Finite Group Databasebeta

← #6559616

Subgroups — #6559616

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#30325C212yesmaximal, complemented
4#30263C211yesmaximal, characteristic, complemented
5#29346C211yesmaximal, characteristic, complemented
6#28927C214yesmaximal, complemented
7#28748C211yesmaximal, characteristic, complemented
8#28606C212yesmaximal, complemented
9#28577C212yesmaximal, complemented
10#28490C212yesmaximal
11#26983C214yesmaximal, complemented
12#25503C212yesmaximal, complemented
13#23835C211yesmaximal, characteristic, complemented
14#23574C212yesmaximal, complemented
15#23222C212yesmaximal, complemented
16#23190C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#533C2212yes—
20C2×#492C2212yes—
21C2×#487C2212yes—
22C2×#487C2212yes—
23C2×#468C2214yes—
24C2×#468C2214yes—
25C2×#440C2214yes—
26C2×#436C2212yes—
27#2868C2211yescharacteristic
28#2858C2211yescharacteristic
29#2857C2212yes—
30#2854C2212yes—
31#2602—22no—
32#2592—21no—
33#2557—22no—
34#2552—21no—
35#2547C2211yescharacteristic
36#2544—21no—
37#2542C2212yes—
38#2541C2211yescharacteristic
39#2538—21no—
40#2491C2212yes—
41#2441—22no—
42#2429C2214yes—
43#2423—22no—
44#2422C2212yes—
45#2367—21no—
46#2366C2211yescharacteristic
47#2365C2211yescharacteristic
48#2365C2212yes—
49#2354—21no—
50Q8×C4⋊C4—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×#121C2212yes—
54C42×D4C2211yescharacteristic
55#2083C2212yes—
56#2083C2211yescharacteristic
57#2083C2211yescharacteristic
58#2071C2212yes—
59#2057C2211yescharacteristic
60#2057C2211yescharacteristic
61#2057C2212yes—
62#2043C2212yes—
63#2043C2212yes—
64#2036C2214yes—
65#2036C2214yes—
66#2023C2212yes—
67#2013C2212yes—
68#2013C2214yes—
69#2013C2212yes—
70#2000C2214yes—
71#2000C2214yes—
72#1988C2212yes—
73#1982C2212yes—
74#1981C2212yes—
75#1981C2212yes—
76#1973C2214yes—
77#1956C2212yes—
78#1935C2214yes—
79#1934C2214yes—
80#1928C2214yes—
81#1894C2214yes—
82#1894C2214yes—
83#1880C2214yes—
84#1877C2214yes—
85#1863C2214yes—
86C4×#107C2212yes—
87C4×#107C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1763C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×SD16C2314yes—
96C22×SD16C2314yes—
97C22×M16C2314yes—
98#559—48no—
99#533—48no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127—21no—
107C2×#127C2311yescharacteristic
108C2×#125—22no—
109C2×#124—21no—
110C2×#124—22no—
111C2×#124—21no—
112C2×#124—21no—
113C2×#123—22no—
114C2×#123—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2314yes—
117C2×C22⋊Q8C2314yes—
118C2×#121C2312yes—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—22no—
124C2×C4×Q8C2314yes—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2312yes—
130C2×C4×D4C2311yescharacteristic
131C2×#117—22no—
132C22×C42C2312yes—
133#492—28no—
134#487—28no—
135#487—44no—
136#468—48no—
137#468—48no—
138#440—48no—
139#436—28no—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C22⋊C8C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2312yes—
157#399—24no—
158#398—24no—
159#397—22no—
160#397—22no—
161#396—24no—
162#394—22no—
163#394—22no—
164#393—22no—
165#392—22no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391C2314yes—
171#390—22no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#389—22no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—24no—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2311yescharacteristic
188#389C2312yes—
189#388—24no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—24no—
196#386—22no—
197#386—22no—
198#386—22no—
199#385—22no—
200#384—22no—
201#384—21no—
202#384—21no—
203#384—21no—
204#384—21no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#380—22no—
208#380—22no—
209#380C2312yes—
210#380C2312yes—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C43C2311yescharacteristic
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8C2414yes—
241C22×D4—22no—
242C22×D4—22no—
243C22×D4—21no—
