Finite Group Databasebeta

← #6502464

Subgroups — #6502464

Order 512.

2154 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic, complemented
2#34206C211yesmaximal, characteristic, complemented
3#28114C212yesmaximal, complemented
4#27613C211yesmaximal, characteristic, complemented
5#27585C211yesmaximal, characteristic, complemented
6#27443C214yesmaximal, complemented
7#27015C214yesmaximal, complemented
8#26904C214yesmaximal, complemented
9#26392C214yesmaximal, complemented
10#24299C212yesmaximal, complemented
11#24060C211yesmaximal, characteristic, complemented
12#23930C211yesmaximal, characteristic
13#22081C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21452C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C2×#520—21no—
20C22×C4×D4C2211yescharacteristic
21C2×#480C2212yes—
22C2×#476C2214yescomplemented
23C2×#447C2212yes—
24C2×#442C2214yes—
25#2452—22no—
26#2452—22no—
27#2452C2214yes—
28#2452C2214yes—
29C2×#396—21no—
30C2×#393—21no—
31#2367—22no—
32#2366C2212yescomplemented
33#2365C2212yescomplemented
34#2354C2212yescomplemented
35#2352—22no—
36D4×C4⋊C4C2212yescomplemented
37#2325—21no—
38#2325—21no—
39#2311—21no—
40C4×#121C2212yes—
41C4×#121C2212yes—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#385—21no—
47#2285C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2212yescomplemented
50#2094C2212yes—
51#2084C2212yescomplemented
52#2071C2212yes—
53#2067C2214yes—
54#2058C2214yescomplemented
55#2057C2212yescomplemented
56#2054C2214yes—
57#2041C2212yescomplemented
58#2023C2212yes—
59#2023C2214yescomplemented
60#2011C2212yes—
61#1987C2214yescomplemented
62#1943C2214yescomplemented
63#1936C2214yescomplemented
64#1926C2214yes—
65#1925C2214yes—
66#1921C2214yescomplemented
67#1919C2214yescomplemented
68#1917C2214yescomplemented
69#1917C2214yescomplemented
70#1879C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1864C2214yes—
76#1864C2214yescomplemented
77#1862C2214yescomplemented
78#1853C2212yes—
79#1853C2212yes—
80#1823C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1786C2212yes—
85#1762C2214yes—
86#1749C2212yes—
87C22×#41—21no—
88C23×Q8—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×D8C2314yes—
92#532—28no—
93C2×C4⋊Q8—22no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2314yes—
97C2×#127C2311yescharacteristic
98C2×#124—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—48no—
102C2×#121—21no—
103C2×#121C2312yes—
104#520—48no—
105C2×C4×Q8—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—21no—
110C2×C4×D4—22no—
111C2×C4×D4—44no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C2×#117—22no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#480—28no—
136#476—48no—
137#447—28no—
138#442—48no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×#97C2314yes—
152#396—24no—
153#396—24no—
154#393—24no—
155#393—24no—
156#391—22no—
157#390—22no—
158#390—21no—
159#390—21no—
160#390—22no—
161#390C2312yes—
162#390C2314yes—
163#390C2312yes—
164#390C2312yes—
165#390C2312yes—
166#389—21no—
167#389—21no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—22no—
179#387—22no—
180#386—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#385—24no—
187#385—24no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#382—21no—
191#380—21no—
192#380—22no—
193#380—22no—
194#380C2312yes—
195C4×C4⋊C4—21no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C2×#41—28no—
223C2×#41—44no—
224C2×#41—44no—
225C22×Q8—21no—
226C22×Q8—24no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8—82no—
230C22×D4—22no—
231C22×D4—24no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—24no—
236C22×D4—42no—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×D8—28no—
257C2×D8—28no—
258C2×D8—44no—
259C22×C8C2414yes—
260C22×C8C2414yes—
261C4⋊Q8—24no—
262C4⋊Q8—28no—
263C4⋊Q8—44no—
264C4⋊Q8—48no—
265#127—24no—
266#124—28no—
267C22⋊Q8—416no—
268C22⋊Q8—416no—
269#121—28no—
270#121—24no—
271#121#4118yes—
272C4×Q8—48no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—28no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—44no—
280C4×D4—48no—
281C4×D4—48no—
282C4×D4—48no—
283C4×D4—48no—
284#117—48no—
285#117—44no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—42no—
