Finite Group Databasebeta

← #6567617

Subgroups — #6567617

Order 512.

1751 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34311C211yesmaximal, characteristic, complemented
3#30059C212yesmaximal, complemented
4#29980C211yesmaximal, characteristic, complemented
5#29552C212yesmaximal
6#29388C212yesmaximal, complemented
7#29208C211yesmaximal, characteristic, complemented
8#28201C211yesmaximal, characteristic, complemented
9#27486C214yesmaximal, complemented
10#27126C214yesmaximal, complemented
11#25056C212yesmaximal, complemented
12#24326C212yesmaximal, complemented
13#23835C212yesmaximal, complemented
14#23676C211yesmaximal, characteristic
15#23060C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17#22807C212yesmaximal, complemented
18C2×#556—21no—
19C2×D4×Q8C2211yescharacteristic
20C2×#519—21no—
21C2×#474C2212yes—
22C2×C4×D8C2212yes—
23#2868C2211yescharacteristic
24#2858C2212yes—
25#2547—21no—
26#2545—22no—
27#2538—21no—
28#2538C2212yes—
29#2512—22no—
30#2512C2214yes—
31#2365C2211yescharacteristic
32#2365C2212yes—
33#2362—21no—
34#2362—22no—
35#2354—21no—
36#2354C2211yescharacteristic
37#2354C2212yes—
38#2354C2212yes—
39Q8×C4⋊C4C2211yescharacteristic
40Q8×C22⋊C4—21no—
41#2323—22no—
42#2323—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×#127C2211yescharacteristic
47#2293C2211yescharacteristic
48#2293C2212yes—
49C42×Q8C2211yescharacteristic
50#2099C2212yes—
51#2093C2212yes—
52#2093C2214yes—
53#2083C2211yescharacteristic
54#2083C2212yes—
55#2083C2214yes—
56#2071C2211yescharacteristic
57#2067C2212yes—
58#2057C2214yes—
59#2057C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2054C2212yes—
63#2041C2212yes—
64#2041C2211yescharacteristic
65#2023C2214yes—
66#2023C2212yes—
67#2011C2211yescharacteristic
68#1990C2212yes—
69#1984C2212yes—
70#1970C2214yes—
71#1947C2212yes—
72#1947C2214yes—
73#1934C2214yes—
74#1925C2214yes—
75#1925C2214yes—
76#1925C2214yes—
77#1925C2214yes—
78#1925C2214yes—
79#1918C2214yes—
80#1918C2214yes—
81#1916C2214yes—
82#1879C2214yes—
83#1862C2214yes—
84#1845C2212yes—
85#1839C2212yes—
86#1831C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89C4×#107C2214yes—
90C4×#107C2214yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1744C2212yes—
95C23×Q8—21no—
96C22×D8C2312yes—
97#556—48no—
98D4×Q8—24no—
99D4×Q8—24no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×#126—24no—
107C2×#124—22no—
108C2×#124—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112#519—48no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2311yescharacteristic
127C2×#117—22no—
128C2×#117—21no—
129C2×#117—22no—
130C2×#117—21no—
131C2×#117—22no—
132#474—28no—
133C4×D8—28no—
134C2×#107C2312yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#397—24no—
151#396—24no—
152#396—24no—
153#391—21no—
154#391—22no—
155#391—21no—
156#391—22no—
157#391—22no—
158#390—21no—
159#390—21no—
160#390—22no—
161#390—22no—
162#390C2312yes—
163#390C2311yescharacteristic
164#390C2311yescharacteristic
165#390C2314yes—
166#390C2314yes—
167#390C2312yes—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2311yescharacteristic
175#389C2314yes—
176#389C2311yescharacteristic
177#389C2312yes—
178#388—22no—
179#388—21no—
180#388—22no—
181#388—21no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—24no—
186#388—24no—
187#387—22no—
188#387—22no—
189#387—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#380—21no—
198#380—22no—
199#380—22no—
200#380—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C43C2311yescharacteristic
219C43C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—28no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8C2×D411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2412yes—
244C22×D4C2411yescharacteristic
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C2×D8—24no—
249C2×D8—24no—
250C2×D8—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—22no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—82no—
