Finite Group Databasebeta

← #6473107

Subgroups — #6473107

Order 512.

2312 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3200C212yesmaximal, complemented
2#35272C211yesmaximal, characteristic
3#34268C211yesmaximal, characteristic, complemented
4#33116C212yesmaximal, complemented
5#32912C212yesmaximal, complemented
6#29758C212yesmaximal, complemented
7#29713C212yesmaximal, complemented
8#28580C212yesmaximal, complemented
9#28574C212yesmaximal, complemented
10#28130C214yesmaximal, complemented
11#27652C212yesmaximal, complemented
12#24768C214yesmaximal, complemented
13#24211C212yesmaximal, complemented
14#21770C212yesmaximal, complemented
15#20630C211yesmaximal, characteristic, complemented
16C2×#536C2211yescharacteristic
17C2×#535C2212yes—
18C22×#127C2211yescharacteristic
19C2×#519C2211yescharacteristic, complemented
20C2×C4×#41C2211yescharacteristic
21#3200—216no—
22C2×#485C2214yes—
23C2×#482C2214yescomplemented
24C2×#468C2214yescomplemented
25C2×#465C2214yescomplemented
26C2×#460C2214yes—
27C2×#448C2214yescomplemented
28C2×#450C2214yes—
29C2×#424C2212yescomplemented
30C2×#408C2212yes—
31#2856C2212yes—
32#2856C2212yes—
33#2619C2212yes—
34#2619C2212yes—
35#2548—22no—
36#2541—22no—
37#2408—22no—
38#2408—22no—
39#2367—22no—
40#2365—22no—
41#2359—22no—
42#2359—22no—
43#2323C2212yes—
44#2323C2212yescomplemented
45#2311C2211yescharacteristic
46#2309C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2212yescomplemented
49#2296C2211yescharacteristic, complemented
50C2×#391C2211yescharacteristic, complemented
51#2285C2211yescharacteristic
52#2274C2211yescharacteristic, complemented
53#2084C2212yescomplemented
54#2083C2212yescomplemented
55#2072C2212yescomplemented
56#2072C2212yescomplemented
57#2067C2214yescomplemented
58#2065C2214yescomplemented
59#2055C2214yescomplemented
60#2055C2214yescomplemented
61#2041C2212yes—
62#2041C2212yes—
63#2014C2212yes—
64#2014C2212yes—
65#1982C2214yes—
66#1949C2214yescomplemented
67#1926C2214yescomplemented
68#1925C2214yescomplemented
69#1895C2214yescomplemented
70#1893C2214yescomplemented
71#1880C2214yescomplemented
72#1880C2214yescomplemented
73#1875C2214yescomplemented
74#1874C2214yescomplemented
75#1825C2214yes—
76#1805C2214yescomplemented
77#1798C2212yes—
78#1798C2212yescomplemented
79#1797C2212yes—
80#1797C2212yescomplemented
81#1786C2214yes—
82#1728C2211yescharacteristic, complemented
83C2×#336C2211yescharacteristic, complemented
84C22×#41C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4—22no—
87C23×D4C2312yes—
88C22×SD16C2314yes—
89C22×SD16C2314yes—
90C22×M16C2312yes—
91C23×C8C2312yes—
92#536—24no—
93#536D418yescomplemented
94#535—28no—
95#535—28no—
96C2×C4⋊Q8—21no—
97C2×#127—22no—
98C2×#127—22no—
99C2×#127—44no—
100C2×#127C2312yes—
101C2×#127C2312yes—
102C2×#124—21no—
103C2×#124—22no—
104C2×#123—21no—
105C2×#123—21no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121—22no—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121—22no—
112C2×#121C2314yes—
113C2×#121C2314yes—
114C2×#120—22no—
115C2×#120—22no—
116#519—44no—
117C4×#41—44no—
118C2×C4×Q8—22no—
119C2×C4×Q8D412yes—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4D412yes—
123C2×C4×D4D412yes—
124C2×C4×D4D412yes—
125C2×#117—22no—
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#485—216no—
137#482—216no—
138#468—216no—
139#465—216no—
140#460—48no—
141#450—216no—
142#448—216no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#424—44no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156#408—28no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159#394—24no—
160#394—24no—
161#392—24no—
162#392—24no—
163#391—21no—
164#391—22no—
165#391C2312yes—
166#391C2312yes—
167#390—21no—
168#390—24no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390C2314yes—
174#390C2314yes—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#386—21no—
188#386—21no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384C2312yes—
193#382C2312yes—
194#380—22no—
195#380C2312yes—
196#380C2312yes—
197#379—24no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C2×#95—22no—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2312yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217C25—22no—
218C2×#41—42no—
219C2×#41—42no—
220C2×#41C2×D414yes—
221C2×#41#4114yes—
222C22×Q8—21no—
223C22×Q8—22no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2×D411yescharacteristic
227C22×Q8C2×D414yescomplemented
228C22×Q8C2412yes—
229C22×D4—24no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—24no—
236C22×D4—24no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4—42no—
240C22×D4—42no—
241C22×D4—48no—
242C22×D4—84no—
243C22×D4—84no—
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C22×D4C2414yes—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4C2×D412yes—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C2×SD16—28no—
257C2×SD16—28no—
258C2×SD16—44no—
259C2×SD16—44no—
260C2×SD16—44no—
261C2×SD16—44no—
262C2×M16—24no—
263C2×M16—42no—
264C2×M16—42no—
265C22×C8—24no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8C2414yes—
269C22×C8C2414yes—
270C4⋊Q8—28no—
271#127—28no—
272#127—44no—
