Finite Group Databasebeta

← #6848559

Subgroups — #6848559

Order 512.

1689 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36177C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3#32954C211yesmaximal, characteristic
4#32184C214yesmaximal, complemented
5#32098C214yesmaximal, complemented
6#29627C212yesmaximal
7#28417C212yesmaximal, complemented
8#27757C212yesmaximal, complemented
9#27589C212yesmaximal
10#25856C211yesmaximal, characteristic
11#25276C212yesmaximal, complemented
12#24632C212yesmaximal, complemented
13#24508C212yesmaximal
14#24212C211yesmaximal, characteristic
15#22603C211yesmaximal, characteristic
16#21811C212yesmaximal
17C4×#451C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#487C2212yes—
22C2×#475C2212yes—
23C2×#462C2212yes—
24C2×#451C2211yescharacteristic
25C2×#451C2212yes—
26C2×C4×Q16C2212yes—
27C2×#432C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2571C2211yescharacteristic
31#2568C2211yescharacteristic
32#2546—21no—
33#2546—21no—
34#2546—22no—
35#2546C2212yes—
36#2546C2212yes—
37#2546C2214yes—
38#2533—22no—
39#2514—21no—
40#2514C2212yes—
41#2509C2212yes—
42#2503—21no—
43#2493—22no—
44C2×#398—22no—
45#2365C2211yescharacteristic
46#2360—21no—
47#2356—21no—
48#2355C2211yescharacteristic
49C4×C22⋊Q8C2211yescharacteristic
50C2×#391C2212yes—
51C2×#389—21no—
52C2×#389—21no—
53#2283C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2097C2214yes—
56#2074C2214yes—
57#2066C2214yes—
58#2056C2214yes—
59#2043C2214yes—
60#2043C2212yes—
61#2043C2214yes—
62#2043C2214yes—
63#2016C2214yes—
64#2016C2214yes—
65#2016C2212yes—
66#1981C2214yes—
67#1981C2214yes—
68#1981C2212yes—
69#1971C2214yes—
70#1960C2214yes—
71#1960C2214yes—
72#1956C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1928C2214yes—
76#1927C2214yes—
77#1917C2214yes—
78#1878C2212yes—
79#1872C2211yescharacteristic
80#1865C2214yes—
81#1863C2212yes—
82#1853C2212yes—
83#1852C2211yescharacteristic
84#1817C2212yes—
85#1803C2212yes—
86#1797C2211yescharacteristic
87#1787C2212yes—
88#1787C2212yes—
89#1787C2211yescharacteristic
90#1787C2212yes—
91#1787C2212yes—
92#1770C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×Q16C2314yes—
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8—44no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2314yes—
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—44no—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#487—44no—
132#475—44no—
133#462—44no—
134#451—24no—
135#451—44no—
136C4×Q16—44no—
137#432—44no—
138C2×#107C2312yes—
139C2×#106C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×#97C2314yes—
159C2×C4×C8C2312yes—
160#401—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#400—22no—
166#398—24no—
167#398—24no—
168#398—24no—
169#398—24no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—22no—
175#391—22no—
176#391C2312yes—
177#391C2314yes—
178#391C2312yes—
179#391C2314yes—
180#391C2311yescharacteristic
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—24no—
186#389—22no—
187#389—24no—
188#389—22no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2311yescharacteristic
194#387—22no—
195#387—22no—
196#387—22no—
197#387—21no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—21no—
203#384C2311yescharacteristic
204#383—22no—
205#383—22no—
206#382—21no—
207#382—21no—
208#382C2312yes—
209#382C2312yes—
210#381—22no—
211#381—22no—
212#380—21no—
213#380C2312yes—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D412yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—21no—
228#376—22no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—24no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8—42no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8#4114yes—
250C22×Q8C2411yescharacteristic
251C22×Q8C2414yes—
252C22×Q8C2412yes—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×Q16—24no—
262C2×Q16—42no—
263C2×Q16—42no—
264C2×Q16—44no—
265C2×Q16—44no—
266C2×Q16—44no—
267C22×C8C2412yes—
268C22×C8C2414yes—
269C22×C8C2412yes—
270C4⋊Q8—28no—
271C4⋊Q8—28no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278C4⋊Q8—48no—
279C4⋊Q8—48no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C22⋊Q8—22no—
283C22⋊Q8—22no—
284C22⋊Q8—44no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—42no—
289C4×Q8—42no—
290C4×Q8—44no—
291C4×Q8—44no—
292C4×Q8—44no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
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⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—48no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—21no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—44no—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354#107—42no—
355#106—44no—
356C4⋊C8—24no—
357C4⋊C8—44no—
358#103—24no—
359#103—42no—
360#103—44no—
361#103—44no—
362#103—44no—
363#103—44no—
364#103—42no—
365#103—42no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—24no—
374#97—44no—
375C4×C8—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24C22×D411yescharacteristic
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—21no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—21no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—48no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—48no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—88no—
431C2×Q8—88no—
432C2×Q8C2×#4114yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—41no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—42no—
464C22×C4#13714yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C2×#4112yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4C2511yescharacteristic
477Q16—48no—
478Q16—88no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4#13714yes—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C42—22no—
526C42—22no—
527C42—24no—
528C42—24no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—44no—
535C42—44no—
536C42—42no—
537C42—42no—
538C42—44no—
539C42—82no—
540C42H(5,𝔽2)14yes—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23#54612yes—
546C23C2×#13712yes—
547C23#57812yes—
548C23C23×D411yescharacteristic
549Q8—24no—
550Q8—24no—
551Q8—48no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—44no—
556Q8—44no—
557Q8—42no—
558Q8—44no—
559Q8—44no—
560Q8—88no—
561Q8—84no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—84no—
617C2×C4—88no—
618C2×C4#53512yes—
619C2×C4#53512yes—
620C2×C4#54612yes—
621C2×C4C2×#13714yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4C2×#13712yes—
625C2×C4C22×#4111yescharacteristic
626C2×C4C2×H(5,𝔽2)12yes—
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4#58512yes—
629C8—42no—
630C8—42no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22#332814yes—
639C22C2×#53511yescharacteristic
640C22C2×#54611yescharacteristic
641C22#348711yescharacteristic
642C22C22×#13711yescharacteristic
643C22C22×#13711yescharacteristic
644C22C2×#57811yescharacteristic
645C22#358611yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—24no—
650C4—44no—
651C4—41no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—84no—
656C4—81no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4—82no—
661C4—81no—
662C4—81no—
663C4#321814yes—
664C4#350512yes—
665C4#354212yes—
666C2—24no—
667C2#3946711yescharacteristic
668C2C2×#321811yescharacteristic
669C2#6081411yescharacteristic
670C2C2×#332811yescharacteristic
671C2#6115711yescharacteristic
672C2#6189311yescharacteristic
673C2#6190011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #32184C2semidirect
5. #32098C2semidirect
7. #28417C2semidirect
8. #27757C2semidirect
11. #25276C2semidirect
12. #24632C2semidirect

Maximal subgroups

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

subgroupindex
#361772
C2×#25462
#329542
#321842
#320982
#296272
#284172
#277572
#275892
#258562
#252762
#246322
#245082
#242122
#226032
#218112
C4×#4512