Finite Group Databasebeta

← #7567481

Subgroups — #7567481

Order 512.

1643 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic, complemented
3#32573C214yesmaximal, complemented
4#29208C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28792C212yesmaximal, complemented
7#27814C212yesmaximal, complemented
8#26463C212yesmaximal, complemented
9#26111C212yesmaximal, complemented
10#26041C212yesmaximal, complemented
11#24812C212yesmaximal, complemented
12#24335C214yesmaximal, complemented
13#23247C211yesmaximal, characteristic, complemented
14#22995C212yesmaximal
15C4⋊C4×D8C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#478C2214yes—
20C2×#476C2211yescharacteristic
21C2×#459C2211yescharacteristic
22C2×C4×D8C2212yes—
23C2×C4×D8C2214yes—
24#2868C2211yescharacteristic
25#2857—21no—
26#2804—22no—
27#2580—22no—
28#2580C2214yes—
29#2571C2212yes—
30#2567C2212yes—
31#2563—21no—
32#2563C2212yes—
33#2552C2212yes—
34#2552C2212yes—
35#2502—22no—
36#2487—22no—
37#2440—22no—
38#2440—22no—
39#2366C2212yes—
40D4×C4⋊C4—21no—
41D4×C4⋊C4C2212yes—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C2×#384—21no—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2087C2214yes—
51#2085C2214yes—
52#2065C2214yes—
53#2059C2214yes—
54#2057C2214yes—
55#2025C2214yes—
56#2023C2214yes—
57#1991C2214yes—
58#1990C2212yes—
59#1987C2211yescharacteristic
60#1984C2214yes—
61#1982C2211yescharacteristic
62#1957C2212yes—
63#1948C2211yescharacteristic
64#1947C2212yes—
65#1947C2212yes—
66#1947C2211yescharacteristic
67#1942C2214yes—
68#1939C2212yes—
69#1939C2212yes—
70#1934C2214yes—
71#1925C2212yes—
72#1920C2212yes—
73#1879C2212yes—
74#1879C2214yes—
75#1864C2212yes—
76#1862C2214yes—
77#1862C2214yes—
78#1862C2214yes—
79#1845C2212yes—
80#1834C2212yes—
81#1804C2214yes—
82#1802C2211yescharacteristic
83#1802C2212yes—
84#1802C2212yes—
85#1802C2211yescharacteristic
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C8×C4⋊C4C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90C4×C4⋊C8C2211yescharacteristic
91C4×#102C2212yes—
92C4×#102C2212yes—
93C22×D8C2312yes—
94#554—44no—
95#554—44no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8D411yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#127C2311yescharacteristic
102C2×#125C2314yes—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—24no—
105C2×C4×Q8—21no—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2314yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2314yes—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C42—21no—
121#478—48no—
122#476—24no—
123#459Q818yescomplemented
124C4×D8—28no—
125C4×D8—48no—
126C2×#107C2312yes—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2312yes—
130C2×#106C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2312yes—
144C2×C4×C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—24no—
148#399—24no—
149#398—24no—
150#398—24no—
151#395—24no—
152#393—24no—
153#393—24no—
154#391—22no—
155#391—21no—
156#391D412yes—
157#390—22no—
158#390—21no—
159#390—21no—
160#390C2312yes—
161#390C2314yes—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—21no—
166#389D412yes—
167#389D412yes—
168#389C2312yes—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#386—24no—
173#385—24no—
174#385—24no—
175#385—24no—
176#385—24no—
177#385—24no—
178#384—22no—
179#384—22no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384D411yescharacteristic
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#382—22no—
189#382C2314yes—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2311yescharacteristic
199C4×C4⋊C4C2311yescharacteristic
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—21no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205C43C2311yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2312yes—
210#340C2314yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2312yes—
217#336C2314yes—
218#336C2314yes—
219#336C2312yes—
220C22×Q8—21no—
221C22×Q8—22no—
222C22×Q8C2×D412yes—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2412yes—
226C22×D4C2411yescharacteristic
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—24no—
233C2×D8—24no—
234C2×D8—24no—
235C2×D8C2×Q818yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2412yes—
240C4⋊Q8—24no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—42no—
244C4⋊Q8—42no—
245C4⋊Q8—82no—
246C4⋊Q8—82no—
247#127—22no—
248#127C2×Q814yes—
