Finite Group Databasebeta

← #6478181

Subgroups — #6478181

Order 512.

1712 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#35005C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic, complemented
3#33148C211yesmaximal, characteristic, complemented
4#33106C211yesmaximal, characteristic, complemented
5#32980C212yesmaximal, complemented
6#29969C211yesmaximal, characteristic, complemented
7#29891C212yesmaximal, complemented
8#29755C211yesmaximal, characteristic, complemented
9#27467C214yesmaximal, complemented
10#27403C214yesmaximal, complemented
11#27339C214yesmaximal, complemented
12#27171C214yesmaximal, complemented
13#22076C212yesmaximal, complemented
14#21902C211yesmaximal, characteristic, complemented
15C4×#480C212yesmaximal, complemented
16C2×#538—21no—
17C22×#117C2211yescharacteristic
18C2×#481C2212yes—
19C2×#480C2212yes—
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2847C2212yes—
23#2639—21no—
24#2617—21no—
25#2559—22no—
26#2557—22no—
27#2496C2212yes—
28#2491C2212yes—
29#2428C2212yes—
30#2422C2212yes—
31#2346—22no—
32#2346—22no—
33#2346C2214yes—
34#2346C2214yes—
35#2325—21no—
36C4×#121C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#388—21no—
40C2×#388—21no—
41C2×#382—21no—
42C2×#380C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×#376—21no—
47#2100C2212yescomplemented
48#2087C2214yescomplemented
49#2084C2212yescomplemented
50#2077C2214yes—
51#2076C2214yes—
52#2065C2214yescomplemented
53#2057C2212yescomplemented
54#2041C2212yescomplemented
55#2034C2214yes—
56#2033C2214yes—
57#1987C2212yescomplemented
58#1982C2212yescomplemented
59#1948C2212yescomplemented
60#1947C2212yescomplemented
61#1941C2214yescomplemented
62#1934C2214yescomplemented
63#1925C2214yescomplemented
64#1925C2214yescomplemented
65#1923C2214yes—
66#1923C2214yes—
67#1920C2214yescomplemented
68#1920C2214yescomplemented
69#1918C2214yescomplemented
70#1875C2214yes—
71#1875C2214yes—
72#1874C2212yes—
73#1874C2212yes—
74#1864C2214yescomplemented
75#1854C2214yes—
76#1854C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1797C2211yescharacteristic
82#1797C2211yescharacteristic
83#1779C2212yes—
84C4×#107C2214yescomplemented
85C4×#102C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C24×C4C2311yescharacteristic
88#538—48no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127C2311yescharacteristic
91C2×#126—22no—
92C2×#124—22no—
93C2×#123—22no—
94C2×C22⋊Q8—21no—
95C2×#121—21no—
96C2×#121—21no—
97C2×#121—22no—
98C2×#121C2312yes—
99C2×#121C2312yes—
100C2×#121C2312yes—
101C2×#120—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2312yes—
104C2×#117—22no—
105C2×#117—24no—
106C2×#117—24no—
107C2×#117C2314yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4D411yescharacteristic
117C22×C22⋊C4D411yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42—21no—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#481—28no—
124#480—28no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×C4×C8C2314yes—
140#400—22no—
141#397—22no—
142#397—22no—
143#394—22no—
144#394—22no—
145#394—22no—
146#393—22no—
147#392—22no—
148#391—21no—
149#390—21no—
150#390—21no—
151#390C2312yes—
152#390C2312yes—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2314yes—
161#389C2312yes—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—22no—
167#388—22no—
168#387—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#386—22no—
173#386—22no—
174#385—22no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#382—24no—
178#382—24no—
179#380—21no—
180#380—22no—
181#380C2312yes—
182#380C2314yes—
183#380C2312yes—
184#380C2312yes—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2314yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4D412yes—
194C4×C22⋊C4D412yes—
195C4×C22⋊C4C2312yes—
196C4×C22⋊C4C2312yes—
197#376—24no—
198#376—24no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95D411yescharacteristic
205C2×#95D411yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217C25C2×D411yescharacteristic
218C22×Q8—21no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D412yes—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—42no—
