Finite Group Databasebeta

← #6493485

Subgroups — #6493485

Order 512.

2106 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic, complemented
2D4×#117C211yesmaximal, characteristic, complemented
3#33160C211yesmaximal, characteristic, complemented
4#33068C211yesmaximal, characteristic, complemented
5#32902C212yesmaximal, complemented
6#29998C211yesmaximal, characteristic, complemented
7#29921C212yesmaximal, complemented
8#29794C211yesmaximal, characteristic, complemented
9#27411C214yesmaximal, complemented
10#27211C214yesmaximal, complemented
11#26174C214yesmaximal, complemented
12#25168C214yesmaximal, complemented
13#22151C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21503C212yesmaximal, complemented
16C2×#540—21no—
17C22×C4×D4C2211yescharacteristic
18C22×#117—21no—
19C2×#459C2212yes—
20C2×#459C2212yes—
21C2×#459C2214yes—
22C2×#447C2214yes—
23C2×#442C2214yes—
24#2867C2211yescharacteristic, complemented
25#2859C2211yescharacteristic, complemented
26#2856C2212yes—
27#2641—21no—
28#2619—21no—
29#2445—22no—
30#2445C2214yescomplemented
31#2428—22no—
32#2428C2214yes—
33#2354C2212yes—
34#2352—22no—
35D4×C4⋊C4C2212yes—
36#2344C2212yes—
37#2342—22no—
38D4×C22⋊C4C2212yes—
39#2328—21no—
40C4×#127C2212yescomplemented
41#2293C2211yescharacteristic
42#2293C2211yescharacteristic
43#2290—21no—
44#2290—21no—
45#2285C2211yescharacteristic
46#2280—21no—
47C42×D4C2211yescharacteristic
48C42×D4C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2094C2214yescomplemented
51#2088C2214yescomplemented
52#2071C2214yescomplemented
53#2065C2214yes—
54#2041C2214yescomplemented
55#2034C2214yes—
56#2034C2214yes—
57#2011C2214yes—
58#1982C2214yescomplemented
59#1982C2212yescomplemented
60#1982C2212yescomplemented
61#1972C2214yes—
62#1948C2212yescomplemented
63#1948C2212yescomplemented
64#1936C2214yescomplemented
65#1926C2214yescomplemented
66#1924C2214yes—
67#1920C2214yescomplemented
68#1917C2214yescomplemented
69#1879C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1879C2214yes—
73#1874C2214yes—
74#1864C2214yes—
75#1864C2214yescomplemented
76#1842C2211yescharacteristic, complemented
77#1831C2211yescharacteristic, complemented
78#1802C2212yes—
79#1802C2212yes—
80#1797C2212yes—
81#1767C2212yescomplemented
82#1762C2214yescomplemented
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C22×D8C2314yes—
86C22×D8C2314yes—
87#540—48no—
88C2×#127—21no—
89C2×#127—21no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2314yes—
93C2×#125—21no—
94C2×#123—24no—
95C2×#121—22no—
96C2×#121—22no—
97C2×#121—24no—
98C2×#121C2314yes—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2311yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2314yes—
112C2×#117—21no—
113C2×#117—22no—
114C2×#117—22no—
115C2×#117—21no—
116C2×#117—21no—
117C2×#117—48no—
118C2×#117D411yescharacteristic
119C2×#117D411yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127#459—28no—
128#459—28no—
129#459—48no—
130#447—48no—
131#442—48no—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2314yes—
146C2×#97C2314yes—
147#394—24no—
148#394—24no—
149#394—24no—
150#392—24no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390C2314yes—
156#390C2312yes—
157#390C2314yes—
158#390C2312yes—
159#389—22no—
160#389C2314yes—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#387—22no—
168#387—22no—
169#386—24no—
170#386—24no—
171#386—22no—
172#386—22no—
173#385—22no—
174#385—22no—
175#384C2312yes—
176#384C2312yes—
177#382—21no—
178#382—21no—
179#381—22no—
180#380—21no—
181#380—22no—
182#380—22no—
183#380C2312yes—
184#379—22no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2311yescharacteristic
190C4×C4⋊C4C2311yescharacteristic
191C4×C4⋊C4C2312yes—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4C2312yes—
200#376D411yescharacteristic
201#376D411yescharacteristic
202C43C2311yescharacteristic
203C43C2311yescharacteristic
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#336C2314yes—
209#336C2314yes—
210#336C2314yes—
211C25—21no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—21no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—42no—
222C22×D4C2414yes—
223C22×D4C2411yescharacteristic
224C22×D4C2411yescharacteristic
225C23×C4—28no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×D8—28no—
240C2×D8—28no—
241C2×D8—28no—
242C2×D8—28no—
243C2×D8—44no—
244C2×D8—44no—
245C22×C8C2414yes—
246C22×C8C2414yes—
247#127—24no—
248#127—24no—
249#127—44no—
250#127—44no—
251#127—44no—
252#127—44no—
253#127—84no—
254#125—44no—
255#123—416no—
256#121—416no—
257#121—48no—
258#121—44no—
