Finite Group Databasebeta

← #6536246

Subgroups — #6536246

Order 512.

1841 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#34324C211yesmaximal, characteristic, complemented
3#30169C212yesmaximal, complemented
4#29983C211yesmaximal, characteristic, complemented
5#29304C212yesmaximal, complemented
6#29208C211yesmaximal, characteristic, complemented
7#28201C211yesmaximal, characteristic, complemented
8#27182C214yesmaximal, complemented
9#26569C214yesmaximal, complemented
10#25011C212yesmaximal, complemented
11#24119C212yesmaximal, complemented
12#24062C212yesmaximal, complemented
13#23678C211yesmaximal, characteristic, complemented
14#23576C212yesmaximal, complemented
15#22968C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic, complemented
17#22671C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#547—21no—
20C2×#532C2212yes—
21C2×#482C2214yes—
22C2×#474C2212yes—
23C2×#440C2214yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2547—21no—
28#2537—21no—
29#2537C2212yes—
30#2432—22no—
31#2422—22no—
32#2422C2214yes—
33#2373—22no—
34#2367—21no—
35#2365C2211yescharacteristic
36#2362C2212yes—
37#2355—21no—
38#2354C2211yescharacteristic
39Q8×C4⋊C4C2211yescharacteristic
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2212yes—
42#2346C2212yes—
43#2334—22no—
44#2326—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2296C2212yes—
49#2295C2211yescharacteristic
50#2276—21no—
51C42×D4C2211yescharacteristic
52#2099C2212yes—
53#2096C2212yes—
54#2084C2211yescharacteristic
55#2083C2212yes—
56#2076C2214yes—
57#2072C2211yescharacteristic
58#2065C2212yes—
59#2058C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2055C2212yes—
63#2041C2212yes—
64#2041C2211yescharacteristic
65#2036C2214yes—
66#2033C2214yes—
67#2024C2212yes—
68#2014C2211yescharacteristic
69#2000C2214yes—
70#1990C2212yes—
71#1954C2212yes—
72#1948C2212yes—
73#1947C2214yes—
74#1944C2214yes—
75#1943C2214yes—
76#1925C2214yes—
77#1924C2214yes—
78#1921C2214yes—
79#1875C2214yes—
80#1864C2214yes—
81#1863C2214yes—
82#1862C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1845C2212yes—
86#1831C2211yescharacteristic
87#1817C2212yes—
88#1803C2212yes—
89#1802C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95C22×#41—22no—
96C22×SD16C2312yes—
97#556—24no—
98#556—24no—
99#547—48no—
100#532—48no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—22no—
107C2×#124—21no—
108C2×#124—22no—
109C2×#123—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2312yes—
112C2×#121—22no—
113C2×#121—22no—
114C2×#121C2314yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—22no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117C2312yes—
129C2×#117C2312yes—
130C22×C4⋊C4—22no—
131C22×C42C2312yes—
132#482—48no—
133#474—28no—
134#440—48no—
135C4×SD16—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2312yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×#97C2314yes—
153C2×C4×C8C2312yes—
154#397—24no—
155#394—24no—
156#394—24no—
157#391—21no—
158#391—21no—
159#391—22no—
160#391C2311yescharacteristic
161#391C2311yescharacteristic
162#390—21no—
163#390—21no—
164#390—21no—
165#390C2312yes—
166#389—21no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2311yescharacteristic
171#389C2311yescharacteristic
172#389C2314yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—21no—
181#387—22no—
182#387—21no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—22no—
188#385—24no—
189#385—24no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#384C2312yes—
197#382—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380C2312yes—
202#380C2314yes—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221#376—21no—
222#376—22no—
223C43C2311yescharacteristic
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2314yes—
235#336C2312yes—
236C2×#41—48no—
237C2×#41—48no—
238C2×#41—84no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8C2411yescharacteristic
242C22×D4—24no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4—24no—
249C23×C4C2×D412yes—
250C23×C4C2×D412yes—
251C23×C4C2412yes—
252C2×SD16—24no—
253C2×SD16—24no—
254C2×SD16—24no—
255C22×C8C2412yes—
256C22×C8C2414yes—
257C22×C8C2412yes—
