Finite Group Databasebeta

← #7567379

Subgroups — #7567379

Order 512.

1621 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59863C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic, complemented
3#32636C214yesmaximal, complemented
4#29141C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#27770C212yesmaximal, complemented
7#27768C212yesmaximal, complemented
8#26023C212yesmaximal, complemented
9#26021C212yesmaximal, complemented
10#25938C214yesmaximal, complemented
11#24731C212yesmaximal
12#24727C212yesmaximal, complemented
13#23247C211yesmaximal, characteristic, complemented
14#22770C212yesmaximal, complemented
15#22766C212yesmaximal, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#488C2214yes—
20C2×#463C2214yes—
21C2×#461C2211yescharacteristic
22C2×#459C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×C4×D8C2212yes—
25#2868C2211yescharacteristic
26#2857—21no—
27#2808—22no—
28#2592C2214yes—
29#2567C2212yes—
30#2563—21no—
31#2552C2212yes—
32#2552C2212yes—
33#2548C2212yes—
34#2547C2212yes—
35#2513—22no—
36#2502—22no—
37#2485—22no—
38#2440—22no—
39#2440—22no—
40#2355C2212yes—
41#2354—21no—
42#2354C2212yes—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#127C2211yescharacteristic
48C2×#384—21no—
49C42×D4C2211yescharacteristic
50#2068C2214yes—
51#2065C2214yes—
52#2042C2214yes—
53#2042C2214yes—
54#2026C2214yes—
55#2025C2214yes—
56#2024C2214yes—
57#2023C2214yes—
58#1989C2214yes—
59#1984C2214yes—
60#1982C2211yescharacteristic
61#1981C2211yescharacteristic
62#1957C2212yes—
63#1956C2212yes—
64#1948C2211yescharacteristic
65#1947C2211yescharacteristic
66#1939C2212yes—
67#1939C2212yes—
68#1934C2214yes—
69#1933C2214yes—
70#1927C2212yes—
71#1925C2212yes—
72#1919C2214yes—
73#1918C2214yes—
74#1879C2212yes—
75#1879C2214yes—
76#1877C2212yes—
77#1877C2214yes—
78#1834C2212yes—
79#1834C2212yes—
80#1803C2212yes—
81#1803C2212yes—
82#1802C2212yes—
83#1802C2212yes—
84#1802C2211yescharacteristic
85#1802C2211yescharacteristic
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#103C2212yes—
91C4×#102C2214yes—
92C4×#102C2212yes—
93C22×SD16C2312yes—
94C22×D8C2312yes—
95#554—44no—
96#554—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8D411yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#125C2314yes—
104C2×#124C2314yes—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—24no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2314yes—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C42—21no—
121#488—48no—
122#463—48no—
123#461—24no—
124#459—24no—
125C4×SD16—28no—
126C4×D8—28no—
127C2×#107C2312yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2314yes—
149#398—24no—
150#398—24no—
151#397—24no—
152#397—24no—
153#395—24no—
154#394—24no—
155#393—24no—
156#391—22no—
157#391D412yes—
158#391C2312yes—
159#391C2314yes—
160#390—22no—
161#390—22no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2314yes—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389D412yes—
171#389D412yes—
172#389C2314yes—
173#389C2314yes—
174#388—24no—
175#388—24no—
176#388—24no—
177#387—24no—
178#385—24no—
179#385—24no—
180#384—22no—
181#384—22no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384D411yescharacteristic
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2312yes—
190#382—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2311yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—21no—
206C4×C22⋊C4—21no—
207C4×C22⋊C4—21no—
208C43C2311yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2314yes—
214#340C2314yes—
215#340C2312yes—
216#340C2312yes—
217#336C2314yes—
218#336C2312yes—
219#336C2314yes—
220#336C2312yes—
221C22×Q8—22no—
222C22×Q8C2×D412yes—
223C22×Q8C2412yes—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C2412yes—
227C22×D4C2411yescharacteristic
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—24no—
233C2×SD16—24no—
234C2×SD16—24no—
235C2×SD16—24no—
236C2×D8—24no—
237C2×D8—24no—
238C2×D8—24no—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4⋊Q8—24no—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248C4⋊Q8—82no—
249C4⋊Q8—82no—
250#127—22no—
251#127—22no—
252#125—28no—
253#125—44no—
