Finite Group Databasebeta

← #6485473

Subgroups — #6485473

Order 512.

1479 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35322C211yesmaximal, characteristic
2#35291C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33255C211yesmaximal, characteristic, complemented
6#33242C211yesmaximal, characteristic, complemented
7#30331C214yesmaximal
8#29509C214yesmaximal, complemented
9#28748C211yesmaximal, characteristic, complemented
10#28719C211yesmaximal, characteristic, complemented
11#28688C211yesmaximal, characteristic, complemented
12#28440C211yesmaximal, characteristic, complemented
13#25674C211yesmaximal, characteristic
14#25672C211yesmaximal, characteristic
15#25600C211yesmaximal, characteristic
16#25600C211yesmaximal, characteristic
17#24664C214yesmaximal
18#22164C214yesmaximal, complemented
19C4×#433C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#494C2211yescharacteristic
25C2×#494C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#487C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#462C2214yes—
31C2×#433C2212yes—
32#2868C2211yescharacteristic
33#2862C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2862C2211yescharacteristic
37#2859C2211yescharacteristic
38#2857C2211yescharacteristic
39#2857C2211yescharacteristic
40#2645C2212yes—
41#2625C2212yes—
42#2621C2212yes—
43#2621C2212yes—
44#2565—21no—
45#2565—21no—
46#2565—21no—
47#2565—21no—
48#2514—21no—
49#2509—21no—
50#2508—21no—
51#2499—21no—
52#2499—21no—
53#2499—21no—
54#2499—21no—
55#2495—21no—
56#2494—21no—
57#2489—21no—
58#2453—21no—
59#2421—21no—
60#2327C2212yes—
61#2327C2212yes—
62#2323C2212yes—
63#2320C2212yes—
64C4×C4⋊Q8C2211yescharacteristic
65C4×C4⋊Q8C2211yescharacteristic
66C4×#117C2211yescharacteristic
67#2085C2214yes—
68#2074C2214yes—
69#2043C2214yes—
70#2036C2214yes—
71#2035C2214yes—
72#2016C2214yes—
73#2001C2214yes—
74#2001C2214yes—
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1981C2214yes—
80#1971C2214yes—
81#1971C2214yes—
82#1969C2214yes—
83#1969C2214yes—
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1949C2214yes—
89#1935C2214yes—
90#1935C2214yes—
91#1933C2214yes—
92#1933C2214yes—
93#1878C2214yes—
94#1878C2214yes—
95#1865C2214yes—
96#1863C2214yes—
97#1805C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109#533—42no—
110#533—42no—
111#533—42no—
112#533—42no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×#125—21no—
122C2×#125—21no—
123C2×#125—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8C2312yes—
128C2×C22⋊Q8C2312yes—
129C2×C22⋊Q8C2312yes—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×#117—21no—
137C2×#117—21no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#500—42no—
142#498—42no—
143#494—42no—
144#494—42no—
145#487—42no—
146#487—42no—
147#487—42no—
148#487—42no—
149#462—216no—
150#433—28no—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2312yes—
172C2×C4×C8C2312yes—
173#401—22no—
174#400—22no—
175#400—22no—
176#400—22no—
177#400—22no—
178#398—22no—
179#398—22no—
180#398—22no—
181#398—22no—
182#398—22no—
183#397—22no—
184#397—22no—
185#397—22no—
186#397—22no—
187#397—22no—
188#397—22no—
189#391—21no—
190#391—21no—
191#391—21no—
192#391—21no—
193#391—21no—
194#391—21no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391C2312yes—
202#391C2312yes—
203#391C2312yes—
204#391C2312yes—
205#391C2312yes—
206#391C2312yes—
207#391C2312yes—
208#391C2312yes—
209#389—22no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#388—22no—
223#388—22no—
224#387—22no—
225#387—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#384—21no—
236#384—21no—
237#384—21no—
238#384—21no—
239#384C2311yescharacteristic
240#384C2311yescharacteristic
241#383—22no—
242#382—21no—
243#382—21no—
244#382—21no—
245#382—21no—
246#382—21no—
247#382—21no—
248#382—21no—
249#382—22no—
250#382C2312yes—
251#382C2312yes—
252#380—21no—
253#380—21no—
254#380—21no—
255#380C2312yes—
256C4×C4⋊C4—21no—
257C4×C4⋊C4—21no—
258C4×C4⋊C4—21no—
259C4×C4⋊C4D411yescharacteristic
260C4×C4⋊C4D411yescharacteristic
261C4×C4⋊C4D411yescharacteristic
262C4×C4⋊C4D411yescharacteristic
263C4×C4⋊C4C2312yes—