258C4⋊Q8—22no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261C4⋊Q8—82no—
262C4⋊Q8#4114yescomplemented
263#127—24no—
264#126—216no—
265#124—28no—
266#124—48no—
267#123—416no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—48no—
271#121—28no—
272#121—44no—
273#121—48no—
274#121—48no—
275C4×Q8—28no—
276C4×Q8—22no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8#4114yescomplemented
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288C4×D4—44no—
289C4×D4—44no—
290C4×D4—48no—
291C4×D4—44no—
292#117—24no—
293#117—24no—
294#117—44no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2414yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—28no—
357#107—44no—
358#107—44no—
359#106—24no—
360#106—44no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363#103—24no—
364#103—44no—
365#102—28no—
366#102—24no—
367#102—24no—
368#102—24no—
369#102—28no—
370C22⋊C8—28no—
371C22⋊C8—44no—
372#97—44no—
373C4×C8—24no—
374#95—21no—
375#95—22no—
376#95—21no—
377#95—21no—
378#95—22no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—24no—
388#95—24no—
389#95C2×D412yes—
390#95C2×D412yes—
391#95C2×D412yes—
392C24—22no—
393C24C22×D412yes—
394#41—832no—
395C2×Q8—24no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—42no—
404C2×Q8C2×#4112yes—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—84no—
422C2×D4H(5,𝔽2)14yes—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—84no—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C2511yescharacteristic
457SD16—216no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C82−1+414yes—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—82no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—82no—
498C4⋊C4C2×#4112yes—
499C4⋊C4C2×#4114yes—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—82no—
512C22⋊C4—82no—
513C42—28no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—42no—
522C42—41no—
523C42—44no—
524C42—44no—
525C42—42no—
526C42—82no—
527C42—82no—
528C42AGL(1,ℤ/8)14yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23C23×D411yescharacteristic
546Q8—24no—
547Q8—48no—
548Q8—42no—
549Q8—88no—
550D4—24no—
551D4—24no—
552D4—48no—
553D4—48no—
554D4—44no—
555D4—42no—
556D4—42no—
557D4—816no—
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—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—84no—
606C2×C4—816no—
607C2×C4—88no—
608C2×C4#52512yes—
609C2×C4#52512yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4#57512yes—
614C2×C4#57812yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C22×#4111yescharacteristic
617C2×C4C2×H(5,𝔽2)12yes—
618C2×C4C2×2−1+412yes—
619C8—24no—
620C8—42no—
621C8—44no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—42no—
630C22—44no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C2×#52511yescharacteristic
637C22#348411yescharacteristic
638C22#349911yescharacteristic
639C22C22×AGL(1,ℤ/8)11yescharacteristic
640C22C2×#57511yescharacteristic
641C22C2×#57811yescharacteristic
642C22#358511yescharacteristic
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—44no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—81no—
655C4—82no—
656C4—82no—
657C4—84no—
658C4—81no—
659C4#311514yes—
660C4#349212yes—
661C4#352212yes—
662C2—82no—
663C2—84no—
664C2#3569311yescharacteristic
665C2C2×#311511yescharacteristic
666C2#6060011yescharacteristic
667C2#6158211yescharacteristic
668C2#6159511yescharacteristic
669C2#6247911yescharacteristic
670C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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
2. #34324C2semidirect
3. #30169C2semidirect
4. #29983C2semidirect
5. #29304C2semidirect
6. #29208C2semidirect
7. #28201C2semidirect
8. #27182C2semidirect
9. #26569C2semidirect
10. #25011C2semidirect
11. #24119C2semidirect
12. #24062C2semidirect
13. #23678C2semidirect
14. #23576C2semidirect
15. #22968C2semidirect
16. #22839C2semidirect
17. #22671C2semidirect
262. C4⋊Q8#41semidirect
280. C4×Q8#41semidirect

Maximal subgroups

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

subgroupindex
#361272
#343242
#301692
#299832
#293042
#292082
#282012
#271822
#265692
#250112
#241192
#240622
#236782
#235762
#229682
#228392
#226712