Finite Group Databasebeta

← #6537765

Subgroups — #6537765

Order 512.

1613 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#36134C211yesmaximal, characteristic
2#34324C211yesmaximal, characteristic
3#30109C212yesmaximal, complemented
4#29799C211yesmaximal, characteristic, complemented
5#29338C212yesmaximal, complemented
6#29276C211yesmaximal, characteristic, complemented
7#28748C211yesmaximal, characteristic, complemented
8#27416C214yesmaximal, complemented
9#26639C214yesmaximal, complemented
10#26231C212yesmaximal
11#25570C212yesmaximal
12#24065C212yesmaximal, complemented
13#23777C211yesmaximal, characteristic, complemented
14#23577C212yesmaximal, complemented
15#23222C212yesmaximal, complemented
16#23004C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×#533C2212yes—
20C2×#487C2212yes—
21C2×#485C2214yes—
22C2×#462C2212yes—
23C2×#441C2214yes—
24#2868C2211yescharacteristic
25#2857C2212yes—
26#2552—21no—
27#2542C2212yes—
28#2538—21no—
29#2441—22no—
30#2429C2214yes—
31#2423—22no—
32#2373—22no—
33#2367—21no—
34#2365C2211yescharacteristic
35#2365C2212yes—
36#2362C2212yes—
37#2355—21no—
38#2354—21no—
39#2354C2211yescharacteristic
40Q8×C4⋊C4—21no—
41Q8×C4⋊C4C2211yescharacteristic
42#2346C2212yes—
43#2334—22no—
44#2326—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2296C2212yes—
49#2295C2211yescharacteristic
50#2276—21no—
51C42×D4C2211yescharacteristic
52#2100C2212yes—
53#2100C2211yescharacteristic
54#2095C2211yescharacteristic
55#2087C2212yes—
56#2068C2212yes—
57#2058C2212yes—
58#2057C2211yescharacteristic
59#2043C2212yes—
60#2043C2212yes—
61#2043C2212yes—
62#2043C2212yes—
63#2036C2214yes—
64#2035C2214yes—
65#2024C2211yescharacteristic
66#2016C2212yes—
67#2016C2212yes—
68#2001C2214yes—
69#2001C2214yes—
70#1988C2212yes—
71#1981C2214yes—
72#1981C2212yes—
73#1956C2212yes—
74#1954C2212yes—
75#1948C2212yes—
76#1935C2214yes—
77#1935C2214yes—
78#1928C2214yes—
79#1924C2214yes—
80#1920C2214yes—
81#1878C2214yes—
82#1873C2214yes—
83#1873C2214yes—
84#1873C2214yes—
85#1865C2214yes—
86#1863C2214yes—
87#1831C2211yescharacteristic
88#1817C2212yes—
89C4×#107C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×Q16C2314yes—
96#556—24no—
97#556—24no—
98#533—48no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#126—22no—
105C2×#125—22no—
106C2×#124—21no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2314yes—
110C2×C22⋊Q8C2314yes—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2314yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×#117—22no—
123C2×#117C2312yes—
124C2×#117C2312yes—
125C22×C42C2312yes—
126#487—28no—
127#485—48no—
128#462—28no—
129#441—48no—
130C2×#107C2312yes—
131C2×#107C2312yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147C2×C4×C8C2312yes—
148#399—24no—
149#398—24no—
150#396—24no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391C2311yescharacteristic
155#391C2314yes—
156#391C2311yescharacteristic
157#390—21no—
158#390—22no—
159#389—22no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389C2312yes—
167#389C2312yes—
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—24no—
178#387—21no—
179#387—22no—
180#387—21no—
181#387—22no—
182#387—22no—
183#387—24no—
184#385—22no—
185#385—22no—
186#385—22no—
187#384—21no—
188#384—22no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384—21no—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376—21no—
221#376—22no—
222C43C2311yescharacteristic
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2314yes—
233#336C2312yes—
234#336C2311yescharacteristic
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8C2414yes—
238C22×Q8C2411yescharacteristic
239C22×D4C2×D411yescharacteristic
240C23×C4C2×D412yes—
241C23×C4C2×D412yes—
242C23×C4C2412yes—
243C2×Q16—28no—
244C2×Q16—44no—
245C2×Q16—44no—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8#4114yes—
