Finite Group Databasebeta

← #196609

Subgroups — #196609

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic, complemented
2#36624C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic, complemented
4C2×#1540C212yesmaximal, complemented
5#9134C212yesmaximal, complemented
6#9133C212yesmaximal, complemented
7#9130C212yesmaximal, complemented
8#9129C212yesmaximal, complemented
9#8039C212yesmaximal, complemented
10C2×#532C2211yescharacteristic
11C2×#525—21no—
12C23×C42C2211yescharacteristic
13C2×#431—21no—
14C2×#431—21no—
15C2×#431—28no—
16C2×#431C2212yes—
17C2×#431C2212yes—
18C22×C4⋊C8C2212yes—
19C22×C4⋊C8C2212yes—
20C22×C22⋊C8—21no—
21C22×C22⋊C8—21no—
22C22×C4×C8—21no—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854—21no—
28#2854C412yes—
29#2850—21no—
30#2850—21no—
31#2847—21no—
32#2847C412yes—
33C4×#121—21no—
34C4×#121—21no—
35C4×#121C412yes—
36C4×#121C412yes—
37#2297—21no—
38#2285C2211yescharacteristic
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43C2×#336—21no—
44#1540—28no—
45C2×#332C2214yescomplemented
46C2×#331C2214yescomplemented
47#1296C2214yescomplemented
48#1296C2214yescomplemented
49C22×#41—21no—
50C24×C4C2×C411yescharacteristic
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C23×C8—22no—
54C23×C8—22no—
55#532—24no—
56#532—24no—
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#124—21no—
60#525—44no—
61#525—44no—
62C2×#123—22no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121C2×C412yes—
71C2×#121C2×C412yes—
72C2×#121C2×C412yes—
73C2×#120—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C412yes—
78C2×#117—21no—
79C22×C42—24no—
80C22×C42—24no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#431—28no—
94#431—28no—
95#431—24no—
96#431—28no—
97#431D418yescomplemented
98C2×C4⋊C8—22no—
99C2×C4⋊C8—28no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—24no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×C22⋊C8—28no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—44no—
111C2×C4×C8—28no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114#391—21no—
115#391—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390C2×C412yes—
121#390C2×C412yes—
122#389—21no—
123#389—21no—
124#389C2×C412yes—
125#389C2×C412yes—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#382—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142#380C2×C412yes—
143#380C2×C412yes—
144#380C2×C412yes—
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×C412yes—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149#336—22no—
150#336—22no—
151#332—28no—
152#331—28no—
153C25C22⋊C411yescharacteristic
154C2×#41—44no—
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—44no—
158C22×Q8—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C412yes—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4D814yes—
185C23×C4SD1614yes—
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C22×C8—28no—
196C22×C8—24no—
197C22×C8—24no—
198C22×C8—44no—
199C22×C8—44no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8—44no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205#127—22no—
206#127—22no—
207#124—44no—
208#123—48no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213#121—28no—
214#121—44no—
215#121—44no—
216#121—42no—
217#121—42no—
218#121—42no—
219#121—42no—
220#121—42no—
221#121—42no—
222#121—44no—
223#121—44no—
224#120—48no—
225C4×D4—28no—
226C4×D4—48no—
227C4×D4—48no—
228C4×D4—42no—
229C4×D4—42no—
230#117—42no—
231#117—42no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D414yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4114yes—
287C2×C42C2411yescharacteristic
288C4⋊C8—28no—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—24no—
292C4⋊C8—44no—
293C4⋊C8C2×D418yes—
294C22⋊C8—28no—
295C22⋊C8—28no—
296C22⋊C8—44no—
297C22⋊C8—44no—
298C4×C8—28no—
299C4×C8—28no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24#10214yes—
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317#41—88no—
318#41—88no—
319C2×Q8—42no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×D4—24no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—216no—
362C22×C4—216no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4#10214yes—
400C22×C4#10214yes—
401C22×C4#10214yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C4×D414yes—
412C22×C4#12012yes—
413C22×C4#12114yes—
414C22×C4#12312yes—
415C22×C4C2×D812yes—
416C22×C4C2×D812yes—
417C22×C4C2×D812yes—
418C22×C4C2×SD1612yes—
419C22×C4C2×SD1612yes—
420C22×C4C2×SD1612yes—
421C22×C4#13514yes—
422C22×C4#13514yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C2×C8—28no—
427C2×C8—416no—
428C2×C8—44no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—44no—
432C2×C8—44no—
433C2×C8—88no—
434C4⋊C4—24no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C42—22no—
459C42—22no—
460C42—21no—
461C42—21no—
462C42—22no—
463C42—22no—
464C42—21no—
465C42—21no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—44no—
473C42C4×D414yes—
474C42#12012yes—
475C42#12114yes—
476C42#12312yes—
477C42C22×D412yes—
478C42C2×#4112yes—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—28no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—28no—
494C23—21no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23#38612yes—
508C23#41312yes—
509C23C2×#10212yes—
510C23C2×#10212yes—
511C23C2×#10212yes—
512C23#41612yes—
513C23#41612yes—
514C23C22×C22⋊C411yescharacteristic
515Q8—82no—
516Q8—82no—
517D4—84no—
518D4—88no—
519D4—88no—
520D4—88no—
521D4—82no—
522D4—82no—
523D4—82no—
524D4—82no—
525C2×C4—28no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—28no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—88no—
577C2×C4#38612yes—
578C2×C4#38612yes—
579C2×C4#38612yes—
580C2×C4#41312yes—
581C2×C4C2×#10212yes—
582C2×C4C2×#10212yes—
583C2×C4C2×#10212yes—
584C2×C4#41612yes—
585C2×C4#41612yes—
586C2×C4#46518yes—
587C2×C4#46618yes—
588C2×C4C2×C4×D412yes—
589C2×C4C2×#12011yescharacteristic
590C2×C4C2×#12112yes—
591C2×C4C2×#12311yescharacteristic
592C2×C4C22×D811yescharacteristic
593C2×C4C22×SD1611yescharacteristic
594C2×C4C2×#13512yes—
595C2×C4C2×#13512yes—
596C2×C4#57512yes—
597C8—48no—
598C8—84no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—416no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22#154014yescomplemented
627C22#189314yes—
628C22C2×#38611yescharacteristic
629C22C2×#41311yescharacteristic
630C22C2×#41311yescharacteristic
631C22C22×#10211yescharacteristic
632C22C2×#41611yescharacteristic
633C22C2×#41611yescharacteristic
634C22#289611yescharacteristic
635C4—22no—
636C4—22no—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—41no—
643C4—41no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4#189314yes—
648C4#300212yes—
649C4C2×#46512yes—
650C4C2×#46612yes—
651C4#309112yes—
652C2—24no—
653C2—24no—
654C2—42no—
655C2—162no—
656C2C2×#154011yescharacteristic
657C2#1770011yescharacteristic
658C2#1770411yescharacteristic
659C2#1771311yescharacteristic
660C2#2090811yescharacteristic
661C2C2×#189311yescharacteristic
662C2#2098911yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#431C2semidirect
3. C2×#1797C2semidirect
4. C2×#1540C2semidirect
5. #9134C2semidirect
6. #9133C2semidirect
7. #9130C2semidirect
8. #9129C2semidirect
9. #8039C2semidirect
45. C2×#332C22semidirect
46. C2×#331C22semidirect
47. #1296C22semidirect
48. #1296C22semidirect
97. #431D4semidirect
626. C22#1540semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
#366242
C2×#17972
C2×#15402
#91342
#91332
#91302
#91292
#80392