Finite Group Databasebeta

← #6494530

Subgroups — #6494530

Order 512.

1511 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35288C211yesmaximal, characteristic
3#33278C211yesmaximal, characteristic, complemented
4#33226C211yesmaximal, characteristic, complemented
5#33174C211yesmaximal, characteristic
6#33049C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#30261C211yesmaximal, characteristic, complemented
9#29843C211yesmaximal, characteristic
10#29808C211yesmaximal, characteristic, complemented
11#28922C214yesmaximal, complemented
12#28901C214yesmaximal, complemented
13#26989C214yesmaximal, complemented
14#26839C214yesmaximal, complemented
15#22268C211yesmaximal, characteristic, complemented
16#22221C211yesmaximal, characteristic, complemented
17#21691C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#488C2214yes—
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#468C2214yes—
26C2×#441C2214yes—
27C2×#431C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2645—21no—
38#2644—21no—
39#2625—21no—
40#2624—21no—
41#2514C2212yes—
42#2513—21no—
43#2509—21no—
44#2508—21no—
45#2495—21no—
46#2494—21no—
47#2490—21no—
48#2489C2212yes—
49#2454—21no—
50#2453C2212yes—
51#2448C2212yes—
52#2447C2212yes—
53#2427C2212yes—
54#2426C2212yes—
55#2421C2212yes—
56#2420—21no—
57#2324—21no—
58#2323—21no—
59#2320—21no—
60#2319—21no—
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×#127C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#2085C2214yes—
67#2065C2214yes—
68#2059C2214yes—
69#2056C2214yes—
70#2034C2214yes—
71#2015C2214yes—
72#2015C2214yes—
73#2001C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1973C2214yes—
77#1971C2214yes—
78#1958C2214yes—
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1936C2214yes—
86#1935C2214yes—
87#1933C2214yes—
88#1933C2214yes—
89#1928C2214yes—
90#1894C2214yes—
91#1880C2214yes—
92#1877C2214yes—
93#1863C2214yes—
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1771C2212yes—
99#1771C2212yes—
100#1762C2214yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109C22×M16C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#124C2312yes—
117C2×#124C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121C2312yes—
124C2×C4×Q8—21no—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4C2312yes—
130C2×#117D411yescharacteristic
131C2×#117D411yescharacteristic
132C22×C42C2311yescharacteristic
133#494—24no—
134#492—24no—
135#488—48no—
136#487—24no—
137#486—24no—
138#468—48no—
139#441—48no—
140#431—24no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C4×M16—24no—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2312yes—
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163#401—22no—
164#400—22no—
165#400—22no—
166#400—22no—
167#400—22no—
168#399—22no—
169#398—22no—
170#397—22no—
171#397—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#396—22no—
177#394—22no—
178#394—22no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#390—22no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390C2312yes—
199#390C2312yes—
200#389—22no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389C2312yes—
218#389C2312yes—
219#389C2312yes—
220#389C2312yes—
221#388—22no—
222#388—22no—
223#388—22no—
224#388—22no—
225#388—22no—
226#387—22no—
227#387—22no—
228#387—22no—
229#386—22no—
230#385—22no—
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#384C2311yescharacteristic
235#383—22no—
236#382—21no—
237#382—21no—
238#382—21no—
239#382—22no—
240#382C2312yes—
241#381—22no—
242#380—21no—
243#380—21no—
244#380—21no—
245#380—21no—
246#380—21no—
247#380—22no—
248#380C2312yes—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4C2311yescharacteristic
262C4×C22⋊C4C2311yescharacteristic
263#376D411yescharacteristic
264#376D411yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8C2412yes—
277C22×Q8C2412yes—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4C2412yes—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286C2×Q16—28no—
287C2×Q16—44no—
288C2×Q16—44no—
289C2×SD16—28no—
290C2×SD16—44no—
291C2×SD16—44no—
292C2×M16—24no—
293C2×M16—24no—
294C2×M16—42no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305#127—22no—
306#127—22no—
307#125—44no—
308#125—82no—
309#124—44no—
310#124—44no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C22⋊Q8—44no—
314#121—44no—
315#121—44no—
316#121—82no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×D4—44no—
321C4×D4—44no—
322C4×D4—82no—
323#117—42no—
324#117—42no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
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—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—44no—
396#106—44no—
397C4⋊C8—24no—
398#103—28no—
399#103—28no—
400#103—28no—
401#103—44no—
402#103—44no—
403#102—28no—
404#102—28no—
405#102—44no—
406C22⋊C8—24no—
407#97—24no—
408#97—24no—
409#97—42no—
410C4×C8—24no—
411C4×C8—22no—
412C4×C8—22no—
413C4×C8—42no—
414C4×C8#4114yes—
415C4×C8#4114yes—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
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#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444C24—22no—
445C24C22×D411yescharacteristic
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—82no—
469C2×D4—82no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C2511yescharacteristic
504Q16—416no—
505SD16—88no—
506M16—48no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8C2×#4114yes—
517C2×C8C2×#4114yes—
518C2×C8H(5,𝔽2)14yes—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—21no—
580C42—21no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—44no—
586C42—82no—
587C42—82no—
588C42C2×#4112yes—
589C42C2×#4112yes—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23C23×D411yescharacteristic
605Q8—44no—
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—84no—
610D4—44no—
611D4—44no—
612D4—84no—
613D4—84no—
614D4—164no—
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—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—44no—
649C2×C4—44no—
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—42no—
672C2×C4—42no—
673C2×C4—164no—
674C2×C4#53212yes—
675C2×C4#53212yes—
676C2×C4#57512yes—
677C2×C4#57512yes—
678C2×C4#57712yes—
679C2×C4#57812yes—
680C2×C4C22×#4111yescharacteristic
681C2×C4C22×#4111yescharacteristic
682C2×C4C2×H(5,𝔽2)12yes—
683C2×C4#58512yes—
684C8—24no—
685C8—24no—
686C8—24no—
687C8—42no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22C2×#53211yescharacteristic
703C22#349811yescharacteristic
704C22#350011yescharacteristic
705C22C2×#57511yescharacteristic
706C22C2×#57511yescharacteristic
707C22C2×#57711yescharacteristic
708C22C2×#57811yescharacteristic
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4#317814yes—
728C4#318314yes—
729C4#346012yes—
730C4#349412yes—
731C2—81no—
732C2—162no—
733C2#3527611yescharacteristic
734C2C2×#317811yescharacteristic
735C2C2×#318311yescharacteristic
736C2#6242511yescharacteristic
737C2#6242711yescharacteristic
738C2#6254411yescharacteristic
739C2#6254511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
1. #35291C2semidirect
3. #33278C2semidirect
4. #33226C2semidirect
6. #33049C2semidirect
7. #30307C2semidirect
8. #30261C2semidirect
10. #29808C2semidirect
11. #28922C2semidirect
12. #28901C2semidirect
13. #26989C2semidirect
14. #26839C2semidirect
15. #22268C2semidirect
16. #22221C2semidirect
17. #21691C2semidirect
18. #21646C2semidirect
19. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352882
#332782
#332262
#331742
#330492
#303072
#302612
#298432
#298082
#289222
#289012
#269892
#268392
#222682
#222212
#216912
#216462
#204722