256#126—216no—
257#125—48no—
258#124—28no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—28no—
262C22⋊Q8—44no—
263C22⋊Q8—48no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C4×Q8—28no—
267C4×Q8—22no—
268C4×Q8—28no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—48no—
274C4×Q8—44no—
275C4×Q8#4114yes—
276C4×D4—24no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—42no—
280#117—24no—
281#117—24no—
282#117—48no—
283#117—44no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4C2×D412yes—
325C2×C22⋊C4C2×D412yes—
326C2×C22⋊C4C2×D412yes—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—44no—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347#107—24no—
348#107—42no—
349#106—28no—
350C4⋊C8—24no—
351C4⋊C8—42no—
352#103—28no—
353#103—28no—
354#103—28no—
355#103—24no—
356#103—44no—
357#103—42no—
358#103—44no—
359#103—44no—
360#103—44no—
361C22⋊C8—44no—
362C22⋊C8—44no—
363#97—44no—
364C4×C8—24no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—21no—
371#95—21no—
372#95—22no—
373#95—24no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24C22×D412yes—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—21no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8C2×#4112yes—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D411yescharacteristic
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2511yescharacteristic
434Q16—416no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—22no—
439C2×C8—24no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—24no—
443C2×C8—22no—
444C4⋊C4—24no—
445C4⋊C4—28no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4C2×#4112yes—
480C4⋊C4C2×#4114yes—
481C4⋊C4H(5,𝔽2)14yes—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—42no—
487C22⋊C4—48no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—84no—
491C42—28no—
492C42—24no—
493C42—22no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42—41no—
501C42—48no—
502C42—44no—
503C42—44no—
504C42—44no—
505C42—42no—
506C42—82no—
507C42—82no—
508C42#13714yes—
509C42C2×#4112yes—
510C42C2×#4112yes—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23C23×D411yescharacteristic
519Q8—24no—
520Q8—48no—
521Q8—48no—
522Q8—42no—
523Q8—44no—
524Q8—44no—
525Q8—44no—
526Q8—44no—
527D4—44no—
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—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4#52512yes—
580C2×C4#52512yes—
581C2×C4C2×#13712yes—
582C2×C4C2×#13712yes—
583C2×C4C2×#13712yes—
584C2×C4#57512yes—
585C2×C4#57812yes—
586C2×C4C22×#4111yescharacteristic
587C2×C4C22×#4111yescharacteristic
588C2×C4C2×H(5,𝔽2)12yes—
589C2×C4#58512yes—
590C8—44no—
591C8—42no—
592C8—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22C2×#52511yescharacteristic
601C22#348411yescharacteristic
602C22#350011yescharacteristic
603C22C22×#13711yescharacteristic
604C22C2×#57511yescharacteristic
605C22C2×#57811yescharacteristic
606C22#358611yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—41no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—42no—
617C4—44no—
618C4—42no—
619C4—81no—
620C4—82no—
621C4—84no—
622C4—82no—
623C4—81no—
624C4#311614yes—
625C4#350612yes—
626C4#351712yes—
627C2—82no—
628C2#3570411yescharacteristic
629C2C2×#311611yescharacteristic
630C2#6059411yescharacteristic
631C2#6158411yescharacteristic
632C2#6159511yescharacteristic
633C2#6253611yescharacteristic
634C2#6258111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #30109C2semidirect
4. #29799C2semidirect
5. #29338C2semidirect
6. #29276C2semidirect
7. #28748C2semidirect
8. #27416C2semidirect
9. #26639C2semidirect
12. #24065C2semidirect
13. #23777C2semidirect
14. #23577C2semidirect
15. #23222C2semidirect
16. #23004C2semidirect

Maximal subgroups

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

subgroupindex
#361342
#343242
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