Finite Group Databasebeta

← #7567211

Subgroups — #7567211

Order 512.

1575 conjugacy classes of subgroups.

The subgroup lattice has 630 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#32237C214yesmaximal, complemented
4#29218C211yesmaximal, characteristic, complemented
5#28716C211yesmaximal, characteristic, complemented
6#28640C212yesmaximal, complemented
7#27814C212yesmaximal, complemented
8#27557C214yesmaximal, complemented
9#26279C212yesmaximal, complemented
10#26021C212yesmaximal, complemented
11#25930C212yesmaximal
12#24812C212yesmaximal, complemented
13#23149C211yesmaximal, characteristic, complemented
14#22766C212yesmaximal, complemented
15#22519C212yesmaximal, complemented
16C4×#476C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×#554C2212yes—
19C2×#486C2212yes—
20C2×#476C2211yescharacteristic
21C2×#459C2211yescharacteristic
22C2×C8×Q8C2212yes—
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2857—21no—
26#2808—22no—
27#2592—22no—
28#2567C2212yes—
29#2563C2212yes—
30#2552C2212yes—
31#2552C2212yes—
32#2548C2212yes—
33#2547—21no—
34#2513C2214yes—
35#2502—22no—
36#2485—22no—
37#2440—22no—
38#2440—22no—
39#2355C2212yes—
40#2354—21no—
41#2354C2212yes—
42D4×C4⋊C4C2212yes—
43D4×C4⋊C4C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C2×#384—21no—
48C42×D4C2211yescharacteristic
49#2101C2214yes—
50#2100C2214yes—
51#2085C2214yes—
52#2071C2214yes—
53#2065C2214yes—
54#2065C2214yes—
55#2025C2214yes—
56#2025C2214yes—
57#1991C2214yes—
58#1990C2212yes—
59#1990C2212yes—
60#1989C2214yes—
61#1987C2212yes—
62#1987C2212yes—
63#1987C2211yescharacteristic
64#1984C2212yes—
65#1982C2211yescharacteristic
66#1948C2211yescharacteristic
67#1947C2211yescharacteristic
68#1939C2212yes—
69#1934C2214yes—
70#1934C2214yes—
71#1925C2212yes—
72#1920C2212yes—
73#1919C2214yes—
74#1918C2214yes—
75#1916C2214yes—
76#1879C2214yes—
77#1879C2214yes—
78#1879C2212yes—
79#1864C2212yes—
80#1834C2212yes—
81#1804C2214yes—
82#1802C2212yes—
83#1802C2212yes—
84#1802C2211yescharacteristic
85#1802C2211yescharacteristic
86C4×#106C2212yes—
87C8×C4⋊C4C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90C4×#102C2212yes—
91C4×#102C2212yes—
92C4×#102C2214yes—
93C22×D8C2312yes—
94#554—28no—
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×#124—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—24no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2312yes—
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2314yes—
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C42—21no—
120#486—44no—
121#476—24no—
122#459—24no—
123C8×Q8—44no—
124C4×D8—28no—
125C2×#107C2312yes—
126C2×#106C2312yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#398—24no—
145#398—24no—
146#397—24no—
147#397—24no—
148#395—24no—
149#394—24no—
150#393—24no—
151#391—22no—
152#391—22no—
153#391—21no—
154#391D412yes—
155#390—22no—
156#390—21no—
157#390—21no—
158#390C2312yes—
159#390C2314yes—
160#390C2314yes—
161#389—22no—
162#389—22no—
163#389—21no—
164#389D412yes—
165#389D412yes—
166#389C2312yes—
167#389C2314yes—
168#389C2314yes—
169#388—24no—
170#388—24no—
171#388—24no—
172#387—24no—
173#385—24no—
174#385—24no—
175#384—22no—
176#384—22no—
177#384—22no—
178#384—21no—
179#384—21no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#382C2314yes—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2311yescharacteristic
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202C4×C22⋊C4—21no—
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2312yes—
207#340C2312yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2312yes—
217#336C2314yes—
218#336C2312yes—
219#336C2314yes—
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—
232C2×D8—24no—
233C2×D8—24no—
234C2×D8—24no—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241C4⋊Q8—22no—
242C4⋊Q8—24no—
243C4⋊Q8—22no—
244C4⋊Q8—82no—
245C4⋊Q8—82no—
246#127—22no—
247#127—22no—
248#125—28no—
249#125—44no—
250#124—84no—
251C22⋊Q8—416no—
252C22⋊Q8—48no—
253C4×Q8—24no—
254C4×Q8—24no—
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—82no—
266C4×D4#4114yes—
267C4×D4#4114yes—
