Finite Group Databasebeta

← #6795515

Subgroups — #6795515

Order 512.

1533 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36188C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32287C214yesmaximal, complemented
5#31938C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#28687C211yesmaximal, characteristic, complemented
8#28529C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27730C212yesmaximal, complemented
11#27679C212yesmaximal, complemented
12#25701C212yesmaximal, complemented
13#24651C212yesmaximal, complemented
14#24119C212yesmaximal
15#22795C211yesmaximal, characteristic, complemented
16#21985C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#556C2212yes—
19C2×#498C2212yes—
20C2×#477C2212yes—
21C2×#474C2212yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2572—22no—
28#2571C2211yescharacteristic
29#2568C2212yes—
30#2563C2211yescharacteristic
31#2563C2211yescharacteristic
32#2547—21no—
33#2547—21no—
34#2537C2211yescharacteristic
35#2499C2214yes—
36#2496—22no—
37#2496—22no—
38#2495—21no—
39#2495C2212yes—
40#2420—21no—
41#2420C2212yes—
42#2398—22no—
43#2365—21no—
44#2365—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#121—21no—
48C4×#121C2212yes—
49#2285C2212yes—
50#2283—21no—
51#2283C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2101C2214yes—
55#2097C2214yes—
56#2092C2214yes—
57#2084C2214yes—
58#2074C2214yes—
59#2068C2214yes—
60#2036C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1981C2212yes—
67#1981C2211yescharacteristic
68#1968C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1933C2214yes—
77#1927C2212yes—
78#1923C2214yes—
79#1877C2212yes—
80#1875C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1846C2214yes—
85#1842C2211yescharacteristic
86#1835C2214yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#556—44no—
97#556—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#126—24no—
104C2×#124—22no—
105C2×C22⋊Q8C2314yes—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117C2314yes—
119C22×C42—21no—
120C22×C42C2312yes—
121#498—44no—
122#477—44no—
123#474—44no—
124#461—24no—
125C4×SD16—44no—
126#431—28no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×C4×C8C2314yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#400—22no—
150#400—22no—
151#397—22no—
152#397—24no—
153#397—24no—
154#397—22no—
155#394—22no—
156#394—22no—
157#393—22no—
158#393—22no—
159#391—22no—
160#391—22no—
161#391—21no—
162#391C2312yes—
163#391C2314yes—
164#390—21no—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—22no—
171#389—22no—
172#389C2312yes—
173#389C2312yes—
174#388—22no—
175#388—22no—
176#387—24no—
177#387—24no—
178#386—22no—
179#386—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#383—22no—
187#383—22no—
188#382—21no—
189#382—21no—
190#382C2312yes—
191#382C2314yes—
192#382C2312yes—
193#381—22no—
194#381—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380C2312yes—
203#380C2312yes—
204#380C2312yes—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376—22no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—44no—
261C4⋊Q8—44no—
262#126—416no—
263#124—48no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266#121—24no—
267#121—42no—
268#121—44no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—48no—
272C4×Q8—82no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—42no—
282C4×D4—41no—
283C4×D4—84no—
284#117—44no—
285#117—44no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—44no—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2412yes—
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—44no—
346#107—44no—
347#106—42no—
348#106—44no—
349#106—44no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352C4⋊C8—22no—
353C4⋊C8—42no—
354C4⋊C8#4114yes—
355#103—24no—
356#103—42no—
357#103—44no—
358#103—42no—
359#103—44no—
360#102—42no—
361#102—44no—
362C22⋊C8—28no—
363C22⋊C8—44no—
364C4×C8—28no—
365C4×C8—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24C22×D412yes—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—82no—
413C2×D42−1+412yes—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—81no—
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
453SD16—48no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8C2×#4114yes—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—82no—
494C4⋊C4—81no—
495C4⋊C4—82no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—41no—
519C42—48no—
520C42—44no—
521C42—42no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42#13514yes—
526C42C2×#4112yes—
527C42H(5,𝔽2)12yes—
528C42H(5,𝔽2)12yes—
529C422−1+412yes—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C23×D411yescharacteristic
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—84no—
550Q8—88no—
551D4—22no—
552D4—22no—
553D4—44no—
554D4—41no—
555D4—41no—
556D4—88no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4#53512yes—
609C2×C4#53512yes—
610C2×C4#54712yes—
611C2×C4#54712yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4#57512yes—
616C2×C4#57712yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C2×C4#58511yescharacteristic
623C8—24no—
624C8—44no—
625C8—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22C2×#53511yescharacteristic
641C22C2×#54711yescharacteristic
642C22#348911yescharacteristic
643C22C22×#13511yescharacteristic
644C22C2×#57511yescharacteristic
645C22C2×#57711yescharacteristic
646C22#358611yescharacteristic
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—41no—
654C4—82no—
655C4—81no—
656C4—81no—
657C4—84no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#321214yes—
663C4#334014yes—
664C4#351711yescharacteristic
665C4#352011yescharacteristic
666C4#354711yescharacteristic
667C4#355211yescharacteristic
668C2—82no—
669C2—82no—
670C2#3892011yescharacteristic
671C2C2×#321211yescharacteristic
672C2#6083611yescharacteristic
673C2C2×#334011yescharacteristic
674C2#6125711yescharacteristic
675C2#6209711yescharacteristic
676C2#6214411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #33195C2semidirect
4. #32287C2semidirect
5. #31938C2semidirect
6. #29926C2semidirect
7. #28687C2semidirect
8. #28529C2semidirect
9. #28284C2semidirect
10. #27730C2semidirect
11. #27679C2semidirect
12. #25701C2semidirect
13. #24651C2semidirect
15. #22795C2semidirect
16. #21985C2semidirect
17. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349432
#331952
#322872
#319382
#299262
#286872
#285292
#282842
#277302
#276792
#257012
#246512
#241192
#227952
#219852
C4×#4612