Finite Group Databasebeta

← #7533925

Subgroups — #7533925

Order 512.

1711 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic, complemented
2#36194C211yesmaximal, characteristic, complemented
3#31547C214yesmaximal, complemented
4#29206C211yesmaximal, characteristic, complemented
5#29139C211yesmaximal, characteristic, complemented
6#28797C212yesmaximal, complemented
7#28273C212yesmaximal, complemented
8#27019C214yesmaximal, complemented
9#26533C212yesmaximal, complemented
10#26024C212yesmaximal, complemented
11#25059C212yesmaximal, complemented
12#24326C212yesmaximal, complemented
13#23153C211yesmaximal, characteristic, complemented
14#23099C212yesmaximal
15#22523C212yesmaximal, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#459C211yesmaximal, characteristic
18C2×D4×Q8—21no—
19C2×D4×Q8C2212yes—
20C2×#474C2211yescharacteristic
21C2×#459C2211yescharacteristic
22C2×#446C2212yes—
23C2×C4×D8C2212yes—
24#2868C2211yescharacteristic
25#2857—21no—
26#2683—22no—
27#2571C2212yes—
28#2563C2212yes—
29#2551C2212yes—
30#2538—21no—
31#2538C2212yes—
32#2538C2212yes—
33#2497—22no—
34#2492—22no—
35#2452—22no—
36#2452C2214yes—
37#2439—22no—
38#2409—22no—
39#2369C2212yes—
40#2366C2212yes—
41#2355C2212yes—
42#2354C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46#2285—21no—
47C42×D4C2211yescharacteristic
48#2092C2214yes—
49#2085C2214yes—
50#2065C2214yes—
51#2065C2214yes—
52#2058C2214yes—
53#2057C2214yes—
54#2023C2214yes—
55#2023C2214yes—
56#1993C2214yes—
57#1990C2212yes—
58#1988C2211yescharacteristic
59#1987C2214yes—
60#1984C2212yes—
61#1982C2211yescharacteristic
62#1972C2214yes—
63#1957C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1947C2211yescharacteristic
67#1947C2211yescharacteristic
68#1944C2214yes—
69#1939C2212yes—
70#1926C2214yes—
71#1925C2212yes—
72#1920C2214yes—
73#1918C2212yes—
74#1917C2214yes—
75#1879C2214yes—
76#1879C2212yes—
77#1862C2214yes—
78#1862C2212yes—
79#1844C2212yes—
80#1834C2212yes—
81#1804C2214yes—
82#1802C2212yes—
83#1802C2211yescharacteristic
84#1802C2212yes—
85#1802C2211yescharacteristic
86C4×#107C2212yes—
87#1770C2212yes—
88C4×C4⋊C8C2211yescharacteristic
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93C23×Q8—22no—
94C22×D8C2312yes—
95D4×Q8—28no—
96D4×Q8—44no—
97D4×Q8—48no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8D411yescharacteristic
102C2×C4⋊Q8C2314yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2314yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C22×C42—21no—
123#474—24no—
124#459Q818yescomplemented
125#446—44no—
126C4×D8—44no—
127C2×#107C2312yes—
128C2×#107C2312yes—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2311yescharacteristic
133C2×C4⋊C8C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2311yescharacteristic
137C2×#102C2312yes—
138C2×#102C2314yes—
139C2×#102C2312yes—
140C2×#102C2312yes—
141C2×#102C2314yes—
142C2×#102C2312yes—
143C2×#102C2312yes—
144C2×#102C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148#400—24no—
149#396—24no—
150#396—24no—
151#396—24no—
152#395—24no—
153#393—24no—
154#393—24no—
155#391—22no—
156#391—21no—
157#391—24no—
158#391D412yes—
159#390—22no—
160#390—21no—
161#390—21no—
162#390C2312yes—
163#390C2314yes—
164#390C2314yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—21no—
169#389D412yes—
170#389D412yes—
171#389C2312yes—
172#389C2314yes—
173#389C2314yes—
174#388—24no—
175#387—24no—
176#386—24no—
177#385—24no—
178#384—21no—
179#384—21no—
180#384D411yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382C2314yes—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2314yes—
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#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2314yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#336C2314yes—
217#336C2312yes—
218#336C2314yes—
219#336C2312yes—
220C22×Q8—21no—
221C22×Q8—24no—
222C22×Q8—22no—
223C22×Q8—48no—
224C22×Q8—48no—
225C22×Q8—84no—
226C22×Q8C2×D412yes—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4C2412yes—
230C22×D4C2411yescharacteristic
231C23×C4—21no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C2×D8—42no—
237C2×D8—42no—
238C2×D8C2×Q818yescomplemented
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C4⋊Q8—28no—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—42no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8—48no—
250C4⋊Q8—82no—
251C4⋊Q8—82no—
252C4⋊Q8#4114yes—
253#127—22no—
254#127C2×Q814yes—
255C22⋊Q8—416no—
256C22⋊Q8—48no—
257C22⋊Q8—416no—
258C22⋊Q8—48no—
259C22⋊Q8—84no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—44no—
