Finite Group Databasebeta

← #6494525

Subgroups — #6494525

Order 512.

1593 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic, complemented
2#35288C211yesmaximal, characteristic, complemented
3#33277C211yesmaximal, characteristic, complemented
4#33226C211yesmaximal, characteristic, complemented
5#33175C211yesmaximal, characteristic
6#33049C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#30260C211yesmaximal, characteristic, complemented
9#29842C211yesmaximal, characteristic, complemented
10#29807C211yesmaximal, characteristic, complemented
11#28924C214yesmaximal, complemented
12#28897C214yesmaximal, complemented
13#26982C214yesmaximal, complemented
14#26837C214yesmaximal, complemented
15#22269C211yesmaximal, characteristic, complemented
16#22218C211yesmaximal, characteristic, complemented
17#21692C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#493C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#488C2214yes—
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#469C2214yes—
26C2×#440C2214yes—
27C2×#431C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2856C2211yescharacteristic
36#2856C2211yescharacteristic
37#2644—21no—
38#2644—21no—
39#2624—21no—
40#2624—21no—
41#2513—21no—
42#2513C2212yes—
43#2490—21no—
44#2490C2212yes—
45#2454—21no—
46#2454C2212yes—
47#2448—21no—
48#2448C2212yes—
49#2447—21no—
50#2447C2212yes—
51#2427—21no—
52#2427C2212yes—
53#2426—21no—
54#2426C2212yes—
55#2420—21no—
56#2420C2212yes—
57#2324—21no—
58#2324—21no—
59#2319—21no—
60#2319—21no—
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C4×#127C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#2085C2214yes—
67#2067C2214yes—
68#2059C2214yes—
69#2055C2214yes—
70#2033C2214yes—
71#2012C2214yes—
72#2012C2214yes—
73#2000C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1973C2214yes—
77#1972C2214yes—
78#1958C2214yes—
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1936C2214yes—
86#1935C2214yes—
87#1934C2214yes—
88#1934C2214yes—
89#1927C2214yes—
90#1894C2214yes—
91#1879C2214yes—
92#1879C2214yes—
93#1865C2214yes—
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×SD16C2314yes—
108C22×D8C2314yes—
109C22×M16C2312yes—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127C2311yescharacteristic
113C2×#127C2311yescharacteristic
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#124—21no—
117C2×#124—21no—
118C2×#124C2312yes—
119C2×#124C2312yes—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2312yes—
122C2×#121—21no—
123C2×#121—21no—
124C2×#121—21no—
125C2×#121—21no—
126C2×#121C2312yes—
127C2×#121C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×#117D411yescharacteristic
137C2×#117D411yescharacteristic
138C22×C42C2311yescharacteristic
139#493—24no—
140#492—24no—
141#488—48no—
142#487—24no—
143#486—24no—
144#469—48no—
145#440—48no—
146#431—24no—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2312yes—
159C4×M16—24no—
160C2×#97C2312yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#399—22no—
170#399—22no—
171#397—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#397—22no—
176#397—22no—
177#397—22no—
178#397—22no—
179#396—22no—
180#396—22no—
181#394—22no—
182#394—22no—
183#394—22no—
184#394—22no—
185#391—21no—
186#391—21no—
187#391C2312yes—
188#391C2312yes—
189#390—22no—
190#390—22no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390—21no—
203#390—21no—
204#390—21no—
205#390C2312yes—
206#390C2312yes—
207#390C2312yes—
208#390C2312yes—
209#390C2312yes—
210#390C2312yes—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—21no—
220#389—21no—
221#389—21no—
222#389—21no—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#388—22no—
228#388—22no—
229#388—22no—
230#388—22no—
231#388—22no—
232#388—22no—
233#386—22no—
234#386—22no—
235#385—22no—
236#385—22no—
237#384C2311yescharacteristic
238#384C2311yescharacteristic
239#384C2311yescharacteristic
240#384C2311yescharacteristic
241#382—21no—
242#382C2312yes—
243#381—22no—
244#381—22no—
245#380—21no—
246#380—21no—
247#380—21no—
248#380—22no—
249#380—22no—
250#380C2312yes—
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×D4—22no—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4C2412yes—
284C22×D4C2412yes—
285C23×C4—22no—
286C23×C4—22no—
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D411yescharacteristic
289C23×C4C2411yescharacteristic
290C2×SD16—28no—
291C2×SD16—44no—
292C2×SD16—44no—
293C2×D8—28no—
294C2×D8—44no—
295C2×D8—44no—
296C2×M16—24no—
297C2×M16—24no—
298C2×M16—42no—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307#127—22no—
308#127—22no—
309#127—22no—
310#127—22no—
311#125—44no—
312#125—82no—
313#124—44no—
314#124—44no—
315#124—82no—
