Finite Group Databasebeta

← #6793571

Subgroups — #6793571

Order 512.

1443 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36134C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#31441C214yesmaximal, complemented
5#31373C214yesmaximal, complemented
6#29936C212yesmaximal, complemented
7#29290C211yesmaximal, characteristic, complemented
8#29219C211yesmaximal, characteristic, complemented
9#28798C212yesmaximal, complemented
10#27824C212yesmaximal, complemented
11#26297C212yesmaximal
12#24970C212yesmaximal
13#24710C212yesmaximal, complemented
14#24303C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic
16#21839C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#533C2212yes—
19C2×#477C2211yescharacteristic
20C2×#475C2212yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2574—22no—
25#2567C2211yescharacteristic
26#2563C2211yescharacteristic
27#2552C2211yescharacteristic
28#2552C2212yes—
29#2542—21no—
30#2538—21no—
31#2538C2211yescharacteristic
32#2508—21no—
33#2497C2212yes—
34#2454—21no—
35#2441C2214yes—
36#2429—22no—
37#2423—22no—
38#2423C2212yes—
39#2396—22no—
40#2365—21no—
41#2354C2212yes—
42Q8×C4⋊C4—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#125—21no—
46C4×C22⋊Q8C2212yes—
47#2293—21no—
48#2285C2212yes—
49C42×Q8C2211yescharacteristic
50C42×D4C2211yescharacteristic
51#2099C2214yes—
52#2099C2214yes—
53#2087C2214yes—
54#2083C2214yes—
55#2077C2214yes—
56#2075C2214yes—
57#2075C2214yes—
58#2059C2214yes—
59#2036C2214yes—
60#2026C2214yes—
61#1992C2214yes—
62#1992C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1988C2212yes—
67#1987C2211yescharacteristic
68#1987C2212yes—
69#1987C2212yes—
70#1954C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1947C2212yes—
75#1947C2212yes—
76#1943C2214yes—
77#1935C2214yes—
78#1924C2214yes—
79#1921C2212yes—
80#1873C2214yes—
81#1863C2212yes—
82#1854C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1825C2214yes—
86#1825C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#533—28no—
96#533—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125—21no—
103C2×#125C2314yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117C2314yes—
120C22×C42C2312yes—
121#477—24no—
122#475—44no—
123#432—44no—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2314yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#399—22no—
142#399—24no—
143#398—22no—
144#398—22no—
145#398—24no—
146#396—22no—
147#396—22no—
148#396—24no—
149#395—22no—
150#393—22no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391C2312yes—
155#390—21no—
156#390—22no—
157#390—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—24no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#388—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#385—22no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#384C2312yes—
188#382—21no—
189#382—22no—
190#380—22no—
191#380—22no—
192#380C2312yes—
193#380C2312yes—
194#380C2314yes—
195#380C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43—21no—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8C2411yescharacteristic
237C22×D4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2412yes—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—44no—
250C4⋊Q8—82no—
251C4⋊Q8#4114yes—
252#125—28no—
253#125—44no—
254#125—44no—
255C22⋊Q8—24no—
256C22⋊Q8—42no—
257C22⋊Q8—48no—
258C22⋊Q8—48no—
259C22⋊Q8—84no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—41no—
266C4×Q8—48no—
267C4×Q8—48no—
268C4×Q8—44no—
269C4×Q8—82no—
270C4×D4—24no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—42no—
274#117—28no—
275#117—44no—
276#117—84no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—24no—
344#107—42no—
345#107—44no—
346#106—44no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—24no—
350C4⋊C8—24no—
351C4⋊C8—42no—
352#103—24no—
353#103—24no—
354#103—44no—
355C22⋊C8—44no—
356C22⋊C8—44no—
357#97—44no—
358C4×C8—24no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—24no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D412yes—
378C2×Q8—24no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q82−1+412yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—24no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—81no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C2×#4114yes—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—84no—
489C42—24no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—41no—
498C42—48no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—44no—
503C42—42no—
504C42—42no—
505C42—44no—
506C42—81no—
507C42#13714yes—
508C42C2×#4112yes—
509C42H(5,𝔽2)12yes—
510C422−1+412yes—
511C422−1+412yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23C23×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—48no—
523Q8—41no—
524Q8—41no—
525Q8—84no—
526Q8—88no—
527D4—44no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—48no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—84no—
579C2×C4#53612yes—
580C2×C4#53612yes—
581C2×C4#54612yes—
582C2×C4#54612yes—
583C2×C4C2×#13712yes—
584C2×C4C2×#13712yes—
585C2×C4C2×#13712yes—
586C2×C4#57512yes—
587C2×C4#57612yes—
588C2×C4C22×#4111yescharacteristic
589C2×C4C2×H(5,𝔽2)11yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4C2×2−1+411yescharacteristic
593C2×C4#58511yescharacteristic
594C8—42no—
595C8—42no—
596C8—44no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22C2×#53611yescharacteristic
605C22C2×#54611yescharacteristic
606C22#348911yescharacteristic
607C22C22×#13711yescharacteristic
608C22C2×#57511yescharacteristic
609C22C2×#57611yescharacteristic
610C22#358611yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—81no—
621C4—82no—
622C4—82no—
623C4—81no—
624C4—82no—
625C4—84no—
626C4—81no—
627C4#323714yes—
628C4#333314yes—
629C4#351811yescharacteristic
630C4#352311yescharacteristic
631C4#355211yescharacteristic
632C4#355411yescharacteristic
633C2—82no—
634C2#3891111yescharacteristic
635C2C2×#323711yescharacteristic
636C2#6088811yescharacteristic
637C2C2×#333311yescharacteristic
638C2#6115911yescharacteristic
639C2#6211311yescharacteristic
640C2#6213611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33196C2semidirect
4. #31441C2semidirect
5. #31373C2semidirect
6. #29936C2semidirect
7. #29290C2semidirect
8. #29219C2semidirect
9. #28798C2semidirect
10. #27824C2semidirect
13. #24710C2semidirect
14. #24303C2semidirect
16. #21839C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#349412
#331962
#314412
#313732
#299362
#292902
#292192
#287982
#278242
#262972
#249702
#247102
#243032
#227992
#218392
C4×#4772