Finite Group Databasebeta

← #6560759

Subgroups — #6560759

Order 512.

1473 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic, complemented
2#36270C211yesmaximal, characteristic
3#30383C212yesmaximal, complemented
4#30230C211yesmaximal, characteristic, complemented
5#29580C212yesmaximal, complemented
6#29521C212yesmaximal, complemented
7#29312C211yesmaximal, characteristic, complemented
8#29032C214yesmaximal
9#28509C211yesmaximal, characteristic
10#28000C214yesmaximal, complemented
11#25477C212yesmaximal, complemented
12#25477C212yesmaximal, complemented
13#23815C211yesmaximal, characteristic
14#23780C212yesmaximal, complemented
15#23067C212yesmaximal, complemented
16#22969C212yesmaximal, complemented
17#20570C211yesmaximal, characteristic, complemented
18C2×#552—21no—
19C2×#476C2212yes—
20C2×#476C2212yes—
21C2×#446C2212yes—
22#2868C2211yescharacteristic
23#2860C2212yes—
24#2859C2211yescharacteristic
25#2859C2212yes—
26#2601—22no—
27#2582—21no—
28#2567C2212yes—
29#2562—21no—
30#2558—22no—
31#2552—21no—
32#2550—22no—
33#2538C2211yescharacteristic
34#2538C2211yescharacteristic
35#2536—21no—
36#2507C2214yes—
37#2506C2212yes—
38#2451—22no—
39#2445C2212yes—
40#2369C2212yes—
41#2355—21no—
42#2354C2211yescharacteristic
43#2352—21no—
44Q8×C4⋊C4—21no—
45D4×C4⋊C4C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#127C2211yescharacteristic
49#2293C2212yes—
50C42×Q8C2211yescharacteristic
51#2100C2212yes—
52#2100C2214yes—
53#2100C2212yes—
54#2099C2212yes—
55#2094C2212yes—
56#2094C2212yes—
57#2088C2214yes—
58#2088C2214yes—
59#2083C2212yes—
60#2071C2211yescharacteristic
61#2071C2211yescharacteristic
62#2067C2214yes—
63#2067C2214yes—
64#2057C2212yes—
65#2041C2212yes—
66#2023C2211yescharacteristic
67#2023C2211yescharacteristic
68#1990C2212yes—
69#1990C2212yes—
70#1957C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1944C2214yes—
74#1943C2214yes—
75#1943C2214yes—
76#1942C2214yes—
77#1936C2214yes—
78#1936C2214yes—
79#1936C2214yes—
80#1925C2214yes—
81#1920C2214yes—
82#1919C2214yes—
83#1862C2214yes—
84#1844C2212yes—
85#1776C2214yes—
86#1771C2214yes—
87#1770C2212yes—
88#1767C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91C4×#102C2212yes—
92C4×#102C2212yes—
93#1745C2212yes—
94#1744C2211yescharacteristic
95#552—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#126—22no—
102C2×#125—22no—
103C2×#125—21no—
104C2×#125—22no—
105C2×#123—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—22no—
120C22×C4⋊C4—21no—
121#476—28no—
122#476—28no—
123#446—28no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×#102C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#97C2314yes—
137C2×#97C2314yes—
138C2×C4×C8C2312yes—
139C2×C4×C8C2312yes—
140#400—22no—
141#398—24no—
142#398—22no—
143#398—24no—
144#396—22no—
145#396—22no—
146#395—24no—
147#394—22no—
148#394—22no—
149#391—22no—
150#391—22no—
151#391—21no—
152#391—24no—
153#390—21no—
154#390—21no—
155#390C2311yescharacteristic
156#390C2311yescharacteristic
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—22no—
166#389—21no—
167#389—24no—
168#389C2314yes—
169#389C2312yes—
170#389C2314yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2314yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—24no—
184#387—22no—
185#386—22no—
186#385—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#384C2312yes—
196#382—21no—
197#382—22no—
198#380—21no—
199#380—21no—
200#380—21no—
201#380—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D412yes—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—22no—
215C43C2311yescharacteristic
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2411yescharacteristic
238C22×D4C2411yescharacteristic
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—82no—
249C4⋊Q8—82no—
250#127—22no—
251#127—41no—
252#126—48no—
253#125—48no—
254#125—44no—
255#125—84no—
256#123—48no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—44no—
263C4×Q8—84no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—42no—
268C4×D4—44no—
269C4×D4—44no—
270#117—48no—
271#117—44no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—28no—
339#107—44no—
340#106—28no—
341#106—28no—
342C4⋊C8—28no—
343C4⋊C8#4118yes—
344#102—24no—
345#102—28no—
346#102—28no—
347#102—24no—
348#102—24no—
349#102—42no—
350#97—28no—
351#97—44no—
352C4×C8—24no—
353C4×C8—42no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—24no—
359#95—22no—
360#95—24no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—21no—
370C24—21no—
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—82no—
379C2×Q8—82no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D42−1+412yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—82no—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2511yescharacteristic
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8C2×#4118yes—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—84no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C42−1+414yes—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—22no—
487C42—21no—
488C42—22no—
489C42—21no—
490C42—22no—
491C42—42no—
492C42—42no—
493C42—41no—
494C42—41no—
495C42—44no—
496C42—44no—
497C42—84no—
498C42AGL(1,ℤ/8)14yes—
499C42#13714yes—
500C42C2×#4114yes—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C23×D411yescharacteristic
516Q8—48no—
517Q8—44no—
518Q8—164no—
519D4—24no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—84no—
524D4—84no—
525C2×C4—21no—
526C2×C4—21no—
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—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—88no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4#53212yes—
572C2×C4#53212yes—
573C2×C4C2×AGL(1,ℤ/8)12yes—
574C2×C4C2×AGL(1,ℤ/8)12yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4C2×#13712yes—
579C2×C4C22×#4112yes—
580C2×C4C2×2−1+412yes—
581C2×C4C2×2−1+411yescharacteristic
582C2×C4#58511yescharacteristic
583C8—28no—
584C8—42no—
585C8—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22C2×#53211yescharacteristic
601C22#348611yescharacteristic
602C22#348911yescharacteristic
603C22C22×AGL(1,ℤ/8)11yescharacteristic
604C22C22×#13711yescharacteristic
605C22#358511yescharacteristic
606C22#358511yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—84no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#318214yes—
624C4#351512yes—
625C4#352312yes—
626C2—82no—
627C2—161no—
628C2#3594911yescharacteristic
629C2C2×#318211yescharacteristic
630C2#6072511yescharacteristic
631C2#6182511yescharacteristic
632C2#6183011yescharacteristic
633C2#6210811yescharacteristic
634C2#6211011yescharacteristic
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
4. #30230C2semidirect
5. #29580C2semidirect
6. #29521C2semidirect
7. #29312C2semidirect
10. #28000C2semidirect
11. #25477C2semidirect
12. #25477C2semidirect
14. #23780C2semidirect
15. #23067C2semidirect
16. #22969C2semidirect
17. #20570C2semidirect

Maximal subgroups

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

subgroupindex
#365632
#362702
#303832
#302302
#295802
#295212
#293122
#290322
#285092
#280002
#254772
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