263C4×D4—28no—
264C4×D4—24no—
265C4×D4—22no—
266C4×D4—24no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—44no—
272C4×D4—48no—
273C4×D4—44no—
274C4×D4—82no—
275C4×D4C2×Q814yescomplemented
276#117—84no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—24no—
320C2×C42—44no—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2414yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—24no—
330#107—42no—
331#107—42no—
332#106—44no—
333#106—44no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—44no—
338C4⋊C8C2×Q814yescomplemented
339#102—24no—
340#102—42no—
341#102—42no—
342#102—44no—
343#102—44no—
344#102—42no—
345#102—42no—
346#102C2×Q818yescomplemented
347#97—44no—
348C4×C8—24no—
349C4×C8—24no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—24no—
354#95—24no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—21no—
363C24—22no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—44no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—84no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—832no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—21no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D4C22×Q814yes—
394C2×D4C22×Q812yes—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—84no—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D414yes—
419C22×C4C22×D414yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C2511yescharacteristic
426D8—48no—
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×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8C22×Q814yes—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—28no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C22×Q812yes—
472C4⋊C4C2×#4114yes—
473C22⋊C4—24no—
474C22⋊C4—48no—
475C22⋊C4—42no—
476C22⋊C4—48no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—82no—
483C22⋊C4—84no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—24no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—44no—
495C42—84no—
496C42AGL(1,ℤ/8)14yes—
497C42C22×Q812yes—
498C42C2×#4112yes—
499C42H(5,𝔽2)12yes—
500C422−1+412yes—
501C23—22no—
502C23—24no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—42no—
515C23C23×D411yescharacteristic
516Q8—48no—
517Q8—48no—
518Q8—816no—
519Q8—164no—
520D4—22no—
521D4—24no—
522D4—22no—
523D4—24no—
524D4—48no—
525D4—41no—
526D4—42no—
527D4—41no—
528D4—42no—
529D4—84no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—24no—
543C2×C4—21no—
544C2×C4—24no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—84no—
568C2×C4—816no—
569C2×C4—84no—
570C2×C4—88no—
571C2×C4#54612yes—
572C2×C4#54612yes—
573C2×C4D4×Q812yes—
574C2×C4D4×Q812yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4#57612yes—
579C2×C4#57712yes—
580C2×C4C23×Q811yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4#58511yescharacteristic
585C2×C4#58511yescharacteristic
586C8—24no—
587C8—42no—
588C8—42no—
589C8—42no—
590C22—42no—
591C22—44no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—82no—
597C22—81no—
598C22—81no—
599C22—82no—
600C22—82no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22C2×#54611yescharacteristic
605C22C2×D4×Q811yescharacteristic
606C22#348711yescharacteristic
607C22C22×AGL(1,ℤ/8)11yescharacteristic
608C22C2×#57611yescharacteristic
609C22C2×#57711yescharacteristic
610C22#358611yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—44no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—41no—
619C4—82no—
620C4—81no—
621C4—81no—
622C4—82no—
623C4—84no—
624C4#332514yes—
625C4#338414yes—
626C4Q8×#4111yescharacteristic
627C4#350111yescharacteristic
628C4#351811yescharacteristic
629C4#354711yescharacteristic
630C2—82no—
631C2—84no—
632C2#5903611yescharacteristic
633C2C2×#332511yescharacteristic
634C2#6120211yescharacteristic
635C2C2×#338411yescharacteristic
636C2#6138111yescharacteristic
637C2#6188811yescharacteristic
638C2#6193711yescharacteristic
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. #59817C2semidirect
2. #36194C2semidirect
3. #31547C2semidirect
4. #29206C2semidirect
5. #29139C2semidirect
6. #28797C2semidirect
7. #28273C2semidirect
8. #27019C2semidirect
9. #26533C2semidirect
10. #26024C2semidirect
11. #25059C2semidirect
12. #24326C2semidirect
13. #23153C2semidirect
15. #22523C2semidirect
16. C4×#474C2semidirect
124. #459Q8semidirect
238. C2×D8C2×Q8semidirect
275. C4×D4C2×Q8semidirect
338. C4⋊C8C2×Q8semidirect
346. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#361942
#315472
#292062
#291392
#287972
#282732
#270192
#265332
#260242
#250592
#243262
#231532
#230992
#225232
C4×#4742
C4×#4592