254#124—44no—
255#124—44no—
256C22⋊Q8—416no—
257C22⋊Q8—48no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4—28no—
265C4×D4—24no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—24no—
269C4×D4—44no—
270C4×D4—44no—
271C4×D4—82no—
272C4×D4#4114yes—
273C4×D4#4114yes—
274#117—84no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—84no—
293C2×C4⋊C4—84no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2414yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—24no—
332#106—24no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—44no—
338#103—24no—
339#103—24no—
340#103—24no—
341#103—44no—
342#103—44no—
343#102—24no—
344#102—24no—
345#102—24no—
346#102—24no—
347#102—24no—
348#102—44no—
349#102—44no—
350C4×C8—24no—
351C4×C8—24no—
352C4×C8—44no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95C2×D414yes—
364#95C2×D414yes—
365C24—21no—
366C24—22no—
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4C2×#4112yes—
395C2×D4C2×#4112yes—
396C2×D4H(5,𝔽2)12yes—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—42no—
408C22×C4—48no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4C22×D414yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D414yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2511yescharacteristic
428SD16—48no—
429D8—48no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4C2×#4112yes—
474C4⋊C4C2×#4112yes—
475C4⋊C4H(5,𝔽2)12yes—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—24no—
479C22⋊C4—48no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C22⋊C4—82no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—24no—
491C42—21no—
492C42—21no—
493C42—41no—
494C42—44no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—84no—
499C42#13514yes—
500C42C2×#4112yes—
501C42C2×#4112yes—
502C422−1+412yes—
503C422−1+412yes—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23C23×D411yescharacteristic
519Q8—42no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—84no—
524Q8—164no—
525D4—22no—
526D4—22no—
527D4—22no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—84no—
533D4—84no—
534D4#55514yes—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—21no—
552C2×C4—24no—
553C2×C4—48no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—48no—
561C2×C4—48no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—88no—
575C2×C4—168no—
576C2×C4#54812yes—
577C2×C4#54812yes—
578C2×C4#54812yes—
579C2×C4#54812yes—
580C2×C4C2×#13512yes—
581C2×C4C2×#13512yes—
582C2×C4C2×#13512yes—
583C2×C4#57612yes—
584C2×C4#57712yes—
585C2×C4C22×#4111yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C2×C4C2×H(5,𝔽2)11yescharacteristic
589C2×C4C2×2−1+411yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C8—42no—
592C8—42no—
593C8—42no—
594C8—42no—
595C22—24no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22C2×#54811yescharacteristic
610C22C2×#54811yescharacteristic
611C22#348411yescharacteristic
612C22C22×#13511yescharacteristic
613C22C2×#57611yescharacteristic
614C22C2×#57711yescharacteristic
615C22#358611yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—41no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4—82no—
629C4#336014yes—
630C4#336114yes—
631C4C2×#55511yescharacteristic
632C4#351111yescharacteristic
633C4#351111yescharacteristic
634C4#354411yescharacteristic
635C2—44no—
636C2—162no—
637C2#5913511yescharacteristic
638C2C2×#336011yescharacteristic
639C2C2×#336111yescharacteristic
640C2#6134511yescharacteristic
641C2#6134611yescharacteristic
642C2#6156511yescharacteristic
643C2#6166811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#459C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59863C2semidirect
2. #36180C2semidirect
3. #32636C2semidirect
4. #29141C2semidirect
5. #29139C2semidirect
6. #27770C2semidirect
7. #27768C2semidirect
8. #26023C2semidirect
9. #26021C2semidirect
10. #25938C2semidirect
12. #24727C2semidirect
13. #23247C2semidirect
14. #22770C2semidirect
15. #22766C2semidirect
16. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#361802
#326362
#291412
#291392
#277702
#277682
#260232
#260212
#259382
#247312
#247272
#232472
#227702
#227662
C4×#4612
C4×#4592