Finite Group Databasebeta

← #6795706

Subgroups — #6795706

Order 512.

1485 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#34945C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#31708C214yesmaximal, complemented
5#31103C214yesmaximal, complemented
6#29941C212yesmaximal, complemented
7#29286C211yesmaximal, characteristic, complemented
8#29208C211yesmaximal, characteristic, complemented
9#28835C212yesmaximal, complemented
10#27737C212yesmaximal, complemented
11#27648C212yesmaximal, complemented
12#26298C212yesmaximal
13#25016C212yesmaximal
14#24710C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic
16#21987C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#554—21no—
19C2×#484C2214yes—
20C2×#475C2211yescharacteristic
21C2×#475C2212yes—
22C2×#432C2212yes—
23#2868C2211yescharacteristic
24#2857C2212yes—
25#2573—22no—
26#2571C2211yescharacteristic
27#2568C2211yescharacteristic
28#2567—21no—
29#2552—21no—
30#2552C2212yes—
31#2547C2211yescharacteristic
32#2538C2211yescharacteristic
33#2514—21no—
34#2502—22no—
35#2494C2212yes—
36#2447—21no—
37#2440—22no—
38#2440C2214yes—
39#2421C2212yes—
40#2401—22no—
41D4×C4⋊C4—21no—
42D4×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#124—21no—
46C4×C22⋊Q8C2212yes—
47#2295—21no—
48C2×#384C2212yes—
49#2283C2212yes—
50C42×Q8C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2101C2214yes—
53#2086C2214yes—
54#2085C2214yes—
55#2083C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2076C2214yes—
59#2072C2214yes—
60#2057C2214yes—
61#2036C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1988C2212yes—
65#1988C2211yescharacteristic
66#1988C2212yes—
67#1954C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1935C2214yes—
77#1924C2214yes—
78#1919C2212yes—
79#1865C2212yes—
80#1853C2214yes—
81#1852C2214yes—
82#1847C2214yes—
83#1845C2212yes—
84#1842C2211yescharacteristic
85#1836C2214yes—
86#1824C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91C4×#106C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#554—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125—22no—
102C2×#124—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×D4—22no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4C2314yes—
118C22×C42C2312yes—
119#484—48no—
120#475—24no—
121#475—44no—
122#432—28no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2311yescharacteristic
134C2×#103C2314yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2314yes—
140C2×C4×C8C2312yes—
141#401—22no—
142#400—22no—
143#400—22no—
144#400—22no—
145#398—24no—
146#398—24no—
147#397—22no—
148#397—22no—
149#397—22no—
150#396—22no—
151#395—24no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—22no—
158#391C2312yes—
159#390—21no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#385—22no—
177#385—24no—
178#385—24no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#384C2314yes—
188#383—22no—
189#382—21no—
190#382C2312yes—
191#382C2312yes—
192#381—22no—
193#380—22no—
194#380C2312yes—
195#380C2314yes—
196#380C2312yes—
197#380C2312yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
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⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217#376—22no—
218C43—21no—
219C43C2311yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2411yescharacteristic
237C22×D4C2×D411yescharacteristic
238C23×C4—24no—
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C22×C8C2412yes—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253#125—48no—
254#124—44no—
255C22⋊Q8—24no—
256C22⋊Q8—416no—
257C22⋊Q8—42no—
258C22⋊Q8—48no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—41no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—48no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4—84no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337#107—44no—
338#107—44no—
339#107—44no—
340#106—24no—
341#106—42no—
342#106—44no—
343C4⋊C8—24no—
344C4⋊C8—24no—
345C4⋊C8—22no—
346C4⋊C8—42no—
347C4⋊C8#4114yes—
348#103—28no—
349#103—24no—
350#103—24no—
351C22⋊C8—28no—
352C22⋊C8—44no—
353#97—28no—
354C4×C8—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24C22×D412yes—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×Q8H(5,𝔽2)12yes—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4#13714yes—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—42no—
417C22×C4—48no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2511yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—24no—
437C2×C8—22no—
438C2×C8C2×#4114yes—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—81no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—82no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—21no—
494C42—22no—
495C42—21no—
496C42—41no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—44no—
501C42—44no—
502C42—42no—
503C42—42no—
504C42—81no—
505C42AGL(1,ℤ/8)14yes—
506C42C2×#4112yes—
507C422−1+412yes—
508C422−1+412yes—
509C422−1+412yes—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23C2×#13714yes—
517C23C23×D411yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522Q8—84no—
523Q8—84no—
524Q8—164no—
525D4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—48no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—168no—
578C2×C4#53612yes—
579C2×C4#53612yes—
580C2×C4#54712yes—
581C2×C4#54712yes—
582C2×C4C2×AGL(1,ℤ/8)12yes—
583C2×C4C2×AGL(1,ℤ/8)12yes—
584C2×C4C2×AGL(1,ℤ/8)12yes—
585C2×C4C2×#13712yes—
586C2×C4#57812yes—
587C2×C4C22×#4111yescharacteristic
588C2×C4C2×H(5,𝔽2)11yescharacteristic
589C2×C4C2×2−1+411yescharacteristic
590C2×C4C2×2−1+411yescharacteristic
591C2×C4C2×2−1+411yescharacteristic
592C2×C4#58511yescharacteristic
593C8—24no—
594C8—44no—
595C8—42no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22C2×#53611yescharacteristic
604C22C2×#54711yescharacteristic
605C22#348811yescharacteristic
606C22C22×AGL(1,ℤ/8)11yescharacteristic
607C22C22×#13711yescharacteristic
608C22C2×#57811yescharacteristic
609C22#358511yescharacteristic
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4—81no—
622C4—82no—
623C4—81no—
624C4—82no—
625C4—81no—
626C4#323914yes—
627C4#335314yes—
628C4#351511yescharacteristic
629C4#352211yescharacteristic
630C4#354411yescharacteristic
631C4#355511yescharacteristic
632C2—44no—
633C2#3892111yescharacteristic
634C2C2×#323911yescharacteristic
635C2#6089311yescharacteristic
636C2C2×#335311yescharacteristic
637C2#6125411yescharacteristic
638C2#6196311yescharacteristic
639C2#6198411yescharacteristic
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. #31708C2semidirect
5. #31103C2semidirect
6. #29941C2semidirect
7. #29286C2semidirect
8. #29208C2semidirect
9. #28835C2semidirect
10. #27737C2semidirect
11. #27648C2semidirect
14. #24710C2semidirect
16. #21987C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349452
#331962
#317082
#311032
#299412
#292862
#292082
#288352
#277372
#276482
#262982
#250162
#247102
#227992
#219872
C4×#4752