260C4⋊Q8#4114yes—
261#127—42no—
262#126—416no—
263#124—48no—
264#124—84no—
265C22⋊Q8—28no—
266C22⋊Q8—216no—
267C22⋊Q8—48no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—42no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×Q8—48no—
276C4×Q8—48no—
277C4×Q8D818yescomplemented
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4#4114yes—
286#117—44no—
287#117—48no—
288#117—48no—
289#117—44no—
290#117—48no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D412yes—
347C2×C42C2×D412yes—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—24no—
357#107—28no—
358#107—44no—
359#107—44no—
360#107—44no—
361C4⋊C8—24no—
362C4⋊C8—24no—
363#102—24no—
364#102—24no—
365#102—24no—
366#102—28no—
367#102—44no—
368#102—42no—
369#102—44no—
370C4×C8—28no—
371C4×C8—24no—
372#95—21no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—24no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—21no—
383#95C2×D412yes—
384#95C2×D412yes—
385#95C2×D412yes—
386C24—21no—
387C24—22no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—432no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8C2×D814yes—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4C2×#4112yes—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2511yescharacteristic
444D8—48no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C82−1+414yes—
454C4⋊C4—28no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—42no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4C2×D814yes—
486C4⋊C4C2×D818yes—
487C4⋊C4C2×#4114yes—
488C4⋊C4C2×#4112yes—
489C22⋊C4—28no—
490C22⋊C4—24no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—42no—
500C22⋊C4—82no—
501C22⋊C4—81no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—22no—
505C42—22no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—42no—
513C42—42no—
514C42—41no—
515C42—44no—
516C42—44no—
517C42—44no—
518C42—44no—
519C42—42no—
520C42—48no—
521C42C2×D814yes—
522C42C2×D818yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C23×D411yescharacteristic
540Q8—28no—
541Q8—416no—
542Q8—48no—
543Q8—88no—
544D4—24no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—88no—
597C2×C4#52512yes—
598C2×C4#52512yes—
599C2×C4C22×D814yes—
600C2×C4C22×D812yes—
601C2×C4C22×D814yes—
602C2×C4C22×D812yes—
603C2×C4C22×D812yes—
604C2×C4#57512yes—
605C2×C4#57812yes—
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×2−1+412yes—
609C2×C4#58512yes—
610C8—24no—
611C8—42no—
612C8—44no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22C2×#52511yescharacteristic
628C22#348711yescharacteristic
629C22#349911yescharacteristic
630C22C23×D811yescharacteristic
631C22C2×#57511yescharacteristic
632C22C2×#57811yescharacteristic
633C22#358511yescharacteristic
634C4—24no—
635C4—22no—
636C4—24no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—42no—
641C4—41no—
642C4—44no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—82no—
647C4—82no—
648C4—81no—
649C4—84no—
650C4#310014yes—
651C4#349512yes—
652C4#351812yes—
653C2—82no—
654C2—162no—
655C2#3603011yescharacteristic
656C2C2×#310011yescharacteristic
657C2#6060011yescharacteristic
658C2#6188011yescharacteristic
659C2#6190511yescharacteristic
660C2#6247111yescharacteristic
661C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2011C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36594C2semidirect
2. #34311C2semidirect
3. #30059C2semidirect
4. #29980C2semidirect
6. #29388C2semidirect
7. #29208C2semidirect
8. #28201C2semidirect
9. #27486C2semidirect
10. #27126C2semidirect
11. #25056C2semidirect
12. #24326C2semidirect
13. #23835C2semidirect
15. #23060C2semidirect
16. #22828C2semidirect
17. #22807C2semidirect
277. C4×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#365942
#343112
#300592
#299802
#295522
#293882
#292082
#282012
#274862
#271262
#250562
#243262
#238352
#236762
#230602
#228282
#228072