Finite Group Databasebeta

← #6567618

Subgroups — #6567618

Order 512.

1593 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36594C211yesmaximal, characteristic, complemented
2#34317C211yesmaximal, characteristic, complemented
3#30066C212yesmaximal, complemented
4#29981C211yesmaximal, characteristic, complemented
5#29552C212yesmaximal
6#29393C212yesmaximal, complemented
7#29208C211yesmaximal, characteristic, complemented
8#28202C211yesmaximal, characteristic, complemented
9#27495C214yesmaximal, complemented
10#27135C214yesmaximal, complemented
11#25057C212yesmaximal, complemented
12#24327C212yesmaximal, complemented
13#23838C212yesmaximal, complemented
14#23676C211yesmaximal, characteristic
15#23061C212yesmaximal, complemented
16#22828C211yesmaximal, characteristic, complemented
17#22807C212yesmaximal, complemented
18C2×#556—21no—
19C2×#554C2211yescharacteristic
20C2×#476C2212yes—
21C2×C4×D8C2212yes—
22#2868C2211yescharacteristic
23#2858C2212yes—
24#2547—21no—
25#2545—22no—
26#2538—21no—
27#2538C2212yes—
28#2512—22no—
29#2512C2214yes—
30#2370C2212yes—
31#2369C2211yescharacteristic
32#2362—22no—
33#2362—21no—
34#2357C2212yes—
35#2355C2211yescharacteristic
36#2354—21no—
37#2354C2211yescharacteristic
38#2354C2212yes—
39#2344—21no—
40#2332—22no—
41#2323—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×#127C2211yescharacteristic
45C4×#125C2212yes—
46#2293C2211yescharacteristic
47#2293C2212yes—
48#2274—21no—
49C42×Q8C2211yescharacteristic
50#2100C2212yes—
51#2094C2212yes—
52#2092C2214yes—
53#2085C2214yes—
54#2084C2212yes—
55#2083C2211yescharacteristic
56#2071C2211yescharacteristic
57#2067C2212yes—
58#2059C2214yes—
59#2058C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2054C2212yes—
63#2041C2212yes—
64#2041C2211yescharacteristic
65#2025C2214yes—
66#2023C2212yes—
67#2011C2211yescharacteristic
68#1990C2212yes—
69#1984C2212yes—
70#1970C2214yes—
71#1948C2212yes—
72#1947C2214yes—
73#1934C2214yes—
74#1926C2214yes—
75#1926C2214yes—
76#1925C2214yes—
77#1925C2214yes—
78#1925C2214yes—
79#1921C2214yes—
80#1920C2214yes—
81#1916C2214yes—
82#1879C2214yes—
83#1864C2214yes—
84#1845C2212yes—
85#1839C2212yes—
86#1831C2211yescharacteristic
87#1802C2212yes—
88#1802C2212yes—
89C4×#107C2214yes—
90C4×#106C2214yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1744C2212yes—
95C22×D8C2312yes—
96#556—48no—
97#554—24no—
98#554—24no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#127C2311yescharacteristic
104C2×#126—24no—
105C2×#125C2312yes—
106C2×#124—22no—
107C2×#124—22no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117—22no—
125C22×C4⋊C4—21no—
126#476—28no—
127C4×D8—28no—
128C2×#107C2312yes—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#397—24no—
145#396—24no—
146#396—24no—
147#391—21no—
148#391—21no—
149#391—22no—
150#391—22no—
151#390—21no—
152#390—21no—
153#390—22no—
154#390—22no—
155#390C2312yes—
156#390C2311yescharacteristic
157#390C2311yescharacteristic
158#390C2314yes—
159#390C2314yes—
160#390C2312yes—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—22no—
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—21no—
174#388—21no—
175#388—22no—
176#388—22no—
177#388—24no—
178#388—24no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382C2312yes—
191#381—22no—
192#380—21no—
193#380—22no—
194#380—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D412yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—22no—
217C43C2312yes—
218C43C2311yescharacteristic
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2314yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2×D411yescharacteristic
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2412yes—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—22no—
243C2×D8—24no—
244C2×D8—24no—
