Finite Group Databasebeta

← #6480602

Subgroups — #6480602

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic
2#35291C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33255C211yesmaximal, characteristic, complemented
6#33242C211yesmaximal, characteristic, complemented
7#30351C214yesmaximal, complemented
8#29565C214yesmaximal, complemented
9#28656C211yesmaximal, characteristic, complemented
10#28653C211yesmaximal, characteristic, complemented
11#28616C211yesmaximal, characteristic, complemented
12#28616C211yesmaximal, characteristic, complemented
13#25904C211yesmaximal, characteristic
14#25856C211yesmaximal, characteristic
15#25812C211yesmaximal, characteristic
16#25722C211yesmaximal, characteristic
17#24514C214yesmaximal
18#22054C214yesmaximal
19C4×#433C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#494C2211yescharacteristic
25C2×#494C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#487C2211yescharacteristic
28C2×#487C2211yescharacteristic
29C2×#487C2211yescharacteristic
30C2×#451C2214yes—
31C2×#433C2212yes—
32#2862C2211yescharacteristic
33#2862C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2862C2211yescharacteristic
37#2859C2211yescharacteristic
38#2857C2211yescharacteristic
39#2857C2211yescharacteristic
40#2645C2212yes—
41#2642C2212yes—
42#2642C2212yes—
43#2625C2212yes—
44#2568—21no—
45#2568—21no—
46#2568—21no—
47#2568—21no—
48#2514—21no—
49#2509—21no—
50#2508—21no—
51#2503—21no—
52#2503—21no—
53#2503—21no—
54#2503—21no—
55#2495—21no—
56#2494—21no—
57#2489—21no—
58#2453—21no—
59#2421—21no—
60#2323C2212yes—
61#2320C2212yes—
62#2310C2212yes—
63#2310C2212yes—
64C4×C4⋊Q8C2211yescharacteristic
65C4×C4⋊Q8C2211yescharacteristic
66C4×#117C2211yescharacteristic
67#2068C2214yes—
68#2066C2214yes—
69#2059C2214yes—
70#2056C2214yes—
71#2056C2214yes—
72#2043C2214yes—
73#2026C2214yes—
74#2016C2214yes—
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1971C2214yes—
80#1971C2214yes—
81#1969C2214yes—
82#1969C2214yes—
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1935C2214yes—
88#1935C2214yes—
89#1933C2214yes—
90#1933C2214yes—
91#1928C2214yes—
92#1927C2214yes—
93#1878C2214yes—
94#1878C2214yes—
95#1873C2214yes—
96#1854C2214yes—
97#1787C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8C2312yes—
108C22×Q16C2314yes—
109C22×Q16C2314yes—
110#533—42no—
111#533—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—42no—
115C2×C4⋊Q8—42no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×#125—21no—
124C2×#125C2312yes—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8C2312yes—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8C2312yes—
133C2×#117C2312yes—
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C42C2311yescharacteristic
139#500—42no—
140#498—42no—
141#494—42no—
142#494—42no—
143#487—42no—
144#487—42no—
145#487—42no—
146#487—42no—
147#451—216no—
148#433—28no—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2312yes—
170C2×C4×C8C2312yes—
171#401—22no—
172#400—22no—
173#400—22no—
174#400—22no—
175#400—22no—
176#400—22no—
177#400—22no—
178#400—22no—
179#400—22no—
180#400—22no—
181#400—22no—
182#400—22no—
183#400—22no—
184#398—22no—
185#397—22no—
186#397—22no—
187#391—22no—
188#391—22no—
189#391—22no—
190#391—22no—
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#391—21no—
202#391—21no—
203#391C2312yes—
204#391C2312yes—
205#391C2312yes—
206#391C2312yes—
207#391C2312yes—
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#389—22no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—21no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#388—22no—
229#388—22no—
230#387—22no—
231#387—22no—
232#387—22no—
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#383—22no—
236#382—21no—
237#382—21no—
238#382—21no—
239#382—21no—
240#382—21no—
241#382—21no—
242#382—21no—
243#382—21no—
244#382—21no—
245#382—21no—
246#382—21no—
247#382—22no—
248#381—22no—
249#381—22no—
250#381—22no—
251#381—22no—
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×Q8—22no—
284C22×Q8—28no—
285C22×Q8—42no—
286C22×Q8—42no—
287C22×Q8—44no—
288C22×Q8C2412yes—
289C22×Q8C2412yes—
290C22×Q8C2412yes—
291C23×C4—22no—
292C23×C4C2×D411yescharacteristic