268#117—84no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—84no—
286C2×C4⋊C4—84no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D412yes—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2411yescharacteristic
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2414yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325#107—24no—
326#106—24no—
327#106—24no—
328C4⋊C8—22no—
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—44no—
333#102—24no—
334#102—24no—
335#102—24no—
336#102—24no—
337#102—24no—
338#102—24no—
339#102—44no—
340#102—44no—
341C4×C8—24no—
342C4×C8—24no—
343C4×C8—44no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—24no—
348#95—22no—
349#95—24no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95C2×D414yes—
355#95C2×D414yes—
356C24—21no—
357C24—22no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—44no—
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×Q8—82no—
366C2×Q8—82no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4C2×#4112yes—
383C2×D4C2×#4112yes—
384C2×D42−1+412yes—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—48no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4C22×D414yes—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D414yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2511yescharacteristic
416D8—48no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C4⋊C4—24no—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4C2×#4112yes—
462C4⋊C4C2×#4112yes—
463C4⋊C42−1+412yes—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—82no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C42—22no—
476C42—22no—
477C42—24no—
478C42—24no—
479C42—21no—
480C42—21no—
481C42—41no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—44no—
486C42—84no—
487C42#13514yes—
488C42C2×#4112yes—
489C42C2×#4112yes—
490C42H(5,𝔽2)12yes—
491C422−1+412yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C23×D411yescharacteristic
507Q8—48no—
508Q8—84no—
509Q8—164no—
510D4—22no—
511D4—22no—
512D4—22no—
513D4—42no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—84no—
518D4—84no—
519D4#55614yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—21no—
537C2×C4—24no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—88no—
560C2×C4—168no—
561C2×C4#54612yes—
562C2×C4#54612yes—
563C2×C4#54812yes—
564C2×C4#54812yes—
565C2×C4C2×#13512yes—
566C2×C4C2×#13512yes—
567C2×C4C2×#13512yes—
568C2×C4#57612yes—
569C2×C4#57712yes—
570C2×C4C22×#4111yescharacteristic
571C2×C4C22×#4111yescharacteristic
572C2×C4C2×H(5,𝔽2)11yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C2×C4C2×2−1+411yescharacteristic
575C2×C4C2×2−1+411yescharacteristic
576C8—42no—
577C8—42no—
578C8—42no—
579C8—42no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22C2×#54611yescharacteristic
595C22C2×#54811yescharacteristic
596C22#348611yescharacteristic
597C22C22×#13511yescharacteristic
598C22C2×#57611yescharacteristic
599C22C2×#57711yescharacteristic
600C22#358511yescharacteristic
601C4—22no—
602C4—22no—
603C4—22no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—44no—
609C4—42no—
610C4—84no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#330314yes—
615C4#336114yes—
616C4C2×#55611yescharacteristic
617C4#350911yescharacteristic
618C4#351111yescharacteristic
619C4#355211yescharacteristic
620C2—44no—
621C2—162no—
622C2#5913411yescharacteristic
623C2C2×#330311yescharacteristic
624C2#6120511yescharacteristic
625C2C2×#336111yescharacteristic
626C2#6134711yescharacteristic
627C2#6181911yescharacteristic
628C2#6186811yescharacteristic
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. #32237C2semidirect
4. #29218C2semidirect
5. #28716C2semidirect
6. #28640C2semidirect
7. #27814C2semidirect
8. #27557C2semidirect
9. #26279C2semidirect
10. #26021C2semidirect
12. #24812C2semidirect
13. #23149C2semidirect
14. #22766C2semidirect
15. #22519C2semidirect
16. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#598632
#361802
#322372
#292182
#287162
#286402
#278142
#275572
#262792
#260212
#259302
#248122
#231492
#227662
#225192
C4×#4762
C4×#4592