245C2×D8—42no—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C4⋊Q8—22no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253C4⋊Q8#4114yes—
254#127—42no—
255#126—416no—
256#125—24no—
257#125—24no—
258#124—48no—
259#124—84no—
260C22⋊Q8—28no—
261C22⋊Q8—216no—
262C22⋊Q8—48no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—48no—
268C4×Q8—82no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—22no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4#4114yes—
277#117—44no—
278#117—48no—
279#117—44no—
280#117—84no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—21no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346#107—24no—
347#107—44no—
348#107—44no—
349#106—28no—
350#106—44no—
351C4⋊C8—24no—
352C4⋊C8—24no—
353#102—24no—
354#102—24no—
355#102—24no—
356#102—28no—
357#102—42no—
358#102—44no—
359#102—44no—
360C4×C8—28no—
361C4×C8—24no—
362#95—22no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—21no—
379C24—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4C2×#4112yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C2511yescharacteristic
432D8—48no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C82−1+414yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—82no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4AGL(1,ℤ/8)14yes—
474C4⋊C4C2×#4114yes—
475C4⋊C4C2×#4112yes—
476C22⋊C4—28no—
477C22⋊C4—24no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—82no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C42—24no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—42no—
502C42—41no—
503C42—42no—
504C42—44no—
505C42—48no—
506C42—82no—
507C42—82no—
508C42—82no—
509C42#13514yes—
510C42C2×#4112yes—
511C42C2×#4112yes—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C23×D411yescharacteristic
527Q8—44no—
528Q8—84no—
529Q8—88no—
530D4—24no—
531D4—42no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—84no—
537D4—84no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—48no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—88no—
583C2×C4—88no—
584C2×C4#52512yes—
585C2×C4#52512yes—
586C2×C4C2×#13512yes—
587C2×C4C2×#13512yes—
588C2×C4C2×#13512yes—
589C2×C4C2×AGL(1,ℤ/8)14yes—
590C2×C4C2×AGL(1,ℤ/8)12yes—
591C2×C4#57612yes—
592C2×C4C22×#4111yescharacteristic
593C2×C4C22×#4111yescharacteristic
594C2×C4C2×2−1+412yes—
595C2×C4#58512yes—
596C8—24no—
597C8—42no—
598C8—44no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22C2×#52511yescharacteristic
614C22#348711yescharacteristic
615C22#349911yescharacteristic
616C22C22×#13511yescharacteristic
617C22C22×AGL(1,ℤ/8)11yescharacteristic
618C22C2×#57611yescharacteristic
619C22#358611yescharacteristic
620C4—22no—
621C4—24no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—41no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—44no—
631C4—81no—
632C4—82no—
633C4—81no—
634C4—82no—
635C4—84no—
636C4#310614yes—
637C4#349512yes—
638C4#351812yes—
639C2—82no—
640C2—162no—
641C2#3603011yescharacteristic
642C2C2×#310611yescharacteristic
643C2#6059611yescharacteristic
644C2#6188211yescharacteristic
645C2#6190711yescharacteristic
646C2#6247411yescharacteristic
647C2#6250311yescharacteristic
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. #34317C2semidirect
3. #30066C2semidirect
4. #29981C2semidirect
6. #29393C2semidirect
7. #29208C2semidirect
8. #28202C2semidirect
9. #27495C2semidirect
10. #27135C2semidirect
11. #25057C2semidirect
12. #24327C2semidirect
13. #23838C2semidirect
15. #23061C2semidirect
16. #22828C2semidirect
17. #22807C2semidirect

Maximal subgroups

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

subgroupindex
#365942
#343172
#300662
#299812
#295522
#293932
#292082
#282022
#274952
#271352
#250572
#243272
#238382
#236762
#230612
#228282
#228072