Finite Group Databasebeta

← #6566103

Subgroups — #6566103

Order 512.

2103 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3353C212yesmaximal, complemented
2#36179C211yesmaximal, characteristic
3#34324C211yesmaximal, characteristic, complemented
4#29866C211yesmaximal, characteristic, complemented
5#29758C212yesmaximal, complemented
6#28689C211yesmaximal, characteristic, complemented
7#28580C212yesmaximal, complemented
8#28201C211yesmaximal, characteristic, complemented
9#27648C212yesmaximal, complemented
10#25449C212yesmaximal, complemented
11#25217C214yesmaximal, complemented
12#24813C212yesmaximal, complemented
13#24594C214yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23658C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22500C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#547—21no—
20C2×#547—21no—
21C2×#547C2212yes—
22#3353—216no—
23C2×#500C2212yes—
24C2×#484C2214yes—
25C2×#482C2214yes—
26C2×#475C2212yes—
27C2×#474C2212yes—
28C2×#468C2214yes—
29C2×C4×SD16C2212yes—
30C2×#436C2212yes—
31C2×#428C2214yes—
32#2868C2211yescharacteristic
33#2856C2212yes—
34#2542—21no—
35#2542—21no—
36#2542C2212yes—
37#2501—22no—
38#2501—22no—
39#2501C2214yes—
40#2373—22no—
41#2367—21no—
42#2365C2211yescharacteristic
43#2362C2212yes—
44#2355—21no—
45#2354C2211yescharacteristic
46Q8×C4⋊C4C2211yescharacteristic
47#2346C2212yes—
48#2334—22no—
49#2326—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53#2296C2212yes—
54#2295C2211yescharacteristic
55C2×#384C2212yes—
56#2276—21no—
57C42×D4C2211yescharacteristic
58#2084C2212yes—
59#2084C2214yes—
60#2084C2212yes—
61#2084C2211yescharacteristic
62#2083C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2083C2214yes—
66#2072C2211yescharacteristic
67#2072C2212yes—
68#2072C2212yes—
69#2067C2214yes—
70#2065C2212yes—
71#2065C2214yes—
72#2055C2212yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2014C2211yescharacteristic
76#1990C2212yes—
77#1988C2212yes—
78#1947C2214yes—
79#1947C2214yes—
80#1939C2212yes—
81#1925C2214yes—
82#1925C2214yes—
83#1895C2214yes—
84#1880C2214yes—
85#1875C2214yes—
86#1851C2214yes—
87#1831C2211yescharacteristic
88#1824C2214yes—
89#1803C2214yes—
90#1802C2212yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×#41—22no—
97C22×SD16C2312yes—
98C22×M16C2314yes—
99#556—24no—
100#556D418yescomplemented
101#547—28no—
102#547—28no—
103#547—48no—
104#547—48no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#126—22no—
111C2×#124—21no—
112C2×#123—24no—
113C2×#123—24no—
114C2×#123—24no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8C2312yes—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121C2314yes—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4D411yescharacteristic
130C2×C4×D4D411yescharacteristic
131C2×C4×D4C2312yes—
132C2×C4×D4C2311yescharacteristic
133C2×#117C2312yes—
134C2×#117C2312yes—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4C2314yes—
138C22×C42C2312yes—
139#500—28no—
140#484—216no—
141#482—216no—
142#475—28no—
143#474—28no—
144#468—216no—
145C4×SD16—28no—
146#436—28no—
147#428—216no—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C4×C8C2312yes—
165#399—24no—
166#399—24no—
167#399—24no—
168#391—21no—
169#391—21no—
170#391C2311yescharacteristic
171#391C2311yescharacteristic
172#390—21no—
173#390—21no—
174#390—21no—
175#390—22no—
176#390—22no—
177#390C2312yes—
178#390C2314yes—
179#389—21no—
180#389—22no—
181#389C2312yes—
182#389C2311yescharacteristic
183#389C2312yes—
184#389C2312yes—
185#389C2311yescharacteristic
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—21no—
192#387—22no—
193#387—21no—
194#387—22no—
195#387—22no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384—22no—
203#384—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#384C2314yes—
207#384C2312yes—
208#384C2312yes—
209#382—22no—
210#380C2312yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C22⋊C4—22no—
226C4×C22⋊C4D412yes—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229#376—21no—
230#376—22no—
231C43C2311yescharacteristic
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2314yes—
244C2×#41—48no—
245C2×#41—48no—
246C2×#41—48no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8C2411yescharacteristic
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4C2×D412yes—
258C23×C4C2×D412yes—
259C23×C4C2412yes—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—42no—
