Finite Group Databasebeta

← #6561637

Subgroups — #6561637

Order 512.

1478 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#36463C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic, complemented
3#30365C212yesmaximal, complemented
4#30276C211yesmaximal, characteristic, complemented
5#29354C211yesmaximal, characteristic, complemented
6#29226C211yesmaximal, characteristic, complemented
7#29157C212yesmaximal, complemented
8#28854C214yesmaximal, complemented
9#28705C212yesmaximal, complemented
10#28599C212yesmaximal
11#28085C214yesmaximal, complemented
12#27856C212yesmaximal, complemented
13#23838C211yesmaximal, characteristic, complemented
14#23761C212yesmaximal, complemented
15#23223C212yesmaximal, complemented
16#23162C212yesmaximal, complemented
17#20558C211yesmaximal, characteristic, complemented
18C2×#500C2212yes—
19C2×#486C2212yes—
20C2×C8×D4C2212yes—
21#2868C2211yescharacteristic
22#2858C2211yescharacteristic
23#2857C2212yes—
24#2854C2212yes—
25#2571—21no—
26#2563C2211yescharacteristic
27#2563C2212yes—
28#2557—22no—
29#2541C2211yescharacteristic
30#2538—21no—
31#2513—21no—
32#2497C2214yes—
33#2492—22no—
34#2490C2212yes—
35#2459—22no—
36#2445—21no—
37#2439—22no—
38#2420C2212yes—
39#2369C2212yes—
40#2369C2211yescharacteristic
41#2366—21no—
42#2366C2211yescharacteristic
43#2357—21no—
44#2355—21no—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C4×#121C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2099C2212yes—
52#2099C2212yes—
53#2093C2212yes—
54#2093C2214yes—
55#2093C2212yes—
56#2085C2212yes—
57#2084C2211yescharacteristic
58#2083C2211yescharacteristic
59#2076C2214yes—
60#2076C2214yes—
61#2071C2212yes—
62#2059C2212yes—
63#2058C2211yescharacteristic
64#2057C2211yescharacteristic
65#2034C2214yes—
66#2034C2214yes—
67#2023C2212yes—
68#1992C2212yes—
69#1987C2212yes—
70#1987C2212yes—
71#1987C2212yes—
72#1982C2212yes—
73#1943C2214yes—
74#1936C2214yes—
75#1936C2214yes—
76#1925C2214yes—
77#1918C2214yes—
78#1918C2214yes—
79#1893C2214yes—
80#1893C2214yes—
81#1862C2214yes—
82#1836C2212yes—
83#1808C2214yes—
84#1808C2214yes—
85#1776C2214yes—
86C4×#107C2212yes—
87C4×#106C2212yes—
88#1770C2212yes—
89#1767C2211yescharacteristic
90#1762C2214yes—
91#1752C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94C2×#336C2212yes—
95C23×C8C2314yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×#127—21no—
102C2×#127C2311yescharacteristic
103C2×#125C2311yescharacteristic
104C2×#124—21no—
105C2×#124—21no—
106C2×#123—22no—
107C2×C22⋊Q8—22no—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×C4×Q8—21no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2311yescharacteristic
117C2×#117—22no—
118C22×C42C2312yes—
119#500—28no—
120#486—28no—
121C8×D4—28no—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#106C2312yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2312yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2312yes—
135C2×#97C2314yes—
136C2×C4×C8C2312yes—
137#400—24no—
138#397—22no—
139#397—22no—
140#396—24no—
141#394—22no—
142#394—22no—
143#393—24no—
144#393—22no—
145#392—22no—
146#391—21no—
147#391—22no—
148#391—24no—
149#391—22no—
150#390—22no—
151#390—21no—
152#390—21no—
153#390—21no—
154#390—22no—
155#390—21no—
156#390—21no—
157#390C2312yes—
158#390C2312yes—
159#390C2312yes—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389C2311yescharacteristic
165#389C2311yescharacteristic
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#388—24no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#387—21no—
175#387—21no—
176#387—24no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2312yes—
186#382—22no—
187#382—21no—
188#382—22no—
189#382C2311yescharacteristic
190#381—22no—
191#380—22no—
192#380—22no—
193#380C2312yes—
194#380C2312yes—
195#380C2314yes—
196#380C2314yes—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215C43C2311yescharacteristic
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4C2×D411yescharacteristic
239C22×D4C2412yes—
240C23×C4—22no—
241C23×C4C2×D412yes—
242C23×C4C2×D412yes—
243C23×C4C2412yes—
