Finite Group Databasebeta

← #6557825

Subgroups — #6557825

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36469C211yesmaximal, characteristic
2#36134C211yesmaximal, characteristic, complemented
3#30364C212yesmaximal, complemented
4#30203C211yesmaximal, characteristic, complemented
5#29208C211yesmaximal, characteristic, complemented
6#29162C212yesmaximal, complemented
7#29005C214yesmaximal, complemented
8#28374C211yesmaximal, characteristic, complemented
9#25503C212yesmaximal, complemented
10#25395C212yesmaximal
11#25348C212yesmaximal, complemented
12#25045C214yesmaximal, complemented
13#23816C211yesmaximal, characteristic, complemented
14#23574C212yesmaximal, complemented
15#23003C212yesmaximal, complemented
16#22970C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#559—21no—
19C2×#533C2212yes—
20C2×#475C2212yes—
21C2×#468C2214yes—
22C2×#460C2212yes—
23C2×#440C2214yes—
24C2×#436C2212yes—
25#2868C2211yescharacteristic
26#2858C2211yescharacteristic
27#2857C2212yes—
28#2854C2212yes—
29#2602—22no—
30#2592—21no—
31#2557—22no—
32#2552—21no—
33#2547C2211yescharacteristic
34#2544—21no—
35#2542—21no—
36#2541C2211yescharacteristic
37#2538C2212yes—
38#2491C2212yes—
39#2441—22no—
40#2429—22no—
41#2423C2214yes—
42#2422C2212yes—
43#2367—21no—
44#2366C2211yescharacteristic
45#2365—21no—
46#2365C2211yescharacteristic
47#2354—21no—
48Q8×C4⋊C4C2212yes—
49C4×C4⋊Q8C2211yescharacteristic
50C4×C4⋊Q8C2211yescharacteristic
51C4×#121C2212yes—
52C42×D4C2211yescharacteristic
53#2100C2211yescharacteristic
54#2100C2211yescharacteristic
55#2094C2212yes—
56#2077C2214yes—
57#2072C2212yes—
58#2071C2214yes—
59#2057C2212yes—
60#2057C2212yes—
61#2043C2211yescharacteristic
62#2043C2211yescharacteristic
63#2036C2214yes—
64#2033C2214yes—
65#2024C2212yes—
66#2023C2212yes—
67#2023C2212yes—
68#2013C2212yes—
69#2000C2214yes—
70#1990C2212yes—
71#1990C2212yes—
72#1968C2214yes—
73#1957C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1944C2214yes—
77#1936C2214yes—
78#1919C2214yes—
79#1895C2214yes—
80#1894C2214yes—
81#1880C2214yes—
82#1863C2214yes—
83#1862C2214yes—
84#1845C2212yes—
85#1809C2214yes—
86#1770C2212yes—
87C4×C4⋊C8C2211yescharacteristic
88#1763C2214yes—
89C4×#103C2212yes—
90C4×#103C2212yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×SD16C2314yes—
96C22×M16C2314yes—
97#559—48no—
98#533—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127—21no—
106C2×#127C2311yescharacteristic
107C2×#125—22no—
108C2×#124—21no—
109C2×#124—22no—
110C2×#124—21no—
111C2×#124—21no—
112C2×#123—22no—
113C2×#123—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2314yes—
117C2×#121C2312yes—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—22no—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×#117—22no—
131C22×C42C2312yes—
132#475—28no—
133#468—48no—
134#460—28no—
135#440—48no—
136#436—28no—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×C22⋊C8C2314yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2312yes—
153#399—24no—
154#398—24no—
155#397—22no—
156#397—22no—
157#396—24no—
158#394—22no—
159#394—22no—
160#393—22no—
161#392—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#391C2312yes—
167#390—22no—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—24no—
181#389C2311yescharacteristic
182#389C2311yescharacteristic
183#389C2312yes—
184#389C2314yes—
185#388—24no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#387—22no—
191#387—24no—
192#386—22no—
193#386—22no—
194#386—22no—
195#385—22no—
196#384—22no—
197#384—21no—
198#384—21no—
199#384C2311yescharacteristic
200#384C2311yescharacteristic
201#384C2312yes—
202#384C2312yes—
203#380—22no—
204#380C2312yes—
205#380C2314yes—
206#380C2312yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4D412yes—
220C4×C22⋊C4C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2314yes—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C23×C4—22no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×SD16—28no—
250C2×SD16—28no—
251C2×SD16—44no—
252C2×M16—28no—
253C2×M16—44no—
