Finite Group Databasebeta

← #6560758

Subgroups — #6560758

Order 512.

1553 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#36560C211yesmaximal, characteristic, complemented
2#36270C211yesmaximal, characteristic
3#30384C212yesmaximal, complemented
4#30228C211yesmaximal, characteristic, complemented
5#29578C212yesmaximal, complemented
6#29518C212yesmaximal, complemented
7#29313C211yesmaximal, characteristic, complemented
8#29036C214yesmaximal
9#28508C211yesmaximal, characteristic
10#28005C214yesmaximal, complemented
11#25478C212yesmaximal, complemented
12#25478C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic
14#23781C212yesmaximal, complemented
15#23066C212yesmaximal, complemented
16#22968C212yesmaximal, complemented
17#20570C211yesmaximal, characteristic, complemented
18C2×#557—21no—
19C2×#552—21no—
20C2×#474C2212yes—
21C2×#474C2212yes—
22C2×#446C2212yes—
23#2868C2211yescharacteristic
24#2860C2212yes—
25#2859C2211yescharacteristic
26#2859C2212yes—
27#2601—22no—
28#2568—21no—
29#2565—22no—
30#2562—21no—
31#2558—22no—
32#2546—21no—
33#2546C2212yes—
34#2538C2211yescharacteristic
35#2538C2211yescharacteristic
36#2536—21no—
37#2506—22no—
38#2506C2212yes—
39#2506C2214yes—
40#2445C2212yes—
41#2365—21no—
42#2365C2212yes—
43#2354C2211yescharacteristic
44#2352—21no—
45D4×C4⋊C4C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#127C2211yescharacteristic
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51#2099C2212yes—
52#2099C2212yes—
53#2099C2214yes—
54#2099C2212yes—
55#2093C2212yes—
56#2093C2212yes—
57#2087C2214yes—
58#2087C2214yes—
59#2084C2212yes—
60#2071C2211yescharacteristic
61#2071C2211yescharacteristic
62#2065C2214yes—
63#2065C2214yes—
64#2058C2212yes—
65#2041C2212yes—
66#2023C2211yescharacteristic
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1957C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1944C2214yes—
74#1944C2214yes—
75#1943C2214yes—
76#1941C2214yes—
77#1936C2214yes—
78#1936C2214yes—
79#1936C2214yes—
80#1926C2214yes—
81#1921C2214yes—
82#1921C2214yes—
83#1864C2214yes—
84#1845C2212yes—
85#1776C2214yes—
86#1771C2214yes—
87#1770C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95#557—48no—
96#552—48no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#127C2311yescharacteristic
103C2×#126—22no—
104C2×#125—22no—
105C2×#125—24no—
106C2×#125—21no—
107C2×#125—22no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C4×Q8—22no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C2×#117—22no—
124C22×C4⋊C4—21no—
125#474—28no—
126#474—28no—
127#446—28no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2312yes—
134C2×#102C2312yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#97C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#400—22no—
145#400—24no—
146#398—24no—
147#398—24no—
148#398—22no—
149#396—22no—
150#396—22no—
151#394—22no—
152#394—22no—
153#391—22no—
154#391—22no—
155#391—21no—
156#391—22no—
157#390—21no—
158#390—21no—
159#390C2311yescharacteristic
160#390C2311yescharacteristic
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#389—21no—
165#389—22no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—24no—
186#387—24no—
187#387—22no—
188#386—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#384C2312yes—
198#382—22no—
199#382—21no—
200#380—21no—
201#380—21no—
202#380—21no—
203#380—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C22⋊C4—22no—
216C43C2311yescharacteristic
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2312yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#336C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2×D411yescharacteristic
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2411yescharacteristic
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—48no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252#127—22no—
253#127—41no—
254#126—48no—
255#125—48no—
256#125—48no—
257#125—416no—
258#125—44no—
259#123—48no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×Q8—44no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—24no—
271C4×D4—42no—
272C4×D4—44no—
273C4×D4—44no—
274#117—48no—
275#117—44no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342#107—28no—
343#107—28no—
344#106—28no—
345#106—44no—
346C4⋊C8—28no—
347C4⋊C8#4118yes—
348#102—24no—
349#102—28no—
350#102—28no—
351#102—24no—
352#102—24no—
353#102—42no—
354#97—28no—
355#97—44no—
356C4×C8—24no—
357C4×C8—42no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24—21no—
374C24—21no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—42no—
380C2×Q8—44no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—81no—
396C2×D4—81no—
397C2×D42−1+412yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—82no—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2511yescharacteristic
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8C2×#4118yes—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C42−1+414yes—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—21no—
492C42—22no—
493C42—21no—
494C42—22no—
495C42—41no—
496C42—42no—
497C42—41no—
498C42—42no—
499C42—44no—
500C42—48no—
501C42—44no—
502C42AGL(1,ℤ/8)14yes—
503C42AGL(1,ℤ/8)14yes—
504C42C2×#4114yes—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C23×D411yescharacteristic
520Q8—48no—
521Q8—44no—
522Q8—88no—
523D4—24no—
524D4—42no—
525D4—42no—
526D4—42no—
527D4—84no—
528D4—84no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—88no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4#53212yes—
576C2×C4#53212yes—
577C2×C4C2×AGL(1,ℤ/8)12yes—
578C2×C4C2×AGL(1,ℤ/8)12yes—
579C2×C4C2×AGL(1,ℤ/8)12yes—
580C2×C4C2×AGL(1,ℤ/8)12yes—
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4C2×AGL(1,ℤ/8)12yes—
583C2×C4C22×#4112yes—
584C2×C4C2×2−1+412yes—
585C2×C4C2×2−1+411yescharacteristic
586C2×C4#58511yescharacteristic
587C8—28no—
588C8—42no—
589C8—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22C2×#53211yescharacteristic
605C22#348611yescharacteristic
606C22#348911yescharacteristic
607C22C22×AGL(1,ℤ/8)11yescharacteristic
608C22C22×AGL(1,ℤ/8)11yescharacteristic
609C22#358611yescharacteristic
610C22#358611yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—84no—
624C4—82no—
625C4—82no—
626C4—82no—
627C4#318414yes—
628C4#351512yes—
629C4#352312yes—
630C2—82no—
631C2—161no—
632C2#3594911yescharacteristic
633C2C2×#318411yescharacteristic
634C2#6072611yescharacteristic
635C2#6182711yescharacteristic
636C2#6182811yescharacteristic
637C2#6210711yescharacteristic
638C2#6210911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36560C2semidirect
3. #30384C2semidirect
4. #30228C2semidirect
5. #29578C2semidirect
6. #29518C2semidirect
7. #29313C2semidirect
10. #28005C2semidirect
11. #25478C2semidirect
12. #25478C2semidirect
14. #23781C2semidirect
15. #23066C2semidirect
16. #22968C2semidirect
17. #20570C2semidirect

Maximal subgroups

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

subgroupindex
#365602
#362702
#303842
#302282
#295782
#295182
#293132
#290362
#285082
#280052
#254782
#238152
#237812
#230662
#229682
#205702