Finite Group Databasebeta

← #6556036

Subgroups — #6556036

Order 512.

1563 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#36188C211yesmaximal, characteristic, complemented
2#34321C211yesmaximal, characteristic
3#30166C212yesmaximal, complemented
4#29864C211yesmaximal, characteristic, complemented
5#29304C212yesmaximal, complemented
6#29273C211yesmaximal, characteristic, complemented
7#28687C211yesmaximal, characteristic, complemented
8#27615C212yesmaximal, complemented
9#27482C214yesmaximal, complemented
10#27054C214yesmaximal, complemented
11#26303C212yesmaximal, complemented
12#25700C212yesmaximal
13#23774C211yesmaximal, characteristic, complemented
14#23761C212yesmaximal, complemented
15#23163C212yesmaximal, complemented
16#22968C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#555C2211yescharacteristic
20C2×#498C2212yes—
21C2×#474C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2568—21no—
25#2547—21no—
26#2547C2212yes—
27#2499—22no—
28#2496—22no—
29#2496C2214yes—
30#2369C2211yescharacteristic
31#2367—22no—
32#2365—21no—
33#2365C2212yes—
34#2363C2212yes—
35#2361—21no—
36#2354C2211yescharacteristic
37D4×C4⋊C4C2211yescharacteristic
38#2347—21no—
39#2347C2212yes—
40#2337—22no—
41#2324—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#121C2212yes—
45#2296C2212yes—
46#2293C2211yescharacteristic
47#2285C2212yes—
48#2274—21no—
49C42×D4C2211yescharacteristic
50#2099C2212yes—
51#2099C2211yescharacteristic
52#2093C2211yescharacteristic
53#2086C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2077C2214yes—
59#2076C2214yes—
60#2071C2212yes—
61#2071C2212yes—
62#2067C2212yes—
63#2058C2212yes—
64#2057C2211yescharacteristic
65#2034C2214yes—
66#2034C2214yes—
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1988C2212yes—
70#1987C2212yes—
71#1954C2212yes—
72#1948C2214yes—
73#1948C2212yes—
74#1944C2214yes—
75#1943C2214yes—
76#1924C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1864C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1831C2211yescharacteristic
86#1825C2214yes—
87#1817C2212yes—
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1762C2214yes—
93C4×#102C2212yes—
94#1749C2212yes—
95#556—48no—
96#555—24no—
97#555—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#126—24no—
104C2×#124—22no—
105C2×#124—22no—
106C2×C22⋊Q8—22no—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×D4—22no—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—22no—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4—21no—
124C22×C42C2312yes—
125#498—28no—
126#474—28no—
127C2×#107C2312yes—
128C2×#107C2314yes—
129C2×#107C2312yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#397—24no—
144#397—24no—
145#391—22no—
146#391—22no—
147#391—21no—
148#391—21no—
149#391—22no—
150#391—22no—
151#390—21no—
152#390—22no—
153#390C2311yescharacteristic
154#390C2311yescharacteristic
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389C2312yes—
160#389C2312yes—
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2314yes—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—21no—
170#388—22no—
171#388—21no—
172#388—22no—
173#387—24no—
174#387—24no—
175#386—22no—
176#386—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#384—21no—
181#384—21no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#383—22no—
187#382—22no—
188#380—22no—
189#380—22no—
190#380—21no—
191#380C2312yes—
192#380C2314yes—
193#380C2314yes—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—22no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4C2312yes—
216C43C2311yescharacteristic
217#340C2312yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#336C2311yescharacteristic
229#336C2312yes—
230#336C2312yes—
231#336C2311yescharacteristic
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×D4—22no—
237C22×D4C2×D411yescharacteristic
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—22no—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C22×C8C2412yes—
247C4⋊Q8—22no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—42no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8#4114yes—
254#126—416no—
255#124—216no—
256#124—48no—
257C22⋊Q8—48no—
258#121—28no—
259#121—24no—
260#121—24no—
261C4×Q8—48no—
262C4×Q8—44no—
263C4×Q8—82no—
264C4×D4—24no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—22no—
268C4×D4—42no—
269C4×D4—44no—
270C4×D4—84no—
271C4×D4#4114yes—
272#117—28no—
273#117—24no—
274#117—24no—
275#117—48no—
276#117—44no—
277#117—44no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—24no—
342#107—28no—
343#107—42no—
344#106—28no—
345#106—28no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349#102—28no—
350#102—24no—
351#102—42no—
352C22⋊C8—28no—
353C22⋊C8—44no—
354#97—44no—
355C4×C8—24no—
356#95—22no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—24no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—24no—
370#95—24no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C24—21no—
375C24C22×D412yes—
376C2×Q8—24no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×D4—28no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—82no—
393C2×D4—82no—
394C2×D4C2×#4112yes—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—44no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C2511yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C82−1+414yes—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—82no—
469C4⋊C4—81no—
470C4⋊C4C2×#4114yes—
471C4⋊C4C2×#4112yes—
472C22⋊C4—28no—
473C22⋊C4—28no—
474C22⋊C4—24no—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—28no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—24no—
492C42—22no—
493C42—42no—
494C42—42no—
495C42—41no—
496C42—48no—
497C42—42no—
498C42—44no—
499C42—82no—
500C42—82no—
501C42—82no—
502C42AGL(1,ℤ/8)14yes—
503C42C2×#4112yes—
504C42C2×#4112yes—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C23×D411yescharacteristic
520Q8—84no—
521Q8—84no—
522Q8—88no—
523D4—24no—
524D4—44no—
525D4—42no—
526D4—88no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—48no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—48no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
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—42no—
574C2×C4—42no—
575C2×C4—84no—
576C2×C4—164no—
577C2×C4#52512yes—
578C2×C4#52512yes—
579C2×C4C2×AGL(1,ℤ/8)12yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4#57512yes—
583C2×C4#57712yes—
584C2×C4C22×#4111yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×2−1+412yes—
587C2×C4#58512yes—
588C8—24no—
589C8—44no—
590C8—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22C2×#52511yescharacteristic
606C22#348511yescharacteristic
607C22#349911yescharacteristic
608C22C22×AGL(1,ℤ/8)11yescharacteristic
609C22C2×#57511yescharacteristic
610C22C2×#57711yescharacteristic
611C22#358611yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—44no—
622C4—82no—
623C4—81no—
624C4—81no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4#311514yes—
629C4#350712yes—
630C4#351612yes—
631C2—82no—
632C2—82no—
633C2#3584611yescharacteristic
634C2C2×#311511yescharacteristic
635C2#6059111yescharacteristic
636C2#6171711yescharacteristic
637C2#6173911yescharacteristic
638C2#6247711yescharacteristic
639C2#6250711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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
1. #36188C2semidirect
3. #30166C2semidirect
4. #29864C2semidirect
5. #29304C2semidirect
6. #29273C2semidirect
7. #28687C2semidirect
8. #27615C2semidirect
9. #27482C2semidirect
10. #27054C2semidirect
11. #26303C2semidirect
13. #23774C2semidirect
14. #23761C2semidirect
15. #23163C2semidirect
16. #22968C2semidirect
17. #22839C2semidirect

Maximal subgroups

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

subgroupindex
#361882
#343212
#301662
#298642
#293042
#292732
#286872
#276152
#274822
#270542
#263032
#257002
#237742
#237612
#231632
#229682
#228392