Finite Group Databasebeta

← #6616133

Subgroups — #6616133

Order 512.

1708 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#36115C211yesmaximal, characteristic
3#33238C211yesmaximal, characteristic, complemented
4#33051C212yesmaximal, complemented
5#29225C211yesmaximal, characteristic, complemented
6#29054C214yesmaximal, complemented
7#29048C214yesmaximal, complemented
8#28754C211yesmaximal, characteristic, complemented
9#28600C212yesmaximal, complemented
10#28597C212yesmaximal
11#27894C212yesmaximal, complemented
12#25679C212yesmaximal, complemented
13#23222C212yesmaximal, complemented
14#23220C212yesmaximal, complemented
15C4×#501C211yesmaximal, characteristic, complemented
16#21406C212yesmaximal, complemented
17D4×#97C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#529C2212yes—
20C2×#501C2211yescharacteristic
21C2×#489C2212yes—
22C2×#487C2212yes—
23C2×#486C2212yes—
24C2×C8×D4C2212yes—
25C22×#97C2212yes—
26#2868C2211yescharacteristic
27#2858C2211yescharacteristic
28#2857C2212yes—
29#2850C2212yes—
30#2568—21no—
31#2547—21no—
32#2547C2212yes—
33#2542C2211yescharacteristic
34#2541C2211yescharacteristic
35#2512—21no—
36#2499—22no—
37#2496—22no—
38#2496C2214yes—
39#2471—22no—
40#2467—22no—
41#2445—21no—
42#2427C2212yes—
43#2426C2212yes—
44#2366—21no—
45#2366C2211yescharacteristic
46#2365—21no—
47#2365C2212yes—
48#2365C2211yescharacteristic
49#2354—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#125C2211yescharacteristic
52#2285C2212yes—
53C42×D4C2211yescharacteristic
54#2100C2212yes—
55#2100C2212yes—
56#2042C2214yes—
57#2035C2214yes—
58#2035C2214yes—
59#2033C2214yes—
60#2033C2214yes—
61#1993C2211yescharacteristic
62#1992C2212yes—
63#1992C2211yescharacteristic
64#1988C2212yes—
65#1988C2212yes—
66#1982C2212yes—
67#1981C2212yes—
68#1959C2212yes—
69#1959C2212yes—
70#1959C2212yes—
71#1959C2214yes—
72#1944C2212yes—
73#1943C2212yes—
74#1936C2214yes—
75#1935C2214yes—
76#1928C2214yes—
77#1926C2214yes—
78#1893C2214yes—
79#1893C2214yes—
80#1892C2214yes—
81#1892C2214yes—
82#1864C2214yes—
83#1863C2214yes—
84#1847C2211yescharacteristic
85#1836C2212yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89#1767C2211yescharacteristic
90#1763C2214yes—
91#1762C2214yes—
92#1752C2214yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127—21no—
103C2×#127C2311yescharacteristic
104#529—28no—
105#529—44no—
106C2×#126—24no—
107C2×#126—22no—
108C2×#125C2311yescharacteristic
109C2×#124—22no—
110C2×#124C2312yes—
111C2×#124C2312yes—
112C2×#123—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×#121C2312yes—
117C2×#121C2312yes—
118C2×C4×Q8—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—22no—
129C22×C42C2312yes—
130#501—24no—
131#489—44no—
132#487—28no—
133#486—28no—
134C8×D4—28no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#106C2312yes—
138C2×#106C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C22⋊C8C2314yes—
149C2×#97—44no—
150C2×#97C2312yes—
151C2×#97C2311yescharacteristic
152C2×#97C2311yescharacteristic
153C2×#97C2314yes—
154C2×#97C2312yes—
155C2×C4×C8C2312yes—
156#400—24no—
157#399—22no—
158#397—24no—
159#397—24no—
160#397—22no—
161#396—22no—
162#394—22no—
163#394—22no—
164#392—22no—
165#391—22no—
166#391—22no—
167#391—21no—
168#391—22no—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—22no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#389—22no—
182#389C2311yescharacteristic
183#389C2311yescharacteristic
184#389C2312yes—
185#389C2314yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#387—21no—
190#387—21no—
191#387—24no—
192#387—22no—
193#387—24no—
194#386—22no—
195#386—22no—
196#386—22no—
197#386—22no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2312yes—
203#384C2312yes—
204#382—22no—
205#382C2311yescharacteristic
206#380—22no—
207#380—22no—
208#380—22no—
209#380C2314yes—