263C2×M16—28no—
264C2×M16—28no—
265C2×M16—44no—
266C22×C8C2414yes—
267C22×C8C2412yes—
268C22×C8C2412yes—
269C4⋊Q8—24no—
270C4⋊Q8—22no—
271C4⋊Q8—24no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8C2×D414yes—
275#127—42no—
276#126—216no—
277#124—28no—
278#123—416no—
279#123—416no—
280#123—416no—
281C22⋊Q8—24no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284C22⋊Q8—48no—
285C22⋊Q8C2×D418yes—
286#121—28no—
287#121—28no—
288#121—48no—
289#121—48no—
290C4×Q8—28no—
291C4×Q8—22no—
292C4×Q8—44no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×Q8C2×D414yes—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—44no—
301C4×D4—44no—
302C4×D4C2×D414yes—
303C4×D4#4114yes—
304#117—24no—
305#117—44no—
306#117C2×D418yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4—48no—
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—44no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—28no—
374#107—28no—
375#107—28no—
376#106—24no—
377#106—28no—
378C4⋊C8—24no—
379C4⋊C8—24no—
380C4⋊C8—44no—
381#103—24no—
382#103—28no—
383#102—24no—
384#102—24no—
385#102—28no—
386#102—42no—
387#102—44no—
388C22⋊C8—28no—
389C4×C8—24no—
390#95—21no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—21no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—22no—
407C24C22×D412yes—
408#41—832no—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8C22×D412yes—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—42no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4#12114yes—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C2×#4114yes—
470C22×C4C2511yescharacteristic
471SD16—48no—
472M16—416no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C82−1+414yes—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—81no—
510C4⋊C4—81no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4#12114yes—
514C4⋊C4AGL(1,ℤ/8)14yes—
515C4⋊C4C22×D412yes—
516C4⋊C4C22×D414yes—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—44no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4#12118yes—
530C42—28no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—24no—
536C42—22no—
537C42—42no—
538C42—42no—
539C42—41no—
540C42—44no—
541C42—44no—
542C42—42no—
543C42—82no—
544C42—82no—
545C42AGL(1,ℤ/8)14yes—
546C42C22×D412yes—
547C42C2×#4112yes—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23C2×#12114yes—
562C23C23×D411yescharacteristic
563Q8—24no—
564Q8—42no—
565Q8—48no—
566Q8—88no—
567D4—28no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—84no—
573D4—84no—
574D4—816no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—816no—
622C2×C4—88no—
623C2×C4C2×#12114yes—
624C2×C4C2×#12112yes—
625C2×C4C2×#12112yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)14yes—
628C2×C4C2×AGL(1,ℤ/8)12yes—
629C2×C4C2×AGL(1,ℤ/8)12yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4#57812yes—
632C2×C4C23×D411yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×2−1+412yes—
635C2×C4#58512yes—
636C8—24no—
637C8—42no—
638C8—44no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22C22×#12111yescharacteristic
654C22#348511yescharacteristic
655C22#349911yescharacteristic
656C22C22×AGL(1,ℤ/8)11yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57811yescharacteristic
659C22#358611yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—24no—
665C4—24no—
666C4—42no—
667C4—42no—
668C4—41no—
669C4—42no—
670C4—44no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4—84no—
676C4#303814yes—
677C4#348912yes—
678C4#351512yes—
679C2—44no—
680C2—162no—
681C2#3602011yescharacteristic
682C2C2×#303811yescharacteristic
683C2#6043211yescharacteristic
684C2#6171711yescharacteristic
685C2#6173711yescharacteristic
686C2#6247911yescharacteristic
687C2#6250911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
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. C2×#3353C2semidirect
3. #34324C2semidirect
4. #29866C2semidirect
5. #29758C2semidirect
6. #28689C2semidirect
7. #28580C2semidirect
8. #28201C2semidirect
9. #27648C2semidirect
10. #25449C2semidirect
11. #25217C2semidirect
12. #24813C2semidirect
13. #24594C2semidirect
14. #23678C2semidirect
15. #23658C2semidirect
16. #22700C2semidirect
17. #22500C2semidirect
100. #556D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33532
#361792
#343242
#298662
#297582
#286892
#285802
#282012
#276482
#254492
#252172
#248132
#245942
#236782
#236582
#227002
#225002