
#243262
#222232
C4×#4962
#213452