293C23×C4C2×D411yescharacteristic
294C23×C4C2411yescharacteristic
295C2×Q16—28no—
296C2×Q16—28no—
297C2×Q16—44no—
298C2×Q16—44no—
299C2×Q16—44no—
300C2×Q16—44no—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—22no—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C4⋊Q8—42no—
318C4⋊Q8—42no—
319C4⋊Q8—44no—
320C4⋊Q8—44no—
321#125—42no—
322#125—42no—
323#125—82no—
324C22⋊Q8—28no—
325C22⋊Q8—44no—
326C22⋊Q8—44no—
327C22⋊Q8—82no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×Q8—44no—
331C4×Q8—82no—
332#117—24no—
333#117—42no—
334#117—42no—
335#117—42no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
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—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—44no—
360C2×C4⋊C4—44no—
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2×D411yescharacteristic
364C2×C4⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398#107—44no—
399#106—44no—
400C4⋊C8—24no—
401C4⋊C8—24no—
402#103—28no—
403#103—28no—
404#103—44no—
405#103—44no—
406#103—44no—
407#103—44no—
408#103—44no—
409#103—44no—
410C22⋊C8—24no—
411C22⋊C8#4118yes—
412#97—24no—
413#97—42no—
414C4×C8—24no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—22no—
419C4×C8—42no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
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—44no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—42no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—42no—
481C2×Q8—42no—
482C2×Q8—42no—
483C2×Q8—88no—
484C2×Q8—88no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
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×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C2×#4114yes—
519C22×C4C2511yescharacteristic
520Q16—88no—
521Q16—88no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8C2×#4114yes—
533C2×C8C2×#4114yes—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—24no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
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—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—41no—
572C22⋊C4—41no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—82no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—21no—
589C42—21no—
590C42—22no—
591C42—22no—
592C42—44no—
593C42—44no—
594C42—82no—
595C42H(5,𝔽2)12yes—
596C42H(5,𝔽2)12yes—
597C23—21no—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23#53212yes—
604C23C23×D411yescharacteristic
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—44no—
611Q8—88no—
612Q8—84no—
613Q8—84no—
614Q8—84no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—24no—
618C2×C4—24no—
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—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—44no—
650C2×C4—44no—
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—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—88no—
676C2×C4#53212yes—
677C2×C4#53512yes—
678C2×C4#53512yes—
679C2×C4#57712yes—
680C2×C4#57812yes—
681C2×C4#57812yes—
682C2×C4#57812yes—
683C2×C4C22×#4112yes—
684C2×C4C2×H(5,𝔽2)11yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4#58512yes—
687C8—24no—
688C8—24no—
689C8—42no—
690C8—42no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—41no—
695C22—41no—
696C22—41no—
697C22—41no—
698C22C2×#53211yescharacteristic
699C22C2×#53511yescharacteristic
700C22#344911yescharacteristic
701C22C2×#57711yescharacteristic
702C22C2×#57811yescharacteristic
703C22C2×#57811yescharacteristic
704C22C2×#57811yescharacteristic
705C4—22no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4#322314yes—
725C4#322514yes—
726C4#351412yes—
727C4#351412yes—
728C2—42no—
729C2#3513111yescharacteristic
730C2#6071511yescharacteristic
731C2#6072311yescharacteristic
732C2C2×#322311yescharacteristic
733C2C2×#322511yescharacteristic
734C2#6100811yescharacteristic
735C2#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
7. #30351C2semidirect
8. #29565C2semidirect
9. #28656C2semidirect
10. #28653C2semidirect
11. #28616C2semidirect
12. #28616C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#352912
#332872
#332782
#332552
#332422
#303512
#295652
#286562
#286532
#286162
#259042
#258562
#258122
#257222
#245142
#220542
C4×#4332