Finite Group Databasebeta

← #6559614

Subgroups — #6559614

Order 512.

1922 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36476C211yesmaximal, characteristic, complemented
2#36128C211yesmaximal, characteristic
3#30327C212yesmaximal, complemented
4#30259C211yesmaximal, characteristic, complemented
5#29347C211yesmaximal, characteristic, complemented
6#28923C214yesmaximal, complemented
7#28749C211yesmaximal, characteristic, complemented
8#28605C212yesmaximal, complemented
9#28580C212yesmaximal, complemented
10#28490C212yesmaximal, complemented
11#26987C214yesmaximal, complemented
12#25501C212yesmaximal, complemented
13#23835C211yesmaximal, characteristic, complemented
14#23576C212yesmaximal, complemented
15#23223C212yesmaximal, complemented
16#23190C212yesmaximal, complemented
17#20568C211yesmaximal, characteristic, complemented
18C2×#550—21no—
19C2×#532C2212yes—
20C2×#492C2212yes—
21C2×#486C2212yes—
22C2×#486C2212yes—
23C2×#468C2214yes—
24C2×#468C2214yes—
25C2×#440C2214yes—
26C2×#436C2212yes—
27#2867C2211yescharacteristic
28#2862C2211yescharacteristic
29#2856C2212yes—
30#2855C2212yes—
31#2607—22no—
32#2589—22no—
33#2580—21no—
34#2570—21no—
35#2563C2211yescharacteristic
36#2548—21no—
37#2542C2211yescharacteristic
38#2541—21no—
39#2541C2212yes—
40#2492C2212yes—
41#2431—22no—
42#2428—22no—
43#2428C2214yes—
44#2423C2212yes—
45#2369C2211yescharacteristic
46#2366—21no—
47#2366C2212yes—
48#2365C2211yescharacteristic
49#2357—21no—
50#2355—21no—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C22⋊Q8C2212yes—
54C42×D4C2211yescharacteristic
55#2083C2212yes—
56#2083C2211yescharacteristic
57#2083C2211yescharacteristic
58#2072C2212yes—
59#2057C2211yescharacteristic
60#2057C2211yescharacteristic
61#2057C2212yes—
62#2041C2212yes—
63#2041C2212yes—
64#2033C2214yes—
65#2033C2214yes—
66#2024C2212yes—
67#2014C2212yes—
68#2014C2214yes—
69#2014C2212yes—
70#2000C2214yes—
71#2000C2214yes—
72#1987C2212yes—
73#1982C2212yes—
74#1982C2212yes—
75#1981C2212yes—
76#1973C2214yes—
77#1957C2212yes—
78#1936C2214yes—
79#1933C2214yes—
80#1925C2214yes—
81#1895C2214yes—
82#1895C2214yes—
83#1880C2214yes—
84#1877C2214yes—
85#1862C2214yes—
86C4×#107C2212yes—
87C4×#107C2212yes—
88#1770C2212yes—
89C4×C4⋊C8C2211yescharacteristic
90#1762C2214yes—
91#1753C2214yes—
92#1749C2212yes—
93#1744C2211yescharacteristic
94#1728C2212yes—
95C22×#41—22no—
96C22×SD16C2314yes—
97C22×SD16C2314yes—
98C22×M16C2314yes—
99#550—48no—
100#532—48no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127—22no—
106C2×#127—21no—
107C2×#127—21no—
108C2×#127—21no—
109C2×#127—21no—
110C2×#127C2312yes—
111C2×#127C2312yes—
112C2×#127C2312yes—
113C2×#124—22no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121—22no—
119C2×#121—24no—
120C2×#121C2314yes—
121C2×#121C2314yes—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8C2312yes—
125C2×C4×D4—22no—
126C2×C4×D4—22no—
127C2×C4×D4D411yescharacteristic
128C2×C4×D4D411yescharacteristic
129C2×C4×D4C2314yes—
130C2×C4×D4C2311yescharacteristic
131C22×C42C2312yes—
132#492—28no—
133#486—28no—
134#486—44no—
135#468—48no—
136#468—48no—
137#440—48no—
138#436—28no—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#107C2312yes—
142C2×#107C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2314yes—
153C2×#97C2314yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2312yes—
156#400—22no—
157#399—22no—
158#399—22no—
159#398—22no—
160#394—24no—
161#393—22no—
162#393—22no—
163#392—24no—
164#392—24no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—22no—
170#391—22no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#390—21no—
175#390—21no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390C2314yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2311yescharacteristic
186#389C2312yes—
187#389C2311yescharacteristic
188#389C2312yes—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#386—24no—
197#386—24no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#382—21no—
204#382—21no—
205#382—21no—
206#381—22no—
207#380C2312yes—
208#380C2312yes—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225C43C2311yescharacteristic
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237C2×#41—48no—
238C2×#41—48no—
239C2×#41—84no—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8C2412yes—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2414yes—
