Finite Group Databasebeta

← #6989358

Subgroups — #6989358

Order 512.

1463 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36460C211yesmaximal, characteristic
2#35275C211yesmaximal, characteristic, complemented
3#32834C212yesmaximal, complemented
4#31376C214yesmaximal, complemented
5#30659C214yesmaximal, complemented
6#30383C212yesmaximal, complemented
7#30082C212yesmaximal, complemented
8#29921C211yesmaximal, characteristic
9#28797C212yesmaximal, complemented
10#28726C212yesmaximal, complemented
11#27973C211yesmaximal, characteristic
12#27778C212yesmaximal, complemented
13#27648C211yesmaximal, characteristic, complemented
14#26526C212yesmaximal, complemented
15#26108C212yesmaximal, complemented
16#23757C211yesmaximal, characteristic, complemented
17#22643C211yesmaximal, characteristic, complemented
18C2×#544—21no—
19C2×#489C2214yes—
20C2×#484C2212yes—
21#2868C2212yes—
22#2860C2212yes—
23#2859C2212yes—
24#2856C2211yescharacteristic
25#2660—22no—
26#2571C2211yescharacteristic
27#2559—22no—
28#2552—21no—
29#2552C2212yes—
30#2551—21no—
31#2551C2212yes—
32#2548C2211yescharacteristic
33#2526—22no—
34#2512—21no—
35#2500—21no—
36#2440—22no—
37#2440C2214yes—
38#2426C2212yes—
39#2420C2212yes—
40#2366C2211yescharacteristic
41#2363—21no—
42#2357—21no—
43Q8×C4⋊C4C2211yescharacteristic
44#2317—21no—
45C4×#124C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2100C2211yescharacteristic
51#2099C2214yes—
52#2095C2211yescharacteristic
53#2094C2212yes—
54#2093C2214yes—
55#2088C2214yes—
56#2085C2214yes—
57#2083C2212yes—
58#2083C2211yescharacteristic
59#2076C2212yes—
60#2072C2212yes—
61#2072C2211yescharacteristic
62#2071C2212yes—
63#2071C2212yes—
64#2071C2212yes—
65#2067C2214yes—
66#2057C2214yes—
67#2042C2214yes—
68#2036C2212yes—
69#2034C2214yes—
70#2023C2214yes—
71#1993C2214yes—
72#1982C2212yes—
73#1957C2212yes—
74#1948C2212yes—
75#1947C2212yes—
76#1944C2214yes—
77#1939C2212yes—
78#1936C2214yes—
79#1924C2214yes—
80#1874C2214yes—
81#1865C2214yes—
82#1862C2214yes—
83#1839C2212yes—
84#1834C2212yes—
85#1824C2212yes—
86#1803C2212yes—
87#1803C2214yes—
88#1802C2214yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95#544—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8C2314yes—
100C2×#127C2311yescharacteristic
101C2×#126—22no—
102C2×#125—22no—
103C2×#125—22no—
104C2×#124C2312yes—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4—21no—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C2×#117D411yescharacteristic
116C2×#117D411yescharacteristic
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121#489—48no—
122#484—44no—
123C2×#107C2312yes—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#398—24no—
142#398—24no—
143#397—22no—
144#396—22no—
145#395—24no—
146#395—24no—
147#394—22no—
148#394—22no—
149#394—22no—
150#391C2311yescharacteristic
151#390—21no—
152#390—21no—
153#390—21no—
154#390—22no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2314yes—
168#389C2311yescharacteristic
169#389C2314yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—21no—
178#386—22no—
179#386—22no—
180#385—21no—
181#385—24no—
182#384—21no—
183#384—22no—
184#384—22no—
185#384—21no—
186#384—21no—
187#384C2314yes—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#382—21no—
194#382—22no—
195#382—21no—
196#382C2312yes—
197#381—22no—
198#380—21no—
199#380C2311yescharacteristic
200#380C2312yes—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#336C2312yes—
227#336C2314yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231C22×Q8—24no—
232C22×Q8C2411yescharacteristic
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2412yes—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C22×C8C2412yes—
243C22×C8C2414yes—
244C22×C8C2412yes—
245C4⋊Q8—44no—
246C4⋊Q8—44no—
247C4⋊Q8—48no—
248C4⋊Q8—48no—
249C4⋊Q8—84no—
250#127—24no—
251#126—84no—
252#125—48no—
253#125—48no—
254#124—24no—
255#124—42no—
256C22⋊Q8—22no—
257C22⋊Q8—416no—
258C22⋊Q8—41no—
259#121—44no—
260#121—44no—
261#121—82no—
262C4×Q8—24no—
263C4×Q8—44no—
264C4×D4—44no—
265C4×D4—44no—
266#117—24no—
267#117—24no—
268#117—44no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—42no—
337#107—42no—
338#107—44no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—28no—
343#103—24no—
344#103—42no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102—44no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351#97—44no—
352C4×C8—24no—
353#95—21no—
354#95—21no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24—22no—
371C24C22×D411yescharacteristic
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—48no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×Q8H(5,𝔽2)12yes—
379C2×D4—22no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—24no—
383C2×D4—44no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—81no—
389C2×D4—81no—
390C2×D4—82no—
391C2×D4—82no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4#13714yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4AGL(1,ℤ/8)14yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—42no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—81no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—22no—
493C42—22no—
494C42—24no—
495C42—22no—
496C42—42no—
497C42—42no—
498C42—44no—
499C42—42no—
500C42—82no—
501C422−1+414yes—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C2×#13712yes—
516C23C23×D411yescharacteristic
517Q8—24no—
518Q8—42no—
519Q8—168no—
520D4—44no—
521D4—44no—
522D4—84no—
523D4—162no—
524D4—164no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—164no—
574C2×C4—164no—
575C2×C4#53612yes—
576C2×C4#53612yes—
577C2×C4C2×AGL(1,ℤ/8)14yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4#57612yes—
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+412yes—
584C2×C4#58512yes—
585C2×C4#58511yescharacteristic
586C8—42no—
587C8—44no—
588C8—42no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22C2×#53611yescharacteristic
604C22#348811yescharacteristic
605C22#350011yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22C22×#13711yescharacteristic
608C22C2×#57611yescharacteristic
609C22#358611yescharacteristic
610C4—24no—
611C4—22no—
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—42no—
617C4—41no—
618C4—42no—
619C4—81no—
620C4—84no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—81no—
625C4—81no—
626C4#323914yes—
627C4#355912yes—
628C4#356512yes—
629C2—42no—
630C2—162no—
631C2#4294411yescharacteristic
632C2C2×#323911yescharacteristic
633C2#6087111yescharacteristic
634C2#6196411yescharacteristic
635C2#6200111yescharacteristic
636C2#6253711yescharacteristic
637C2#6258311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35275C2semidirect
3. #32834C2semidirect
4. #31376C2semidirect
5. #30659C2semidirect
6. #30383C2semidirect
7. #30082C2semidirect
9. #28797C2semidirect
10. #28726C2semidirect
12. #27778C2semidirect
13. #27648C2semidirect
14. #26526C2semidirect
15. #26108C2semidirect
16. #23757C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#364602
#352752
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