316#124—82no—
317C22⋊Q8—44no—
318C22⋊Q8—82no—
319#121—44no—
320#121—44no—
321#121—44no—
322#121—44no—
323#121—82no—
324#121—82no—
325C4×Q8—44no—
326C4×Q8—82no—
327C4×D4—44no—
328C4×D4—44no—
329C4×D4—44no—
330C4×D4—44no—
331C4×D4—82no—
332C4×D4—82no—
333#117—42no—
334#117—42no—
335C2×C4⋊C4—21no—
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—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
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⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C22⋊C4C2×D411yescharacteristic
387C2×C22⋊C4C2×D411yescharacteristic
388C2×C22⋊C4C2×D411yescharacteristic
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409#107—44no—
410#106—44no—
411C4⋊C8—24no—
412#103—28no—
413#103—28no—
414#103—44no—
415#102—28no—
416#102—28no—
417#102—28no—
418#102—44no—
419#102—44no—
420C22⋊C8—24no—
421#97—24no—
422#97—24no—
423#97—42no—
424C4×C8—24no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—42no—
428C4×C8#4114yes—
429C4×C8#4114yes—
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#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450C24—22no—
451C24—22no—
452C24C22×D411yescharacteristic
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—24no—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—42no—
481C2×D4—42no—
482C2×D4—82no—
483C2×D4—82no—
484C2×D4—82no—
485C2×D4—82no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
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—82no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
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×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C2511yescharacteristic
522SD16—416no—
523D8—88no—
524M16—48no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8C2×#4114yes—
535C2×C8C2×#4114yes—
536C2×C8H(5,𝔽2)14yes—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—42no—
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—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C22⋊C4—41no—
567C22⋊C4—41no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—82no—
587C22⋊C4—82no—
588C22⋊C4—82no—
589C22⋊C4—82no—
590C22⋊C4—82no—
591C22⋊C4—82no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—21no—
603C42—21no—
604C42—22no—
605C42—22no—
606C42—22no—
607C42—22no—
608C42—82no—
609C42—82no—
610C42C2×#4112yes—
611C42C2×#4112yes—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23—42no—
626C23—42no—
627C23—42no—
628C23—42no—
629C23—42no—
630C23—42no—
631C23—42no—
632C23—42no—
633C23C23×D411yescharacteristic
634Q8—44no—
635Q8—44no—
636Q8—84no—
637D4—44no—
638D4—44no—
639D4—44no—
640D4—44no—
641D4—84no—
642D4—84no—
643D4—84no—
644D4—84no—
645D4—164no—
646D4—164no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—21no—
675C2×C4—21no—
676C2×C4—21no—
677C2×C4—21no—
678C2×C4—21no—
679C2×C4—21no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—164no—
703C2×C4—164no—
704C2×C4#53212yes—
705C2×C4#53212yes—
706C2×C4#57512yes—
707C2×C4#57512yes—
708C2×C4#57612yes—
709C2×C4#57712yes—
710C2×C4C22×#4111yescharacteristic
711C2×C4C22×#4111yescharacteristic
712C2×C4C2×H(5,𝔽2)12yes—
713C2×C4#58512yes—
714C8—24no—
715C8—24no—
716C8—24no—
717C8—42no—
718C22—41no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—82no—
739C22C2×#53211yescharacteristic
740C22#349811yescharacteristic
741C22#350011yescharacteristic
742C22C2×#57511yescharacteristic
743C22C2×#57511yescharacteristic
744C22C2×#57611yescharacteristic
745C22C2×#57711yescharacteristic
746C4—41no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—41no—
751C4—41no—
752C4—41no—
753C4—41no—
754C4—41no—
755C4—41no—
756C4—41no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4—82no—
763C4#317914yes—
764C4#318314yes—
765C4#346012yes—
766C4#349412yes—
767C2—81no—
768C2—162no—
769C2—162no—
770C2#3527611yescharacteristic
771C2C2×#317911yescharacteristic
772C2C2×#318311yescharacteristic
773C2#6242611yescharacteristic
774C2#6242711yescharacteristic
775C2#6254311yescharacteristic
776C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#33175C2
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. #35288C2semidirect
2. #35288C2semidirect
3. #33277C2semidirect
4. #33226C2semidirect
6. #33049C2semidirect
7. #30307C2semidirect
8. #30260C2semidirect
9. #29842C2semidirect
10. #29807C2semidirect
11. #28924C2semidirect
12. #28897C2semidirect
13. #26982C2semidirect
14. #26837C2semidirect
15. #22269C2semidirect
16. #22218C2semidirect
17. #21692C2semidirect
18. #21646C2semidirect
19. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#332772
#332262
#331752
#330492
#303072
#302602
#298422
#298072
#289242
#288972
#269822
#268372
#222692
#222182
#216922
#216462
#204722