Finite Group Databasebeta

← #6796021

Subgroups — #6796021

Order 512.

1501 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#36188C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#32408C214yesmaximal, complemented
5#31604C214yesmaximal, complemented
6#29937C212yesmaximal, complemented
7#29287C211yesmaximal, characteristic, complemented
8#28706C212yesmaximal
9#28687C211yesmaximal, characteristic, complemented
10#27628C212yesmaximal, complemented
11#26510C212yesmaximal, complemented
12#25453C212yesmaximal, complemented
13#25341C212yesmaximal
14#24838C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic
16#21959C212yesmaximal, complemented
17C4×#475C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#498C2212yes—
20C2×#475C2211yescharacteristic
21C2×#475C2212yes—
22C2×C8×D4C2212yes—
23C22×C4⋊C8C2212yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2574—22no—
27#2568—21no—
28#2567C2211yescharacteristic
29#2563C2211yescharacteristic
30#2552C2211yescharacteristic
31#2547—21no—
32#2547C2212yes—
33#2538C2211yescharacteristic
34#2508—21no—
35#2499—22no—
36#2497C2212yes—
37#2496—22no—
38#2496C2214yes—
39#2454—21no—
40#2423C2212yes—
41#2396—22no—
42#2365—21no—
43#2365C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#125—21no—
47C4×C22⋊Q8C2212yes—
48#2293—21no—
49#2285C2212yes—
50#2285C2212yes—
51C42×Q8C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2100C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2084C2214yes—
57#2084C2214yes—
58#2077C2214yes—
59#2077C2214yes—
60#2077C2214yes—
61#2066C2214yes—
62#2035C2214yes—
63#1990C2212yes—
64#1988C2212yes—
65#1988C2212yes—
66#1988C2211yescharacteristic
67#1987C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2211yescharacteristic
73#1948C2214yes—
74#1948C2214yes—
75#1948C2212yes—
76#1948C2212yes—
77#1939C2212yes—
78#1927C2214yes—
79#1920C2214yes—
80#1919C2212yes—
81#1892C2214yes—
82#1865C2212yes—
83#1850C2214yes—
84#1842C2211yescharacteristic
85#1807C2214yes—
86#1807C2214yes—
87#1807C2214yes—
88#1803C2212yes—
89#1803C2211yescharacteristic
90#1803C2212yes—
91#1797C2212yes—
92C4×#106C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C23×C8C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×#126—24no—
103C2×#125—21no—
104C2×#124—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8C2312yes—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117—22no—
118C22×C42C2312yes—
119#498—44no—
120#475—24no—
121#475—44no—
122C8×D4—44no—
123C2×#106C2312yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2312yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8—28no—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2311yescharacteristic
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C4×C8C2312yes—
141#400—24no—
142#400—22no—
143#399—22no—
144#398—22no—
145#398—22no—
146#397—24no—
147#397—24no—
148#396—22no—
149#396—22no—
150#395—22no—
151#393—22no—
152#391—22no—
153#391—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391C2312yes—
158#390—21no—
159#390—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—24no—
177#387—24no—
178#385—22no—
179#384—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2312yes—
184#384C2312yes—
185#384C2312yes—
186#384C2312yes—
187#382—21no—
188#382—22no—
189#382—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2314yes—
197#380C2314yes—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4—22no—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219C43—21no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8C2411yescharacteristic
238C22×D4C2×D411yescharacteristic
239C23×C4C2×D412yes—
240C23×C4C2×D412yes—
241C23×C4C2412yes—
242C22×C8—28no—
243C22×C8—44no—
244C22×C8—44no—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C4⋊Q8—22no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—44no—
255C4⋊Q8—44no—
256#126—416no—
257#125—44no—
258#124—48no—
259C22⋊Q8—24no—
260C22⋊Q8—42no—
261C22⋊Q8—48no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—41no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×Q8—82no—
271C4×D4—22no—
272C4×D4—22no—
273C4×D4—42no—
274C4×D4SD1618yes—
275#117—48no—
276#117—84no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—21no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#106—24no—
342#106—42no—
343#106—44no—
344#106—44no—
345#106—44no—
346#106—44no—
347C4⋊C8—28no—
348C4⋊C8—24no—
349C4⋊C8—24no—
350C4⋊C8—22no—
351C4⋊C8—42no—
352C4⋊C8#4114yes—
353#103—24no—
354#103—24no—
355#103—44no—
356C22⋊C8—44no—
357C4×C8—24no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—24no—
374#95—24no—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24C22×D412yes—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q82−1+412yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4C2×SD1614yes—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
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×C4C2×SD1618yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2511yescharacteristic
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—88no—
437C2×C8C2×#4114yes—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—22no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—81no—
478C4⋊C4—82no—
479C4⋊C4C2×SD1614yes—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4C2×SD1618yes—
489C42—24no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—41no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—48no—
502C42—44no—
503C42—42no—
504C42—42no—
505C42—44no—
506C42—81no—
507C42C2×SD1614yes—
508C42C2×#4112yes—
509C42H(5,𝔽2)12yes—
510C422−1+412yes—
511C422−1+412yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23C22×SD1614yes—
519C23C23×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524Q8—84no—
525Q8—84no—
526Q8—88no—
527D4—28no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—216no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
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—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—84no—
578C2×C4#53712yes—
579C2×C4#53712yes—
580C2×C4#54712yes—
581C2×C4#54712yes—
582C2×C4C22×SD1614yes—
583C2×C4C22×SD1612yes—
584C2×C4C22×SD1612yes—
585C2×C4C22×SD1612yes—
586C2×C4#57512yes—
587C2×C4#57712yes—
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—24no—
595C8—44no—
596C8—42no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22C2×#53711yescharacteristic
605C22C2×#54711yescharacteristic
606C22#348911yescharacteristic
607C22C23×SD1611yescharacteristic
608C22C2×#57511yescharacteristic
609C22C2×#57711yescharacteristic
610C22#358611yescharacteristic
611C4—24no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—82no—
620C4—81no—
621C4—82no—
622C4—81no—
623C4—82no—
624C4—84no—
625C4—81no—
626C4—82no—
627C4#324614yes—
628C4#333514yes—
629C4#351411yescharacteristic
630C4#352311yescharacteristic
631C4#355211yescharacteristic
632C4#355511yescharacteristic
633C2—44no—
634C2#3892311yescharacteristic
635C2C2×#324611yescharacteristic
636C2#6092311yescharacteristic
637C2C2×#333511yescharacteristic
638C2#6125711yescharacteristic
639C2#6209911yescharacteristic
640C2#6213511yescharacteristic
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. #32408C2semidirect
5. #31604C2semidirect
6. #29937C2semidirect
7. #29287C2semidirect
9. #28687C2semidirect
10. #27628C2semidirect
11. #26510C2semidirect
12. #25453C2semidirect
14. #24838C2semidirect
16. #21959C2semidirect
17. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#349412
#331962
#324082
#316042
#299372
#292872
#287062
#286872
#276282
#265102
#254532
#253412
#248382
D4×C4⋊C82
#219592
C4×#4752