Finite Group Databasebeta

← #6623454

Subgroups — #6623454

Order 512.

1557 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2C4×#556C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic
4#33067C212yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29131C212yesmaximal
7#28255C211yesmaximal, characteristic, complemented
8#28159C212yesmaximal
9#27448C214yesmaximal, complemented
10#26908C214yesmaximal
11#24997C212yesmaximal, complemented
12#24189C212yesmaximal
13#23004C212yesmaximal, complemented
14#22825C211yesmaximal, characteristic, complemented
15#22759C212yesmaximal
16#21506C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×#476C2212yes—
20C2×#462C2212yes—
21C2×#462C2212yes—
22C2×#461C2211yescharacteristic
23C2×#441C2214yes—
24C2×C4×SD16C2212yes—
25#2868C2211yescharacteristic
26#2859C2212yes—
27#2582C2212yes—
28#2567—21no—
29#2552—21no—
30#2550—22no—
31#2507—22no—
32#2451C2214yes—
33#2373—22no—
34#2369—21no—
35#2367C2212yes—
36#2365C2211yescharacteristic
37#2362—21no—
38#2356C2212yes—
39#2355—21no—
40#2355C2211yescharacteristic
41#2354C2211yescharacteristic
42Q8×C4⋊C4C2212yes—
43#2347—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#126—22no—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50C42×Q8C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53C4×#117—21no—
54#2101C2214yes—
55#2074C2214yes—
56#2056C2214yes—
57#2042C2214yes—
58#2024C2214yes—
59#2015C2214yes—
60#1987C2212yes—
61#1984C2212yes—
62#1981C2211yescharacteristic
63#1956C2212yes—
64#1948C2212yes—
65#1948C2211yescharacteristic
66#1948C2212yes—
67#1936C2214yes—
68#1935C2214yes—
69#1927C2214yes—
70#1927C2212yes—
71#1919C2214yes—
72#1919C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1878C2214yes—
76#1878C2214yes—
77#1877C2212yes—
78#1865C2214yes—
79#1865C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1831C2212yes—
83#1803C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1802C2212yes—
89#1802C2211yescharacteristic
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95C22×Q16C2314yes—
96C22×SD16C2312yes—
97#556—24no—
98#556—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#126—22no—
105C2×#125—22no—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C22×C42—21no—
122#476—44no—
123#462—28no—
124#462—28no—
125#461—24no—
126#441—48no—
127C4×SD16—28no—
128C2×#106C2312yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#97C2314yes—
145C2×C4×C8C2312yes—
146#398—24no—
147#398—24no—
148#395—24no—
149#391—22no—
150#391—21no—
151#391—24no—
152#391C2312yes—
153#391C2312yes—
154#391C2314yes—
155#391C2312yes—
156#390C2312yes—
157#389—22no—
158#389—21no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—24no—
168#389C2314yes—
169#389C2314yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—22no—
179#387—22no—
180#385—22no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#383—22no—
189#382—22no—
190#382C2312yes—
191#380—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221#376—22no—
222C43—22no—
223C43C2311yescharacteristic
224C43C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C2×Q16—28no—
245C2×Q16—28no—
246C2×Q16—44no—
247C2×SD16—24no—
248C2×SD16—24no—
249C2×SD16—42no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8—44no—
258C4⋊Q8—82no—
259C4⋊Q8#4114yes—
260C4⋊Q8#4114yes—
261#126—216no—
262#125—84no—
263#124—24no—
264#124—24no—
265C22⋊Q8—28no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—42no—
277C4×Q8—42no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—41no—
285C4×D4#4114yes—
286#117—28no—
287#117—44no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C22⋊C4—22no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351#106—24no—
352#106—28no—
353C4⋊C8—22no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—42no—
358#103—24no—
359#103—28no—
360#103—28no—
361#103—24no—
362#103—24no—
363#103—44no—
364#103—44no—
365#102—24no—
366#102—44no—
367#97—44no—
368C4×C8—24no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—24no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389C24—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—44no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8C2×#4114yes—
407C2×Q8C2×#4114yes—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C2×#4112yes—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—81no—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C2511yescharacteristic
442Q16—416no—
443SD16—48no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—81no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4C2×#4112yes—
490C4⋊C4H(5,𝔽2)12yes—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—44no—
496C22⋊C4—42no—
497C22⋊C4—82no—
498C42—28no—
499C42—24no—
500C42—22no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—21no—
507C42—21no—
508C42—22no—
509C42—41no—
510C42—42no—
511C42—42no—
512C42—44no—
513C42—44no—
514C42—44no—
515C42—44no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—82no—
520C42—84no—
521C42#13714yes—
522C42C2×#4112yes—
523C42C2×#4112yes—
524C42C2×#4112yes—
525C422−1+412yes—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C23×D411yescharacteristic
534Q8—24no—
535Q8—24no—
536Q8—24no—
537Q8—24no—
538Q8—44no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—164no—
544D4—22no—
545D4—22no—
546D4—41no—
547D4—41no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—48no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4#52512yes—
597C2×C4#52512yes—
598C2×C4#54812yes—
599C2×C4#54812yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4#57512yes—
604C2×C4#57712yes—
605C2×C4C22×#4111yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×H(5,𝔽2)11yescharacteristic
609C2×C4C2×2−1+411yescharacteristic
610C2×C4#58511yescharacteristic
611C8—42no—
612C8—44no—
613C8—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#52511yescharacteristic
622C22C2×#54811yescharacteristic
623C22#348811yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57711yescharacteristic
627C22#358611yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—84no—
641C4—81no—
642C4—82no—
643C4—82no—
644C4#311614yes—
645C4#336914yes—
646C4#347511yescharacteristic
647C4#348111yescharacteristic
648C4#351111yescharacteristic
649C4#351511yescharacteristic
650C2—82no—
651C2#3649211yescharacteristic
652C2C2×#311611yescharacteristic
653C2#6059911yescharacteristic
654C2C2×#336911yescharacteristic
655C2#6132011yescharacteristic
656C2#6196011yescharacteristic
657C2#6199711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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. #36563C2semidirect
4. #33067C2semidirect
5. #29141C2semidirect
7. #28255C2semidirect
9. #27448C2semidirect
11. #24997C2semidirect
13. #23004C2semidirect
14. #22825C2semidirect
16. #21506C2semidirect

Maximal subgroups

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

subgroupindex
#365632
C4×#5562
#331972
#330672
#291412
#291312
#282552
#281592
#274482
#269082
#249972
#241892
#230042
#228252
#227592
#215062
C4×#4612