305C2×C4⋊C4—48no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
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—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—82no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—42no—
345C2×C42—42no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354#107—28no—
355#106—28no—
356#106—44no—
357C4⋊C8—44no—
358#102—28no—
359#102—28no—
360#102—28no—
361#102—28no—
362#102—44no—
363#102—44no—
364C22⋊C8—24no—
365C22⋊C8—24no—
366#97—44no—
367#95—22no—
368#95—22no—
369#95—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24C22×D411yescharacteristic
390#41—432no—
391C2×Q8—24no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—48no—
395C2×Q8—42no—
396C2×Q8—84no—
397C2×Q8—816no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—416no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4C2×#4114yes—
418C2×D42−1+414yes—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—84no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—816no—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4AGL(1,ℤ/8)14yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4114yes—
477C22×C4C2511yescharacteristic
478D8—416no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—24no—
484C2×C8—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—48no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4C2×#4114yes—
510C4⋊C4H(5,𝔽2)14yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—24no—
532C42—24no—
533C42—42no—
534C42—42no—
535C42—44no—
536C42—44no—
537C42—44no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—24no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—84no—
560C23—84no—
561C23—84no—
562C23—84no—
563C23#52512yes—
564C23C2×AGL(1,ℤ/8)12yes—
565C23C2×AGL(1,ℤ/8)12yes—
566C23C23×D411yescharacteristic
567Q8—48no—
568Q8—816no—
569D4—28no—
570D4—28no—
571D4—48no—
572D4—48no—
573D4—416no—
574D4—44no—
575D4—44no—
576D4—84no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—44no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—816no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—164no—
631C2×C4—164no—
632C2×C4#52512yes—
633C2×C4C2×AGL(1,ℤ/8)14yes—
634C2×C4C2×AGL(1,ℤ/8)12yes—
635C2×C4C2×AGL(1,ℤ/8)12yes—
636C2×C4C22×#4112yes—
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4C2×2−1+412yes—
639C8—44no—
640C8—44no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—816no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22#310914yes—
663C22C2×#52511yescharacteristic
664C22#348411yescharacteristic
665C22#348611yescharacteristic
666C22C22×AGL(1,ℤ/8)11yescharacteristic
667C22C22×AGL(1,ℤ/8)11yescharacteristic
668C22#358511yescharacteristic
669C22#358611yescharacteristic
670C4—22no—
671C4—22no—
672C4—22no—
673C4—24no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—41no—
680C4—41no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#352014yes—
686C2—24no—
687C2—84no—
688C2—84no—
689C2#3535711yescharacteristic
690C2C2×#310911yescharacteristic
691C2#6056711yescharacteristic
692C2#6157211yescharacteristic
693C2#6158211yescharacteristic
694C2#6182511yescharacteristic
695C2#6182711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23930C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2452C2semidirect
2. #34206C2semidirect
3. #28114C2semidirect
4. #27613C2semidirect
5. #27585C2semidirect
6. #27443C2semidirect
7. #27015C2semidirect
8. #26904C2semidirect
9. #26392C2semidirect
10. #24299C2semidirect
11. #24060C2semidirect
13. #22081C2semidirect
14. #21916C2semidirect
15. #21452C2semidirect
22. C2×#476C22semidirect
32. #2366C22semidirect
33. #2365C22semidirect
34. #2354C22semidirect
36. D4×C4⋊C4C22semidirect
49. #2100C22semidirect
51. #2084C22semidirect
54. #2058C22semidirect
55. #2057C22semidirect
57. #2041C22semidirect
59. #2023C22semidirect
61. #1987C22semidirect
62. #1943C22semidirect
63. #1936C22semidirect
66. #1921C22semidirect
67. #1919C22semidirect
68. #1917C22semidirect
69. #1917C22semidirect
76. #1864C22semidirect
77. #1862C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
#342062
#281142
#276132
#275852
#274432
#270152
#269042
#263922
#242992
#240602
#239302
#220812
#219162
#214522