Finite Group Databasebeta

← #6557595

Subgroups — #6557595

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#36271C211yesmaximal, characteristic
3#30383C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic
5#29580C212yesmaximal, complemented
6#29500C212yesmaximal, complemented
7#29101C214yesmaximal, complemented
8#28689C211yesmaximal, characteristic, complemented
9#28374C211yesmaximal, characteristic
10#26495C214yesmaximal, complemented
11#25479C212yesmaximal, complemented
12#25303C212yesmaximal
13#23816C211yesmaximal, characteristic, complemented
14#23683C212yesmaximal, complemented
15#23003C212yesmaximal
16#22970C212yesmaximal, complemented
17#20570C211yesmaximal, characteristic, complemented
18C2×#556—21no—
19C2×#500C2212yes—
20C2×#475C2212yes—
21C2×#460C2212yes—
22C2×#446C2212yes—
23C2×#440C2214yes—
24#2868C2211yescharacteristic
25#2860C2212yes—
26#2859C2212yes—
27#2858C2211yescharacteristic
28#2601—22no—
29#2592—21no—
30#2582C2212yes—
31#2567—21no—
32#2560—22no—
33#2552—21no—
34#2550—22no—
35#2548C2211yescharacteristic
36#2546C2211yescharacteristic
37#2545—21no—
38#2512C2212yes—
39#2507—22no—
40#2452C2212yes—
41#2451C2214yes—
42#2369—21no—
43#2367—21no—
44#2366C2211yescharacteristic
45#2365C2211yescharacteristic
46#2355—21no—
47Q8×C4⋊C4C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2212yes—
51#2293C2212yes—
52C42×Q8C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2094C2212yes—
56#2088C2214yes—
57#2083C2212yes—
58#2083C2212yes—
59#2072C2212yes—
60#2071C2212yes—
61#2071C2212yes—
62#2068C2214yes—
63#2067C2214yes—
64#2055C2214yes—
65#2043C2211yescharacteristic
66#2043C2211yescharacteristic
67#2024C2212yes—
68#2023C2214yes—
69#2013C2212yes—
70#1990C2212yes—
71#1990C2212yes—
72#1973C2214yes—
73#1968C2214yes—
74#1957C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1944C2214yes—
78#1943C2214yes—
79#1936C2214yes—
80#1936C2214yes—
81#1935C2214yes—
82#1919C2214yes—
83#1880C2214yes—
84#1863C2214yes—
85#1862C2214yes—
86#1771C2214yes—
87#1770C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95C22×SD16C2314yes—
96#556—48no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#127C2311yescharacteristic
105C2×#126—22no—
106C2×#126—22no—
107C2×#125—22no—
108C2×#124—21no—
109C2×#124—21no—
110C2×#124—21no—
111C2×C22⋊Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4C2312yes—
121C2×#117—22no—
122#500—44no—
123#475—28no—
124#460—28no—
125#446—28no—
126#440—48no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×#106C2314yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2312yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#97C2314yes—
141C2×#97C2314yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2312yes—
144#398—22no—
145#398—24no—
146#398—24no—
147#397—22no—
148#396—22no—
149#396—22no—
150#395—24no—
151#394—22no—
152#393—22no—
153#391—22no—
154#391—21no—
155#391—21no—
156#391—21no—
157#391—24no—
158#391C2314yes—
159#391C2312yes—
160#390—21no—
161#390—21no—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#390C2312yes—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—24no—
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#389C2314yes—
181#389C2312yes—
182#389C2312yes—
183#389C2314yes—
184#389C2312yes—
185#388—22no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—24no—
192#387—22no—
193#386—22no—
194#385—22no—
195#384—22no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2312yes—
202#382—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C43C2311yescharacteristic
220C43C2312yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2412yes—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2412yes—
242C23×C4—22no—
243C2×SD16—28no—
244C2×SD16—28no—
245C2×SD16—44no—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2414yes—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—41no—
253C4⋊Q8—42no—
254C4⋊Q8—42no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C4⋊Q8—82no—
258#127—24no—
259#126—48no—
260#126—48no—
261#125—84no—
262#124—44no—
263#124—44no—
264#124—44no—
265C22⋊Q8—48no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—24no—
276C4×D4—42no—
277C4×D4—44no—
278#117—48no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—28no—
344#107—44no—
345#106—28no—
346C4⋊C8—28no—
347C4⋊C8#4118yes—
348#103—24no—
349#103—24no—
350#103—28no—
351#103—24no—
352#103—42no—
353#102—28no—
354#102—44no—
355#102—44no—
356#97—28no—
357#97—44no—
358C4×C8—24no—
359C4×C8—42no—
360#95—21no—
361#95—22no—
362#95—24no—
363#95—22no—
364#95—24no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377C24—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—81no—
393C2×Q8—82no—
394C2×Q8—82no—
395C2×Q82−1+414yes—
396C2×D4—24no—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—81no—
405C2×D4—81no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—82no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2511yescharacteristic
428SD16—416no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8C2×#4118yes—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—81no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4H(5,𝔽2)12yes—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C42—24no—
489C42—24no—
490C42—24no—
491C42—21no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—41no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—44no—
501C42—44no—
502C42—42no—
503C42—42no—
504C42—84no—
505C42AGL(1,ℤ/8)14yes—
506C42#13714yes—
507C42C2×#4114yes—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23C23×D411yescharacteristic
516Q8—24no—
517Q8—24no—
518Q8—44no—
519Q8—42no—
520Q8—42no—
521Q8—84no—
522Q8—84no—
523Q8—164no—
524D4—44no—
525D4—44no—
526D4—84no—
527D4—162no—
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—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—88no—
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×#13712yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4C22×#4112yes—
584C2×C4C2×H(5,𝔽2)11yescharacteristic
585C2×C4C2×2−1+412yes—
586C2×C4#58511yescharacteristic
587C8—28no—
588C8—42no—
589C8—42no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C2×#53211yescharacteristic
598C22#348611yescharacteristic
599C22#348811yescharacteristic
600C22C22×AGL(1,ℤ/8)11yescharacteristic
601C22C22×#13711yescharacteristic
602C22#358511yescharacteristic
603C22#358611yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—81no—
616C4—84no—
617C4—82no—
618C4—82no—
619C4—81no—
620C4—81no—
621C4#318214yes—
622C4#351612yes—
623C4#352012yes—
624C2—162no—
625C2#3590911yescharacteristic
626C2C2×#318211yescharacteristic
627C2#6072711yescharacteristic
628C2#6182811yescharacteristic
629C2#6183011yescharacteristic
630C2#6196111yescharacteristic
631C2#6196411yescharacteristic
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. #36563C2semidirect
3. #30383C2semidirect
5. #29580C2semidirect
6. #29500C2semidirect
7. #29101C2semidirect
8. #28689C2semidirect
10. #26495C2semidirect
11. #25479C2semidirect
13. #23816C2semidirect
14. #23683C2semidirect
16. #22970C2semidirect
17. #20570C2semidirect

Maximal subgroups

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

subgroupindex
#365632
#362712
#303832
#302032
#295802
#295002
#291012
#286892
#283742
#264952
#254792
#253032
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