254C23×C4—24no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×SD16—28no—
259C2×SD16—28no—
260C2×SD16—44no—
261C2×SD16—44no—
262C2×SD16—44no—
263C2×SD16—44no—
264C2×M16—28no—
265C2×M16—44no—
266C2×M16—44no—
267C22×C8C2414yes—
268C22×C8C2412yes—
269C22×C8C2412yes—
270C4⋊Q8—24no—
271C4⋊Q8—22no—
272C4⋊Q8—24no—
273C4⋊Q8—24no—
274C4⋊Q8—41no—
275#127—24no—
276#127—24no—
277#127—44no—
278#127—42no—
279#127—42no—
280#127—48no—
281#127—44no—
282#127—44no—
283#127—82no—
284#127—82no—
285#124—48no—
286C22⋊Q8—24no—
287C22⋊Q8—42no—
288C22⋊Q8—44no—
289#121—28no—
290#121—416no—
291#121—44no—
292#121—48no—
293#121—48no—
294#121—44no—
295#121—44no—
296#121—84no—
297C4×Q8—44no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×D4—28no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—44no—
305C4×D4—48no—
306C4×D4—48no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D412yes—
343C2×C22⋊C4C2×D412yes—
344C2×C22⋊C4C2×D412yes—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—24no—
362#107—24no—
363#107—24no—
364#107—42no—
365C4⋊C8—24no—
366#103—28no—
367#103—44no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—44no—
372#102—44no—
373#102—44no—
374C22⋊C8—28no—
375#97—44no—
376C4×C8—42no—
377C4×C8#4118yescomplemented
378#95—21no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—24no—
398C24C22×D412yes—
399#41—832no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—81no—
411C2×Q8—81no—
412C2×Q8—81no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4—84no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—84no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2511yescharacteristic
477SD16—416no—
478SD16—88no—
479M16—88no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8C2×#4114yes—
488C2×C8C2×#4114yes—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—24no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4H(5,𝔽2)12yes—
518C4⋊C4H(5,𝔽2)14yes—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C42—21no—
532C42—24no—
533C42—22no—
534C42—21no—
535C42—42no—
536C42—42no—
537C42—41no—
538C42—41no—
539C42—44no—
540C42—42no—
541C42—42no—
542C42—82no—
543C42—82no—
544C42AGL(1,ℤ/8)14yes—
545C42#13714yes—
546C42C2×#4114yes—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23C23×D411yescharacteristic
562Q8—48no—
563Q8—44no—
564Q8—44no—
565Q8—84no—
566Q8—162no—
567D4—44no—
568D4—44no—
569D4—44no—
570D4—44no—
571D4—44no—
572D4—88no—
573D4—88no—
574D4—816no—
575D4—88no—
576D4—164no—
577D4—164no—
578D4—164no—
579D4—168no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—84no—
627C2×C4—816no—
628C2×C4#53212yes—
629C2×C4#53212yes—
630C2×C4C2×AGL(1,ℤ/8)12yes—
631C2×C4C2×AGL(1,ℤ/8)12yes—
632C2×C4C2×AGL(1,ℤ/8)12yes—
633C2×C4C2×#13712yes—
634C2×C4C2×#13712yes—
635C2×C4C2×#13712yes—
636C2×C4C22×#4112yes—
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4#58511yescharacteristic
640C8—24no—
641C8—24no—
642C8—44no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—84no—
655C22—84no—
656C22—84no—
657C22C2×#53211yescharacteristic
658C22#348411yescharacteristic
659C22#348811yescharacteristic
660C22C22×AGL(1,ℤ/8)11yescharacteristic
661C22C22×#13711yescharacteristic
662C22#358511yescharacteristic
663C22#358511yescharacteristic
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—42no—
674C4—82no—
675C4—81no—
676C4—82no—
677C4—82no—
678C4—81no—
679C4—81no—
680C4#318214yes—
681C4#350612yes—
682C4#350812yes—
683C2—82no—
684C2—164no—
685C2#3593011yescharacteristic
686C2C2×#318211yescharacteristic
687C2#6071911yescharacteristic
688C2#6158211yescharacteristic
689C2#6158811yescharacteristic
690C2#6195911yescharacteristic
691C2#6196511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36128C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36476C2semidirect
3. #30327C2semidirect
4. #30259C2semidirect
5. #29347C2semidirect
6. #28923C2semidirect
7. #28749C2semidirect
8. #28605C2semidirect
9. #28580C2semidirect
10. #28490C2semidirect
11. #26987C2semidirect
12. #25501C2semidirect
13. #23835C2semidirect
14. #23576C2semidirect
15. #23223C2semidirect
16. #23190C2semidirect
17. #20568C2semidirect
377. C4×C8#41semidirect

Maximal subgroups

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

subgroupindex
#364762
#361282
#303272
#302592
#293472
#289232
#287492
#286052
#285802
#284902
#269872
#255012
#238352
#235762
#232232
#231902
#205682