254C2×M16—44no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C4⋊Q8—24no—
259C4⋊Q8—22no—
260C4⋊Q8—24no—
261C4⋊Q8—41no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—82no—
265#127—24no—
266#127—44no—
267#125—48no—
268#124—48no—
269#124—44no—
270#124—44no—
271#124—44no—
272#123—48no—
273#123—48no—
274C22⋊Q8—28no—
275C22⋊Q8—44no—
276C22⋊Q8—48no—
277C22⋊Q8—84no—
278#121—24no—
279#121—42no—
280#121—44no—
281C4×Q8—28no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—44no—
285C4×Q8—48no—
286C4×Q8—44no—
287C4×Q8—44no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293C4×D4—48no—
294#117—48no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2414yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C22⋊C4—24no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—44no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—44no—
358#107—44no—
359#106—28no—
360C4⋊C8#4118yes—
361#103—24no—
362#103—24no—
363#103—28no—
364#103—24no—
365#103—42no—
366#102—28no—
367#102—44no—
368#102—44no—
369C22⋊C8—28no—
370#97—44no—
371C4×C8—24no—
372C4×C8—42no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—24no—
383#95—22no—
384#95—22no—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—22no—
389C24C22×D412yes—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—42no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8—44no—
402C2×Q8—42no—
403C2×Q8—81no—
404C2×Q8H(5,𝔽2)14yes—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2511yescharacteristic
455SD16—416no—
456M16—88no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8C2×#4118yes—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—81no—
496C4⋊C4—84no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C42−1+412yes—
500C22⋊C4—24no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—82no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—84no—
516C42—24no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—42no—
521C42—42no—
522C42—41no—
523C42—41no—
524C42—48no—
525C42—44no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42AGL(1,ℤ/8)14yes—
530C42#13714yes—
531C42C2×#4114yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C23×D411yescharacteristic
547Q8—24no—
548Q8—24no—
549Q8—42no—
550Q8—42no—
551Q8—84no—
552Q8—88no—
553Q8—84no—
554D4—44no—
555D4—44no—
556D4—44no—
557D4—88no—
558D4—84no—
559D4—162no—
560D4—164no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—84no—
608C2×C4—164no—
609C2×C4#53212yes—
610C2×C4#53212yes—
611C2×C4C2×AGL(1,ℤ/8)12yes—
612C2×C4C2×AGL(1,ℤ/8)12yes—
613C2×C4C2×AGL(1,ℤ/8)12yes—
614C2×C4C2×#13712yes—
615C2×C4C2×#13712yes—
616C2×C4C2×#13712yes—
617C2×C4C22×#4112yes—
618C2×C4C2×H(5,𝔽2)12yes—
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—28no—
622C8—42no—
623C8—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22C2×#53211yescharacteristic
639C22#348411yescharacteristic
640C22#348911yescharacteristic
641C22C22×AGL(1,ℤ/8)11yescharacteristic
642C22C22×#13711yescharacteristic
643C22#358511yescharacteristic
644C22#358611yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—82no—
656C4—82no—
657C4—81no—
658C4—81no—
659C4—84no—
660C4—81no—
661C4#318214yes—
662C4#351312yes—
663C4#352212yes—
664C2—82no—
665C2—162no—
666C2#3591211yescharacteristic
667C2C2×#318211yescharacteristic
668C2#6072711yescharacteristic
669C2#6158411yescharacteristic
670C2#6158711yescharacteristic
671C2#6210911yescharacteristic
672C2#6211211yescharacteristic
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. #30364C2semidirect
4. #30203C2semidirect
5. #29208C2semidirect
6. #29162C2semidirect
7. #29005C2semidirect
8. #28374C2semidirect
9. #25503C2semidirect
11. #25348C2semidirect
12. #25045C2semidirect
13. #23816C2semidirect
14. #23574C2semidirect
15. #23003C2semidirect
16. #22970C2semidirect
17. #20568C2semidirect

Maximal subgroups

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

subgroupindex
#364692
#361342
#303642
#302032
#292082
#291622
#290052
#283742
#255032
#253952
#253482
#250452
#238162
#235742
#230032
#229702
#205682