Finite Group Databasebeta

← #6434456

Subgroups — #6434456

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#30128C214yesmaximal, complemented
4#30075C212yesmaximal, complemented
5#30039C212yesmaximal, complemented
6#29950C212yesmaximal, complemented
7#29909C212yesmaximal, complemented
8#29714C212yesmaximal, complemented
9#29625C212yesmaximal
10#23822C212yesmaximal, complemented
11#23803C212yesmaximal, complemented
12#23684C214yesmaximal, complemented
13#21453C214yesmaximal, complemented
14#20778C211yesmaximal, characteristic, complemented
15C22×#127C2211yescharacteristic
16C22×C4×D4—21no—
17C22×C4×D4—21no—
18C22×C4×D4C2212yes—
19C2×#450C2214yes—
20C2×#440C2218yescomplemented
21#2864—21no—
22#2863—21no—
23#2863C2212yescomplemented
24#2862C2211yescharacteristic, complemented
25#2859C2211yescharacteristic, complemented
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2855—21no—
29#2855C2212yes—
30#2851—21no—
31C4×C4⋊Q8C2211yescharacteristic, complemented
32C4×C4⋊Q8C2211yescharacteristic, complemented
33C4×#127—21no—
34C4×#127—21no—
35C4×#127—21no—
36C4×#127—21no—
37C4×#127C2212yes—
38C4×#127C2212yes—
39C4×#127C2212yes—
40C4×#127C2212yes—
41C4×#125—21no—
42C4×C22⋊Q8—21no—
43C4×C22⋊Q8C2212yes—
44#2295—21no—
45#2295C2212yescomplemented
46C2×#390—21no—
47C2×#390—21no—
48C2×#390C2212yes—
49C2×#384C2211yescharacteristic
50#2283—21no—
51C2×C43C2211yescharacteristic
52#2100C2214yescomplemented
53#2095C2214yescomplemented
54#2093C2214yescomplemented
55#2083C2214yescomplemented
56#2072C2214yescomplemented
57#2071C2214yescomplemented
58#2067C2218yescomplemented
59#2055C2218yescomplemented
60#2041C2214yes—
61#2041C2214yes—
62#2023C2214yescomplemented
63#2014C2214yes—
64#2014C2214yes—
65#2013C2214yescomplemented
66#1877C2218yescomplemented
67#1872C2214yes—
68#1862C2218yescomplemented
69#1852C2214yes—
70#1839C2212yescomplemented
71#1831C2212yescomplemented
72#1817C2212yes—
73#1797C2212yes—
74#1786C2214yes—
75#1767C2212yescomplemented
76#1762C2218yescomplemented
77#1749C2212yes—
78#1745C2212yescomplemented
79C23×D4—21no—
80C23×D4—21no—
81C23×D4C2312yes—
82C24×C4—22no—
83C22×SD16C2318yes—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×#127—28no—
87C2×#127—21no—
88C2×#127—21no—
89C2×#127C2312yes—
90C2×#127C2312yes—
91C2×#125—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8C2312yes—
94C2×C4×Q8—21no—
95C2×C4×Q8C2312yes—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—48no—
107C2×C4×D4—48no—
108C2×C4×D4—48no—
109C2×C4×D4C2314yes—
110C2×C4×D4C2314yes—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42D411yescharacteristic
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#450—216no—
125#440—416no—
126C2×#107C2318yes—
127C2×C4⋊C8C2314yes—
128C2×#103C2318yes—
129C2×#102C2318yes—
130C2×#102C2318yes—
131C2×#102C2318yes—
132C2×C22⋊C8C2314yes—
133C2×#97C2314yes—
134#391—21no—
135#391—21no—
136#391—21no—
137#391—21no—
138#391C2312yes—
139#391C2312yes—
140#391C2312yes—
141#391C2312yes—
142#390—22no—
143#390—22no—
144#390—22no—
145#390—22no—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—22no—
150#390—22no—
151#390—22no—
152#390C2314yes—
153#390C2314yes—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#385—22no—
175#385—22no—
176#384—21no—
177#384—21no—
178#384C2312yes—
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#383—22no—
183#382—21no—
184#382—21no—
185#382—21no—
186#381—22no—
187#380—21no—
188#380—21no—
189#380—21no—
190#380C2312yes—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2312yes—
197C4×C22⋊C4—22no—
198#376—22no—
199C43D411yescharacteristic
200C43D411yescharacteristic
201C43D411yescharacteristic
202C43D411yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2318yes—
208#340C2318yes—
209#336C2314yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25—22no—
214C22×Q8—21no—
215C22×Q8C2412yes—
216C22×D4—28no—
217C22×D4—28no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4—24no—
224C22×D4—44no—
225C22×D4—44no—
226C22×D4—44no—
227C22×D4—44no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4#4118yes—
231C22×D4C2414yes—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—48no—
236C23×C4—48no—
237C23×C4—48no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×SD16—216no—
