Finite Group Databasebeta

← #163899

Subgroups — #163899

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34258C211yesmaximal, characteristic
2#22471C212yesmaximal, complemented
3#7645C211yesmaximal, characteristic, complemented
4#7640C211yesmaximal, characteristic, complemented
5#7631C211yesmaximal, characteristic, complemented
6#7626C211yesmaximal, characteristic, complemented
7#7612C212yesmaximal, complemented
8#7606C212yesmaximal, complemented
9#7592C211yesmaximal, characteristic, complemented
10#7578C212yesmaximal, complemented
11C2×#1270C211yesmaximal, characteristic, complemented
12C22×H(5,𝔽2)C411yescharacteristic
13C2×#520C411yescharacteristic
14C2×C4×#41C2211yescharacteristic
15C2×#435—22no—
16#2341—22no—
17D4×C22⋊C4—22no—
18D4×C22⋊C4—22no—
19D4×C22⋊C4—22no—
20#2311C411yescharacteristic
21#2311C411yescharacteristic
22#2311C411yescharacteristic
23#2309C411yescharacteristic
24#2309C411yescharacteristic
25#2309C411yescharacteristic
26#2297C2211yescharacteristic
27#2297C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32#1985—22no—
33#1870—22no—
34#1869—22no—
35#1817C2212yescomplemented
36#1817C2212yescomplemented
37#1797C2212yescomplemented
38#1749C2212yescomplemented
39#1728C2212yescomplemented
40C2×#336C2212yescomplemented
41#1272C2212yescomplemented
42#1272C2212yescomplemented
43#1270C2212yescomplemented
44#1270C2211yescharacteristic, complemented
45#1270C2211yescharacteristic, complemented
46#1270C2212yescomplemented
47#1270C2212yescomplemented
48#1270C2212yescomplemented
49#1270C2211yescharacteristic, complemented
50#1270C2211yescharacteristic, complemented
51C2×H(5,𝔽2)—24no—
52C2×H(5,𝔽2)—24no—
53C2×H(5,𝔽2)—44no—
54C2×H(5,𝔽2)—44no—
55C22×#41—21no—
56C22×#41C2×C411yescharacteristic
57C22×#41C2×C411yescharacteristic
58C22×#41C2×C411yescharacteristic
59C22×#41C2×C411yescharacteristic
60C23×D4—21no—
61C23×D4—21no—
62C23×D4—21no—
63C23×D4C2×C411yescharacteristic
64C23×D4C2×C411yescharacteristic
65C23×D4C2×C411yescharacteristic
66C22×M16—22no—
67C23×C8—22no—
68#520—44no—
69C4×#41—44no—
70C2×C4×Q8—21no—
71C2×C4×Q8D412yes—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4D412yes—
76C2×C4×D4D412yes—
77C2×C4×D4D412yes—
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C2×#117C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C22⋊C4—22no—
88C22×C22⋊C4—22no—
89C22×C22⋊C4—22no—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#435—48no—
94C2×C4⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×#97—22no—
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2311yescharacteristic
104#390—22no—
105#390—22no—
106#390—22no—
107#390—22no—
108#390—22no—
109#390—22no—
110#390C2×C411yescharacteristic
111#390C2×C411yescharacteristic
112#390C2×C411yescharacteristic
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#390C2311yescharacteristic
120#390C2311yescharacteristic
121#390C2311yescharacteristic
122#386—22no—
123#386—22no—
124#386—22no—
125#386—22no—
126#386—22no—
127#386—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131#379—22no—
132#379—22no—
133#379—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336C2312yes—
142#336C2312yes—
143#336C2312yes—
144#336C2312yes—
145C25—21no—
146C25—21no—
147C25—21no—
148H(5,𝔽2)—432no—
149H(5,𝔽2)—432no—
150C2×#41—28no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—24no—
154C2×#41—24no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—42no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—44no—
167C2×#41—44no—
168C2×#41M1614yescomplemented
169C22×Q8C22⋊C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4C22⋊C411yescharacteristic
198C22×D4C22⋊C411yescharacteristic
199C22×D4C22⋊C411yescharacteristic
200C22×D4M1614yescomplemented
201C22×D4C22×C411yescharacteristic
202C22×D4C22×C411yescharacteristic
203C22×D4C22×C411yescharacteristic
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C2×M16—44no—
217C2×M16—44no—
218C2×M16—82no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—82no—
226C4×Q8—44no—
227C4×Q8—44no—
228C4×D4—28no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—44no—
234#117—42no—
235#117—42no—
236#117—42no—
237#117—42no—
238#117—42no—
239#117—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4C22⋊C412yes—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—82no—