273#127—48no—
274#127—48no—
275#124—28no—
276#124—48no—
277#123—28no—
278#123—28no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8C2×D418yes—
282C22⋊Q8C2×D418yescomplemented
283#121—28no—
284#121—28no—
285#121—28no—
286#121—28no—
287#121—84no—
288#121—84no—
289#121—84no—
290#121—84no—
291#120—48no—
292#120—48no—
293C4×Q8—28no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—48no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300#117—22no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—24no—
305#117—42no—
306#117—48no—
307#117—42no—
308#117C2×D414yescomplemented
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4C2×D412yes—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D412yes—
342C2×C42—21no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—22no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—42no—
351C2×C42—42no—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—28no—
360#106—28no—
361C4⋊C8—28no—
362C4⋊C8—44no—
363#103—28no—
364#103—28no—
365#102—28no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—44no—
370#102—44no—
371C22⋊C8—28no—
372C22⋊C8—28no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—22no—
383#95—22no—
384C24—24no—
385C24—44no—
386C24—44no—
387C24—44no—
388C24—84no—
389C24—84no—
390C24C22×D411yescharacteristic
391C24C22×D411yescharacteristic
392C24C22×D411yescharacteristic
393#41—48no—
394#41—48no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—41no—
400C2×Q8—41no—
401C2×Q8—48no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—48no—
405C2×Q8#12114yes—
406C2×Q8C22×D414yes—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—41no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—84no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—832no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4#12114yes—
443C2×D4#12114yes—
444C2×D4#12114yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—82no—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2511yescharacteristic
484SD16—88no—
485SD16—88no—
486M16—84no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—84no—
494C4⋊C4—28no—
495C4⋊C4—28no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4C22×D414yes—
512C22⋊C4—24no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C42—24no—
529C42—24no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—44no—
538C42—44no—
539C42—82no—
540C42—82no—
541C42H(5,𝔽2)14yes—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—88no—
562C23—816no—
563C23—168no—
564C23—164no—
565C23—164no—
566C23C2×#12112yes—
567C23C2×#12112yes—
568C23C2×#12112yes—
569C23C23×D411yescharacteristic
570Q8—28no—
571Q8—44no—
572Q8—48no—
573Q8—44no—
574D4—44no—
575D4—44no—
576D4—44no—
577D4—44no—
578D4—44no—
579D4—44no—
580D4—44no—
581D4—88no—
582D4—88no—
583D4—88no—
584D4—816no—
585D4—168no—
586D4—168no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—162no—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4#53414yes—
633C2×C4#57714yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4#58512yes—
638C8—44no—
639C8—44no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—84no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—1616no—
663C22C22×#12111yescharacteristic
664C22C2×#53411yescharacteristic
665C22#344911yescharacteristic
666C22C2×#57712yes—
667C22#358512yes—
668C4—24no—
669C4—24no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—44no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—84no—
678C4—82no—
679C4—82no—
680C4#349612yes—
681C4#350512yes—
682C2—42no—
683C2—42no—
684C2—42no—
685C2—164no—
686C2#3500011yescharacteristic
687C2#6043112yes—
688C2#6075712yes—
689C2#6101211yescharacteristic
690C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35272C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3200C2semidirect
3. #34268C2semidirect
4. #33116C2semidirect
5. #32912C2semidirect
6. #29758C2semidirect
7. #29713C2semidirect
8. #28580C2semidirect
9. #28574C2semidirect
10. #28130C2semidirect
11. #27652C2semidirect
12. #24768C2semidirect
13. #24211C2semidirect
14. #21770C2semidirect
15. #20630C2semidirect
19. C2×#519C22semidirect
23. C2×#482C22semidirect
24. C2×#468C22semidirect
25. C2×#465C22semidirect
27. C2×#448C22semidirect
29. C2×#424C22semidirect
44. #2323C22semidirect
48. C4×C22⋊Q8C22semidirect
49. #2296C22semidirect
50. C2×#391C22semidirect
52. #2274C22semidirect
53. #2084C22semidirect
54. #2083C22semidirect
55. #2072C22semidirect
56. #2072C22semidirect
57. #2067C22semidirect
58. #2065C22semidirect
59. #2055C22semidirect
60. #2055C22semidirect
66. #1949C22semidirect
67. #1926C22semidirect
68. #1925C22semidirect
69. #1895C22semidirect
70. #1893C22semidirect
71. #1880C22semidirect
72. #1880C22semidirect
73. #1875C22semidirect
74. #1874C22semidirect
76. #1805C22semidirect
78. #1798C22semidirect
80. #1797C22semidirect
82. #1728C22semidirect
83. C2×#336C22semidirect
93. #536D4semidirect
227. C22×Q8C2×D4semidirect
282. C22⋊Q8C2×D4semidirect
308. #117C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32002
#352722
#342682
#331162
#329122
#297582
#297132
#285802
#285742
#281302
#276522
#247682
#242112
#217702
#206302