244C22×C8—28no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—42no—
254C4⋊Q8—82no—
255C4⋊Q8—82no—
256#127—24no—
257#127—44no—
258#125—22no—
259#125—41no—
260#124—82no—
261#124—82no—
262#123—84no—
263C22⋊Q8—84no—
264#121—24no—
265#121—42no—
266#121—44no—
267C4×Q8—44no—
268C4×D4—24no—
269C4×D4—24no—
270C4×D4—42no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4—84no—
274#117—84no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4C2×D412yes—
316C2×C22⋊C4C2×D412yes—
317C2×C22⋊C4C2×D412yes—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—21no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—44no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—28no—
335#107—28no—
336#107—42no—
337#107—42no—
338#106—24no—
339#106—42no—
340C4⋊C8—24no—
341#102—28no—
342#102—28no—
343#102—44no—
344#102—44no—
345C22⋊C8—28no—
346#97—42no—
347#97—44no—
348C4×C8#4118yes—
349#95—21no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—21no—
354#95—22no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—24no—
359#95—21no—
360#95—22no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24—22no—
369C24C22×D412yes—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—44no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—81no—
376C2×Q8—81no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—82no—
397C2×D4#13714yes—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—82no—
418C22×C4—82no—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
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—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—88no—
437C2×C8C2×#4114yes—
438C2×C8C2×#4114yes—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—84no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C42−1+412yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—84no—
484C22⋊C4—82no—
485C22⋊C4—84no—
486C22⋊C4—82no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—21no—
492C42—21no—
493C42—24no—
494C42—22no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—44no—
500C42—84no—
501C42—82no—
502C42C2×#4114yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23C2×#13714yes—
517C23C23×D411yescharacteristic
518Q8—84no—
519Q8—162no—
520Q8—164no—
521D4—28no—
522D4—44no—
523D4—44no—
524D4—88no—
525D4—84no—
526D4—162no—
527D4—164no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—48no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—84no—
577C2×C4—164no—
578C2×C4#53212yes—
579C2×C4#53212yes—
580C2×C4C2×#13712yes—
581C2×C4#57512yes—
582C2×C4#57612yes—
583C2×C4#57612yes—
584C2×C4C22×#4112yes—
585C2×C4C2×2−1+411yescharacteristic
586C2×C4#58512yes—
587C2×C4#58511yescharacteristic
588C8—24no—
589C8—24no—
590C8—44no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22C2×#53211yescharacteristic
606C22#348511yescharacteristic
607C22#348911yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57511yescharacteristic
610C22C2×#57611yescharacteristic
611C22C2×#57611yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—82no—
623C4—84no—
624C4—81no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4#318114yes—
629C4#350712yes—
630C4#351012yes—
631C2—44no—
632C2—162no—
633C2#3595911yescharacteristic
634C2C2×#318111yescharacteristic
635C2#6071411yescharacteristic
636C2#6172611yescharacteristic
637C2#6174011yescharacteristic
638C2#6212711yescharacteristic
639C2#6213611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2541C22
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
2. #36194C2semidirect
3. #30365C2semidirect
4. #30276C2semidirect
5. #29354C2semidirect
6. #29226C2semidirect
7. #29157C2semidirect
8. #28854C2semidirect
9. #28705C2semidirect
11. #28085C2semidirect
12. #27856C2semidirect
13. #23838C2semidirect
14. #23761C2semidirect
15. #23223C2semidirect
16. #23162C2semidirect
17. #20558C2semidirect

Maximal subgroups

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

subgroupindex
#364632
#361942
#303652
#302762
#293542
#292262
#291572
#288542
#287052
#285992
#280852
#278562
#238382
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