269C2×C22⋊C4—82no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4C22⋊C412yes—
275C2×C22⋊C4C22⋊C412yes—
276C2×C22⋊C4C22⋊C412yes—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2411yescharacteristic
290C4⋊C8—44no—
291C22⋊C8—44no—
292C22⋊C8—44no—
293C22⋊C8—44no—
294C22⋊C8—82no—
295#97—44no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300C24—24no—
301C24—24no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—24no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—82no—
313C24—82no—
314C24—82no—
315C24—82no—
316C24—82no—
317C24—82no—
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321#41—216no—
322#41—216no—
323#41—48no—
324#41—48no—
325#41—48no—
326#41—48no—
327#41—48no—
328#41—48no—
329#41—432no—
330C2×Q8—21no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8C2×M1612yes—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—24no—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—22no—
352C2×D4—24no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—48no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—432no—
370C2×D4—432no—
371C2×D4—432no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4#9914yes—
381C2×D4#10014yes—
382C2×D4C2×M1612yes—
383C2×D4C2×M1612yes—
384C2×D4C2×M1612yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4C2×C22⋊C412yes—
428C22×C4C2×C22⋊C412yes—
429C22×C4C2×C22⋊C412yes—
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×M1612yes—
434C22×C4C2×M1612yes—
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D412yes—
437M16—88no—
438C2×C8—24no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—88no—
451C4⋊C4—24no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C42—24no—
483C42—41no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—42no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—42no—
492C23—22no—
493C23—22no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—48no—
506C23—48no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—48no—
520C23—88no—
521C23—88no—
522C23—88no—
523C23—162no—
524C23—162no—
525C23—164no—
526C23—162no—
527C23—162no—
528C23—164no—
529C23—162no—
530C23—162no—
531C23—164no—
532C23#40712yes—
533C23C2×#9912yes—
534C23C2×#9912yes—
535C23C2×#10012yes—
536C23C2×#10012yes—
537C23C22×C22⋊C411yescharacteristic
538Q8—28no—
539Q8—42no—
540Q8—42no—
541D4—28no—
542D4—28no—
543D4—28no—
544D4—42no—
545D4—42no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—48no—
551D4—48no—
552D4—48no—
553D4—48no—
554D4—48no—
555D4—48no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—162no—
602C2×C4—162no—
603C2×C4—162no—
604C2×C4#40712yes—
605C2×C4C2×#9912yes—
606C2×C4C2×#10012yes—
607C2×C4C22×M1611yescharacteristic
608C2×C4C22×M1611yescharacteristic
609C2×C4#57612yes—
610C2×C4#57712yes—
611C8—44no—
612C8—82no—
613C8—82no—
614C8—82no—
615C22—22no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—41no—
628C22—42no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—168no—
645C22—168no—
646C22—168no—
647C22#146214yes—
648C22C2×#40711yescharacteristic
649C22C22×#9911yescharacteristic
650C22C22×#10011yescharacteristic
651C22#289712yes—
652C22#290112yes—
653C4—22no—
654C4—22no—
655C4—21no—
656C4—21no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—81no—
668C2—24no—
669C2—42no—
670C2—42no—
671C2—82no—
672C2—82no—
673C2—82no—
674C2C2×#146211yescharacteristic
675C2#1745312yes—
676C2#1760312yes—
677C2#1768511yescharacteristic
678C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×H(5,𝔽2)C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22471C2semidirect
3. #7645C2semidirect
4. #7640C2semidirect
5. #7631C2semidirect
6. #7626C2semidirect
7. #7612C2semidirect
8. #7606C2semidirect
9. #7592C2semidirect
10. #7578C2semidirect
11. C2×#1270C2semidirect
35. #1817C22semidirect
36. #1817C22semidirect
37. #1797C22semidirect
38. #1749C22semidirect
39. #1728C22semidirect
40. C2×#336C22semidirect
41. #1272C22semidirect
42. #1272C22semidirect
43. #1270C22semidirect
44. #1270C22semidirect
45. #1270C22semidirect
46. #1270C22semidirect
47. #1270C22semidirect
48. #1270C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect
168. C2×#41M16semidirect
200. C22×D4M16semidirect

Maximal subgroups

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

subgroupindex
#342582
#224712
#76452
#76402
#76312
#76262
#76122
#76062
#75922
#75782
C2×#12702