246C2×SD16—48no—
247C2×SD16—48no—
248C22×C8C2418yes—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251#127—28no—
252#127—28no—
253#127—28no—
254#127—28no—
255#125—44no—
256C22⋊Q8—28no—
257C22⋊Q8—44no—
258C4×Q8—28no—
259C4×Q8—44no—
260C4×D4—216no—
261C4×D4—48no—
262C4×D4—48no—
263C4×D4—48no—
264C4×D4—48no—
265C4×D4—48no—
266C4×D4—48no—
267C4×D4—48no—
268C4×D4—48no—
269C4×D4—48no—
270C4×D4—48no—
271C4×D4—48no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345#107—48no—
346C4⋊C8—44no—
347#103—48no—
348#102—216no—
349#102—48no—
350#102—48no—
351C22⋊C8—28no—
352#97—28no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367C24—28no—
368C24—24no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24C22×D411yescharacteristic
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—24no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—416no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—432no—
396C2×D4—432no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—416no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4C2×#4118yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—832no—
438C22×C4AGL(1,ℤ/8)14yes—
439C22×C4#13714yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2×#4114yes—
454C22×C4C2511yescharacteristic
455SD16—432no—
456C2×C8—28no—
457C2×C8—28no—
458C2×C8—28no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—44no—
516C42—44no—
517C42AGL(1,ℤ/8)14yes—
518C42H(5,𝔽2)14yes—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—28no—
524C23—416no—
525C23—416no—
526C23—416no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—88no—
534C23—88no—
535C23—88no—
536C23—88no—
537C23#52512yes—
538C23C2×AGL(1,ℤ/8)12yes—
539C23C2×#13712yes—
540C23C23×D411yescharacteristic
541Q8—48no—
542Q8—84no—
543D4—216no—
544D4—416no—
545D4—416no—
546D4—416no—
547D4—416no—
548D4—416no—
549D4—48no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—28no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—816no—
601C2×C4—816no—
602C2×C4—88no—
603C2×C4#52512yes—
604C2×C4#53512yes—
605C2×C4#53512yes—
606C2×C4C2×AGL(1,ℤ/8)12yes—
607C2×C4C2×AGL(1,ℤ/8)12yes—
608C2×C4C2×AGL(1,ℤ/8)12yes—
609C2×C4C2×AGL(1,ℤ/8)12yes—
610C2×C4C2×AGL(1,ℤ/8)12yes—
611C2×C4C2×#13712yes—
612C2×C4C2×#13712yes—
613C2×C4#57512yes—
614C2×C4C22×#4112yes—
615C2×C4C2×H(5,𝔽2)12yes—
616C2×C4#58512yes—
617C8—48no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—48no—
625C22—48no—
626C22—48no—
627C22—832no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—84no—
632C22#311014yes—
633C22C2×#52511yescharacteristic
634C22C2×#53511yescharacteristic
635C22#344911yescharacteristic
636C22C22×AGL(1,ℤ/8)11yescharacteristic
637C22C22×AGL(1,ℤ/8)11yescharacteristic
638C22C22×#13711yescharacteristic
639C22C2×#57511yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—22no—
650C4—22no—
651C4—41no—
652C4—41no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4—82no—
658C4#345914yes—
659C2—24no—
660C2—88no—
661C2#3459311yescharacteristic
662C2C2×#311011yescharacteristic
663C2#6055811yescharacteristic
664C2#6079811yescharacteristic
665C2#6080411yescharacteristic
666C2#6095811yescharacteristic
667C2#6096911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #30128C2semidirect
4. #30075C2semidirect
5. #30039C2semidirect
6. #29950C2semidirect
7. #29909C2semidirect
8. #29714C2semidirect
10. #23822C2semidirect
11. #23803C2semidirect
12. #23684C2semidirect
13. #21453C2semidirect
14. #20778C2semidirect
20. C2×#440C22semidirect
23. #2863C22semidirect
24. #2862C22semidirect
25. #2859C22semidirect
31. C4×C4⋊Q8C22semidirect
32. C4×C4⋊Q8C22semidirect
45. #2295C22semidirect
52. #2100C22semidirect
53. #2095C22semidirect
54. #2093C22semidirect
55. #2083C22semidirect
56. #2072C22semidirect
57. #2071C22semidirect
58. #2067C22semidirect
59. #2055C22semidirect
62. #2023C22semidirect
65. #2013C22semidirect
66. #1877C22semidirect
68. #1862C22semidirect
70. #1839C22semidirect
71. #1831C22semidirect
75. #1767C22semidirect
76. #1762C22semidirect
78. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×C4×#1272
#301282
#300752
#300392
#299502
#299092
#297142
#296252
#238222
#238032
#236842
#214532
#207782