244C22×D4—21no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2412yes—
247C23×C4—22no—
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×M16—28no—
258C2×M16—44no—
259C2×M16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—41no—
269C4⋊Q8—44no—
270C4⋊Q8—42no—
271C4⋊Q8—42no—
272#127—24no—
273#127—44no—
274#125—48no—
275#124—48no—
276#124—44no—
277#124—44no—
278#124—44no—
279#123—48no—
280#123—48no—
281C22⋊Q8—28no—
282C22⋊Q8—44no—
283C22⋊Q8—48no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286#121—24no—
287#121—42no—
288#121—44no—
289C4×Q8—28no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—48no—
293C4×Q8—44no—
294C4×Q8—44no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—48no—
301#117—48no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—44no—
354C2×C42C2×D412yes—
355C2×C42C2×D412yes—
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2412yes—
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—24no—
366#107—24no—
367#107—42no—
368C4⋊C8—24no—
369#103—28no—
370#103—28no—
371#103—28no—
372#103—44no—
373#103—44no—
374#103—44no—
375#102—28no—
376#102—44no—
377C22⋊C8—28no—
378#97—44no—
379C4×C8—42no—
380C4×C8#4118yes—
381#95—21no—
382#95—22no—
383#95—24no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396C24—22no—
397C24C22×D412yes—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—81no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4—82no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C2511yescharacteristic
463SD16—416no—
464SD16—88no—
465M16—88no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8C2×#4114yes—
474C2×C8C2×#4114yes—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—84no—
505C4⋊C4—82no—
506C4⋊C4—82no—
507C4⋊C4H(5,𝔽2)14yes—
508C4⋊C4H(5,𝔽2)12yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—82no—
522C22⋊C4—82no—
523C22⋊C4—82no—
524C22⋊C4—84no—
525C42—21no—
526C42—24no—
527C42—22no—
528C42—21no—
529C42—42no—
530C42—42no—
531C42—41no—
532C42—41no—
533C42—48no—
534C42—44no—
535C42—42no—
536C42—44no—
537C42—42no—
538C42AGL(1,ℤ/8)14yes—
539C42#13714yes—
540C42C2×#4114yes—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C23×D411yescharacteristic
556Q8—48no—
557Q8—44no—
558Q8—44no—
559Q8—44no—
560Q8—44no—
561Q8—84no—
562Q8—84no—
563D4—44no—
564D4—44no—
565D4—44no—
566D4—88no—
567D4—84no—
568D4—162no—
569D4—164no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—164no—
618C2×C4#53212yes—
619C2×C4#53212yes—
620C2×C4C2×AGL(1,ℤ/8)12yes—
621C2×C4C2×AGL(1,ℤ/8)12yes—
622C2×C4C2×AGL(1,ℤ/8)12yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4C2×#13712yes—
626C2×C4C22×#4112yes—
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4#58511yescharacteristic
630C8—24no—
631C8—24no—
632C8—44no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C2×#53211yescharacteristic
648C22#348411yescharacteristic
649C22#348811yescharacteristic
650C22C22×AGL(1,ℤ/8)11yescharacteristic
651C22C22×#13711yescharacteristic
652C22#358611yescharacteristic
653C22#358611yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—82no—
665C4—81no—
666C4—82no—
667C4—84no—
668C4—81no—
669C4—81no—
670C4#318214yes—
671C4#350612yes—
672C4#350812yes—
673C2—82no—
674C2—162no—
675C2#3593011yescharacteristic
676C2C2×#318211yescharacteristic
677C2#6072011yescharacteristic
678C2#6158411yescharacteristic
679C2#6158511yescharacteristic
680C2#6196211yescharacteristic
681C2#6196411yescharacteristic
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. #36134C2semidirect
3. #30325C2semidirect
4. #30263C2semidirect
5. #29346C2semidirect
6. #28927C2semidirect
7. #28748C2semidirect
8. #28606C2semidirect
9. #28577C2semidirect
11. #26983C2semidirect
12. #25503C2semidirect
13. #23835C2semidirect
14. #23574C2semidirect
15. #23222C2semidirect
16. #23190C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364692
#361342
#303252
#302632
#293462
#289272
#287482
#286062
#285772
#284902
#269832
#255032
#238352
#235742
#232222
#231902
#205682