245#127—42no—
246#126—48no—
247#124—48no—
248#123—48no—
249C22⋊Q8—44no—
250#121—24no—
251#121—24no—
252#121—48no—
253#121—44no—
254#121—44no—
255#121—44no—
256#121—44no—
257#120—48no—
258C4×D4—44no—
259C4×D4—48no—
260#117—216no—
261#117—28no—
262#117—28no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—82no—
282C2×C4⋊C4—82no—
283C2×C4⋊C4#4114yes—
284C2×C4⋊C4#4114yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D412yes—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—28no—
330#107—44no—
331#106—28no—
332#106—44no—
333C4⋊C8—42no—
334C4⋊C8—42no—
335#102—28no—
336#102—28no—
337#102—44no—
338#102—44no—
339C22⋊C8—24no—
340C22⋊C8—24no—
341C22⋊C8—44no—
342C22⋊C8—44no—
343C4×C8—44no—
344#95—24no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D412yes—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—41no—
366C24—41no—
367C24—41no—
368C24—41no—
369C24C22×D412yes—
370C24C22×D411yescharacteristic
371C2×Q8—42no—
372C2×Q8—81no—
373C2×Q8—81no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—24no—
377C2×D4—24no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4—81no—
389C2×D4—82no—
390C2×D4—82no—
391C2×D4—82no—
392C2×D4—82no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4#13514yes—
440C22×C4AGL(1,ℤ/8)14yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4C2511yescharacteristic
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—81no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—81no—
481C4⋊C4C2×#4114yes—
482C4⋊C4C2×#4114yes—
483C22⋊C4—216no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C22⋊C4—82no—
499C22⋊C4—84no—
500C42—216no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—41no—
505C42—42no—
506C42—41no—
507C42—44no—
508C42—44no—
509C42—48no—
510C42—44no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23#52512yes—
537C23C2×#13512yes—
538C23C2×AGL(1,ℤ/8)12yes—
539C23C23×D411yescharacteristic
540Q8—162no—
541D4—44no—
542D4—44no—
543D4—88no—
544D4—162no—
545D4—162no—
546D4—164no—
547D4—164no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—164no—
604C2×C4#52512yes—
605C2×C4C2×#13512yes—
606C2×C4C2×#13512yes—
607C2×C4C2×AGL(1,ℤ/8)14yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4#58512yes—
611C2×C4#58512yes—
612C8—44no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—88no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22#310614yes—
636C22C2×#52511yescharacteristic
637C22#348511yescharacteristic
638C22#348511yescharacteristic
639C22C22×#13511yescharacteristic
640C22C22×AGL(1,ℤ/8)11yescharacteristic
641C22#358511yescharacteristic
642C22#358611yescharacteristic
643C4—22no—
644C4—22no—
645C4—24no—
646C4—22no—
647C4—42no—
648C4—42no—
649C4—41no—
650C4—41no—
651C4—44no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4#345512yes—
659C4#349312yes—
660C2—24no—
661C2—82no—
662C2—162no—
663C2#3509311yescharacteristic
664C2C2×#310611yescharacteristic
665C2#6056511yescharacteristic
666C2#6169611yescharacteristic
667C2#6170011yescharacteristic
668C2#6170411yescharacteristic
669C2#6172011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2346C2semidirect
3. #33148C2semidirect
4. #33106C2semidirect
5. #32980C2semidirect
6. #29969C2semidirect
7. #29891C2semidirect
8. #29755C2semidirect
9. #27467C2semidirect
10. #27403C2semidirect
11. #27339C2semidirect
12. #27171C2semidirect
13. #22076C2semidirect
14. #21902C2semidirect
15. C4×#480C2semidirect
47. #2100C22semidirect
48. #2087C22semidirect
49. #2084C22semidirect
52. #2065C22semidirect
53. #2057C22semidirect
54. #2041C22semidirect
57. #1987C22semidirect
58. #1982C22semidirect
59. #1948C22semidirect
60. #1947C22semidirect
61. #1941C22semidirect
62. #1934C22semidirect
63. #1925C22semidirect
64. #1925C22semidirect
67. #1920C22semidirect
68. #1920C22semidirect
69. #1918C22semidirect
74. #1864C22semidirect
84. C4×#107C22semidirect
85. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#350052
C2×#23462
#331482
#331062
#329802
#299692
#298912
#297552
#274672
#274032
#273392
#271712
#220762
#219022
C4×#4802