249#125—28no—
250#125—44no—
251C22⋊Q8—416no—
252C22⋊Q8—48no—
253C4×Q8—42no—
254C4×Q8—42no—
255C4×Q8—82no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—28no—
259C4×D4—24no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—24no—
263C4×D4—44no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—82no—
268C4×D4—84no—
269C4×D4C2×Q814yes—
270C4×D4#4114yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—84no—
289C2×C4⋊C4—84no—
290C2×C4⋊C4—84no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—44no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2414yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—24no—
328#107—44no—
329#107—44no—
330#106—24no—
331#106—24no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—44no—
336C4⋊C8C2×Q814yes—
337#102—24no—
338#102—24no—
339#102—24no—
340#102—24no—
341#102—24no—
342#102—44no—
343#102—44no—
344#102C2×Q818yes—
345C4×C8—24no—
346C4×C8—24no—
347C4×C8—44no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—21no—
361C24—22no—
362C2×Q8—24no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—44no—
366C2×Q8—42no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×Q8—82no—
370C2×Q8—82no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4C22×Q814yes—
386C2×D4C22×Q812yes—
387C2×D4C2×#4112yes—
388C2×D4H(5,𝔽2)12yes—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4C22×D414yes—
415C22×C4C22×D414yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2511yescharacteristic
423D8—216no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8C22×Q814yes—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4C22×Q812yes—
470C4⋊C4C2×#4112yes—
471C4⋊C4H(5,𝔽2)12yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—82no—
481C22⋊C4—84no—
482C42—22no—
483C42—22no—
484C42—24no—
485C42—24no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—42no—
490C42—42no—
491C42—42no—
492C42—44no—
493C42#13514yes—
494C42C22×Q812yes—
495C42C2×#4112yes—
496C422−1+412yes—
497C422−1+412yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23C23×D411yescharacteristic
513Q8—84no—
514Q8—84no—
515Q8—164no—
516D4—24no—
517D4—22no—
518D4—24no—
519D4—22no—
520D4—22no—
521D4—48no—
522D4—42no—
523D4—42no—
524D4—84no—
525D4#55414yes—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—48no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—48no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—88no—
566C2×C4—168no—
567C2×C4—168no—
568C2×C4#54812yes—
569C2×C4#54812yes—
570C2×C4D4×Q812yes—
571C2×C4D4×Q812yes—
572C2×C4C2×#13512yes—
573C2×C4C2×#13512yes—
574C2×C4C2×#13512yes—
575C2×C4#57612yes—
576C2×C4#57612yes—
577C2×C4C23×Q811yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C2×H(5,𝔽2)11yescharacteristic
580C2×C4C2×H(5,𝔽2)11yescharacteristic
581C2×C4C2×2−1+411yescharacteristic
582C2×C4C2×2−1+411yescharacteristic
583C8—24no—
584C8—42no—
585C8—42no—
586C8—42no—
587C22—24no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22C2×#54811yescharacteristic
602C22C2×D4×Q811yescharacteristic
603C22#348411yescharacteristic
604C22C22×#13511yescharacteristic
605C22C2×#57611yescharacteristic
606C22C2×#57611yescharacteristic
607C22#358611yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—44no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—41no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4#336114yes—
622C4#338014yes—
623C4C2×#55411yescharacteristic
624C4#350311yescharacteristic
625C4#351111yescharacteristic
626C4#354411yescharacteristic
627C2—44no—
628C2—84no—
629C2#5913611yescharacteristic
630C2C2×#336111yescharacteristic
631C2#6134511yescharacteristic
632C2C2×#338011yescharacteristic
633C2#6138111yescharacteristic
634C2#6156511yescharacteristic
635C2#6166111yescharacteristic
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. #59864C2semidirect
2. #36180C2semidirect
3. #32573C2semidirect
4. #29208C2semidirect
5. #29139C2semidirect
6. #28792C2semidirect
7. #27814C2semidirect
8. #26463C2semidirect
9. #26111C2semidirect
10. #26041C2semidirect
11. #24812C2semidirect
12. #24335C2semidirect
13. #23247C2semidirect
15. C4⋊C4×D8C2semidirect
16. C4×#476C2semidirect
123. #459Q8semidirect

Maximal subgroups

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

subgroupindex
#598642
#361802
#325732
#292082
#291392
#287922
#278142
#264632
#261112
#260412
#248122
#243352
#232472
#229952
C4⋊C4×D82
C4×#4762
C4×#4592