210#380C2314yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2314yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8C2412yes—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2412yes—
246C23×C4—22no—
247C23×C4C2×D412yes—
248C23×C4C2×D412yes—
249C23×C4C2412yes—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—44no—
262C4⋊Q8—44no—
263#127—24no—
264#127—44no—
265#126—48no—
266#126—416no—
267#125—22no—
268#125#4114yes—
269#124—24no—
270#124—24no—
271#124—44no—
272#124—48no—
273#123—48no—
274C22⋊Q8—24no—
275C22⋊Q8—42no—
276C22⋊Q8—48no—
277C22⋊Q8—44no—
278#121—24no—
279#121—42no—
280#121—44no—
281C4×Q8—24no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—48no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—48no—
291C4×D4—44no—
292#117—48no—
293#117—48no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—21no—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—42no—
358#106—24no—
359#106—42no—
360C4⋊C8—24no—
361#103—28no—
362#103—28no—
363#103—44no—
364#103—44no—
365#102—28no—
366#102—28no—
367#102—44no—
368#102—44no—
369C22⋊C8—28no—
370#97—22no—
371#97—22no—
372#97—24no—
373#97—44no—
374#97—42no—
375C4×C8#4118yes—
376#95—22no—
377#95—24no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—24no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—24no—
388#95—24no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393C24—22no—
394C24C22×D412yes—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—24no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—81no—
419C2×D4—81no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4AGL(1,ℤ/8)14yes—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—416no—
463C2×C8C2×#4114yes—
464C2×C8C2×#4114yes—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—82no—
497C4⋊C4C2×#4112yes—
498C4⋊C4C2×#4112yes—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—82no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—41no—
523C42—41no—
524C42—44no—
525C42—48no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42C2×#4114yes—
530C42C2×#4112yes—
531C42H(5,𝔽2)12yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C2×AGL(1,ℤ/8)14yes—
546C23C23×D411yescharacteristic
547Q8—44no—
548Q8—44no—
549Q8—84no—
550Q8—88no—
551D4—28no—
552D4—44no—
553D4—44no—
554D4—88no—
555D4—84no—
556D4—162no—
557D4—164no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4#53212yes—
609C2×C4#53212yes—
610C2×C4#54612yes—
611C2×C4#54612yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4#57512yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C22×#4112yes—
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4#58512yes—
618C8—24no—
619C8—24no—
620C8—44no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22C2×#53211yescharacteristic
636C22C2×#54611yescharacteristic
637C22#350011yescharacteristic
638C22C22×AGL(1,ℤ/8)11yescharacteristic
639C22C2×#57511yescharacteristic
640C22#358611yescharacteristic
641C22#358611yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—41no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4—82no—
658C4#318014yes—
659C4#348012yes—
660C4#350612yes—
661C2—44no—
662C2—162no—
663C2#3641911yescharacteristic
664C2C2×#318011yescharacteristic
665C2#6072011yescharacteristic
666C2#6116111yescharacteristic
667C2#6116511yescharacteristic
668C2#6258311yescharacteristic
669C2#6258711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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
1. #36188C2semidirect
3. #33238C2semidirect
4. #33051C2semidirect
5. #29225C2semidirect
6. #29054C2semidirect
7. #29048C2semidirect
8. #28754C2semidirect
9. #28600C2semidirect
11. #27894C2semidirect
12. #25679C2semidirect
13. #23222C2semidirect
14. #23220C2semidirect
15. C4×#501C2semidirect
16. #21406C2semidirect
17. D4×#97C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#361152
#332382
#330512
#292252
#290542
#290482
#287542
#286002
#285972
#278942
#256792
#232222
#232202
C4×#5012
#214062
D4×#972