259#121—44no—
260#121—84no—
261C4×D4—28no—
262C4×D4—28no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4—48no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—84no—
277C4×D4#4114yes—
278C4×D4#4114yes—
279#117—24no—
280#117—24no—
281#117—44no—
282#117—48no—
283#117—48no—
284#117—44no—
285#117—44no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—48no—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C22⋊C4—28no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—48no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2412yes—
344C2×C42C2411yescharacteristic
345#106—44no—
346#106—44no—
347C4⋊C8—24no—
348C4⋊C8—24no—
349C4⋊C8—44no—
350#102—28no—
351#102—28no—
352#102—28no—
353#102—44no—
354#102—44no—
355#102—44no—
356#102—44no—
357#102—44no—
358C22⋊C8—44no—
359#97—44no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24—21no—
373C24—24no—
374C24—21no—
375C24—24no—
376C24—44no—
377C24—42no—
378C24—42no—
379C24—82no—
380C24C22×D411yescharacteristic
381C2×D4—24no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—48no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—84no—
407C2×D4—84no—
408C2×D4—84no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—84no—
412C2×D4C2×#4112yes—
413C2×D4C2×#4112yes—
414C2×D4H(5,𝔽2)14yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—432no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4AGL(1,ℤ/8)14yes—
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464D8—416no—
465D8—88no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—82no—
490C4⋊C4C2×#4112yes—
491C4⋊C4C2×#4112yes—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—21no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—42no—
523C42—42no—
524C42—44no—
525C42—44no—
526C42—42no—
527C42—42no—
528C42AGL(1,ℤ/8)14yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C23—22no—
532C23—22no—
533C23—21no—
534C23—24no—
535C23—22no—
536C23—21no—
537C23—22no—
538C23—48no—
539C23—48no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—88no—
555C23—84no—
556C23—88no—
557C23C2×AGL(1,ℤ/8)12yes—
558C23C23×D411yescharacteristic
559D4—22no—
560D4—24no—
561D4—24no—
562D4—48no—
563D4—41no—
564D4—44no—
565D4—41no—
566D4—44no—
567D4—44no—
568D4—44no—
569D4—88no—
570D4—88no—
571D4—168no—
572D4—168no—
573D4—168no—
574D4#54014yescomplemented
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—82no—
618C2×C4—88no—
619C2×C4—88no—
620C2×C4—168no—
621C2×C4#52512yes—
622C2×C4#52512yes—
623C2×C4C2×AGL(1,ℤ/8)14yes—
624C2×C4C2×AGL(1,ℤ/8)12yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)12yes—
630C2×C4#58512yes—
631C8—44no—
632C8—44no—
633C22—24no—
634C22—22no—
635C22—24no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—42no—
640C22—41no—
641C22—42no—
642C22—41no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—816no—
647C22—82no—
648C22—82no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22C2×#52511yescharacteristic
657C22#348812yes—
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22C22×AGL(1,ℤ/8)11yescharacteristic
660C22#358512yes—
661C4—24no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—41no—
671C4—84no—
672C4—82no—
673C4—82no—
674C4#310914yes—
675C4C2×#54011yescharacteristic
676C4#345511yescharacteristic
677C4#349412yes—
678C2—42no—
679C2—44no—
680C2—82no—
681C2—164no—
682C2#3526111yescharacteristic
683C2C2×#310911yescharacteristic
684C2#6056811yescharacteristic
685C2#6195912yes—
686C2#6196112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #35278C2semidirect
2. D4×#117C2semidirect
3. #33160C2semidirect
4. #33068C2semidirect
5. #32902C2semidirect
6. #29998C2semidirect
7. #29921C2semidirect
8. #29794C2semidirect
9. #27411C2semidirect
10. #27211C2semidirect
11. #26174C2semidirect
12. #25168C2semidirect
13. #22151C2semidirect
14. #21926C2semidirect
15. #21503C2semidirect
24. #2867C22semidirect
25. #2859C22semidirect
30. #2445C22semidirect
40. C4×#127C22semidirect
50. #2094C22semidirect
51. #2088C22semidirect
52. #2071C22semidirect
54. #2041C22semidirect
58. #1982C22semidirect
59. #1982C22semidirect
60. #1982C22semidirect
62. #1948C22semidirect
63. #1948C22semidirect
64. #1936C22semidirect
65. #1926C22semidirect
67. #1920C22semidirect
68. #1917C22semidirect
75. #1864C22semidirect
76. #1842C22semidirect
77. #1831C22semidirect
81. #1767C22semidirect
82. #1762C22semidirect
574. D4#540semidirect

Maximal subgroups

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

subgroupindex
#352782
D4×#1172
#331602
#330682
#329022
#299982
#299212
#297942
#274112
#272112
#261742
#251682
#221512
#219262
#215032