264C4×C22⋊C4D411yescharacteristic
265C4×C22⋊C4D411yescharacteristic
266C4×C22⋊C4D411yescharacteristic
267C4×C22⋊C4D411yescharacteristic
268#376C2311yescharacteristic
269#376C2311yescharacteristic
270C43C2311yescharacteristic
271C43C2311yescharacteristic
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#340C2314yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279#336C2312yes—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8C2412yes—
284C22×Q8C2412yes—
285C22×Q8C2412yes—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2411yescharacteristic
289C2×Q16—28no—
290C2×Q16—28no—
291C2×Q16—44no—
292C2×Q16—44no—
293C2×Q16—44no—
294C2×Q16—44no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—28no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—44no—
313#125—44no—
314#125—44no—
315#125—82no—
316C22⋊Q8—44no—
317C22⋊Q8—42no—
318C22⋊Q8—42no—
319C22⋊Q8—42no—
320C22⋊Q8—42no—
321C22⋊Q8—44no—
322C22⋊Q8—44no—
323C22⋊Q8—82no—
324C4×Q8—28no—
325C4×Q8—42no—
326C4×Q8—42no—
327C4×Q8—42no—
328C4×Q8—42no—
329C4×Q8—44no—
330C4×Q8—44no—
331C4×Q8—82no—
332#117—24no—
333#117—42no—
334#117—44no—
335#117—44no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402#107—44no—
403#106—44no—
404C4⋊C8—24no—
405C4⋊C8#4118yes—
406#103—28no—
407#103—28no—
408#103—44no—
409#103—44no—
410#103—44no—
411#103—44no—
412#103—44no—
413#103—44no—
414C22⋊C8—24no—
415C22⋊C8—24no—
416#97—24no—
417#97—42no—
418C4×C8—24no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—42no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95—22no—
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455#95C2×D411yescharacteristic
456#95C2×D411yescharacteristic
457#95C2×D411yescharacteristic
458C24C22×D411yescharacteristic
459C2×Q8—24no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—24no—
469C2×Q8—24no—
470C2×Q8—24no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—42no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2511yescharacteristic
512Q16—88no—
513Q16—88no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8C2×#4114yes—
525C2×C8C2×#4114yes—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—41no—
568C22⋊C4—41no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—82no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—21no—
585C42—21no—
586C42—22no—
587C42—22no—
588C42—44no—
589C42—44no—
590C42—44no—
591C42—82no—
592C42C2×#4114yes—
593C42H(5,𝔽2)12yes—
594C42H(5,𝔽2)12yes—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23C23×D411yescharacteristic
603Q8—44no—
604Q8—44no—
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—84no—
610Q8—84no—
611Q8—84no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4#53212yes—
672C2×C4#53212yes—
673C2×C4#53512yes—
674C2×C4#53512yes—
675C2×C4#57712yes—
676C2×C4#57812yes—
677C2×C4#57812yes—
678C2×C4#57812yes—
679C2×C4C22×#4112yes—
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×H(5,𝔽2)11yescharacteristic
682C2×C4#58512yes—
683C8—24no—
684C8—24no—
685C8—42no—
686C8—42no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22C2×#53211yescharacteristic
695C22C2×#53511yescharacteristic
696C22#344911yescharacteristic
697C22C2×#57711yescharacteristic
698C22C2×#57811yescharacteristic
699C22C2×#57811yescharacteristic
700C22C2×#57811yescharacteristic
701C4—22no—
702C4—22no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4#322314yes—
721C4#322514yes—
722C4#351412yes—
723C4#351712yes—
724C2—81no—
725C2#3519111yescharacteristic
726C2#6071511yescharacteristic
727C2#6072311yescharacteristic
728C2C2×#322311yescharacteristic
729C2C2×#322511yescharacteristic
730C2#6100811yescharacteristic
731C2#6101311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33287C2semidirect
4. #33278C2semidirect
5. #33255C2semidirect
6. #33242C2semidirect
8. #29509C2semidirect
9. #28748C2semidirect
10. #28719C2semidirect
11. #28688C2semidirect
12. #28440C2semidirect
18. #22164C2semidirect

Maximal subgroups

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

subgroupindex
#353222
#352912
#332872
#332782
#332552
#332422
#303312
#295092
#287482
#287192
#286882
#284402
#256742
#256722
#256002
#246642
#221642
C4×#4332