Finite Group Databasebeta

← #6459428

Subgroups — #6459428

Order 512.

2910 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic, complemented
2C2×#3007C211yesmaximal, characteristic
3#34203C211yesmaximal, characteristic, complemented
4C2×#2311C211yesmaximal, characteristic
5#28131C218yesmaximal, complemented
6#28123C214yesmaximal, complemented
7#28121C214yesmaximal, complemented
8#23611C212yesmaximal, complemented
9#23545C212yesmaximal, complemented
10#21067C212yesmaximal, complemented
11#20930C212yesmaximal, complemented
12#20700C211yesmaximal, characteristic, complemented
13#20628C211yesmaximal, characteristic, complemented
14#20597C211yesmaximal, characteristic
15C23×#41C2211yescharacteristic
16C22×#137C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C2×#519C2211yescharacteristic, complemented
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#484—28no—
22C2×#484C2214yes—
23C2×#484C2214yescomplemented
24#3007—28no—
25C2×#452C2214yes—
26C2×#451C2212yes—
27C2×#450C2212yes—
28C22×#107C2212yescomplemented
29C2×#424C2212yescomplemented
30C2×#417C2212yes—
31C2×#416C2212yes—
32C2×#416C2212yes—
33C22×#103C2212yescomplemented
34C2×#407C2211yescharacteristic
35C22×C22⋊C8C2211yescharacteristic
36#2362—22no—
37#2360—22no—
38#2344—22no—
39#2342—22no—
40#2323C2214yescomplemented
41#2316C2211yescharacteristic, complemented
42#2311—22no—
43#2311—22no—
44#2311C2212yes—
45#2311C2212yes—
46#2311C2214yes—
47#2310C2211yescharacteristic, complemented
48C4×C22⋊Q8C2214yes—
49C2×#391C2212yes—
50C2×#390C2212yes—
51C2×#386—22no—
52#2280C2211yescharacteristic
53C2×#380—22no—
54#2274C2211yescharacteristic, complemented
55#2068C2218yescomplemented
56#2067C2218yescomplemented
57#2056C2218yescomplemented
58#2055C2218yescomplemented
59#1928C2218yescomplemented
60#1925C2218yescomplemented
61#1880C2218yescomplemented
62#1878C2218yescomplemented
63#1824C2214yes—
64#1823C2214yescomplemented
65#1798C2214yescomplemented
66#1797C2214yes—
67#1788C2214yes—
68#1787C2212yes—
69#1786C2212yes—
70#1736C2212yescomplemented
71#1736C2212yescomplemented
72#1729C2212yescomplemented
73C22×#41—28no—
74C22×#41—21no—
75C22×#41—21no—
76C22×#41—24no—
77C22×#41—24no—
78C22×#41—44no—
79C22×#41C2312yes—
80C22×#41C2312yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×D4—21no—
84C23×D4C2311yescharacteristic
85C24×C4—21no—
86C24×C4—21no—
87C24×C4C2311yescharacteristic
88C2×#137—28no—
89C2×#137—28no—
90C2×#137—44no—
91C22×Q16C2314yes—
92C22×SD16C2314yes—
93C22×M16C2312yes—
94C23×C8C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8D418yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101#519—44no—
102C2×C4×Q8—22no—
103C2×C4×Q8C2314yes—
104C2×#117—22no—
105C2×#117—44no—
106C2×#117D412yes—
107C2×#117D412yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2312yes—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D412yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—24no—
126C22×C22⋊C4D412yes—
127C22×C42C2312yes—
128#484—216no—
129#484—216no—
130#452—216no—
131#451—28no—
132#450—28no—
133C2×#107—28no—
134C2×#107C2318yes—
135#424—44no—
136C2×C4⋊C8C2318yes—
137#417—28no—
138#416—28no—
139#416—44no—
140C2×#103—28no—
141C2×#103C2318yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148#407—24no—
149C2×C22⋊C8—24no—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154#391—22no—
155#391—24no—
156#391C2314yes—
157#391C2314yes—
158#391C2314yes—
159#390—24no—
160#390C2314yes—
161#390C2314yes—
162#389—22no—
163#389—24no—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#387—24no—
168#387—24no—
169#386—24no—
170#386—24no—
171#386—28no—
172#385—24no—
173#381—21no—
174#381—22no—
175#380—24no—
176#380—24no—
177#380—28no—
178#380—22no—
179#380C2312yes—
180#380C2312yes—
181#379—21no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—24no—
184C4×C4⋊C4C2314yes—
185C4×C22⋊C4—24no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—22no—
189C2×#95—24no—
190#340C2318yes—
191#340C2318yes—
192#336C2318yes—
193C25—21no—
194C25C2×D411yescharacteristic
195C2×#41—28no—
196C2×#41—28no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—24no—
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—24no—
206C2×#41—42no—
207C2×#41—42no—
208C2×#41—432no—
209C2×#41—432no—
210C2×#41—44no—
211C2×#41—44no—
212C2×#41—44no—
213C2×#41—44no—
214C2×#41#4118yes—
215C22×Q8—24no—
216C22×Q8—21no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—24no—
221C22×Q8—22no—
222C22×Q8—24no—
223C22×Q8—42no—
224C22×Q8C2×D414yes—
225C22×Q8C2412yes—
226C22×Q8C2412yes—
227C22×D4—28no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—24no—
231C22×D4—21no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—42no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4C2412yes—
238C23×C4—24no—
239C23×C4—24no—
240C23×C4—28no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—44no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—84no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D414yes—
257C23×C4C2×D414yes—
258C23×C4#4114yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262#137—432no—
263C2×Q16—28no—
264C2×Q16—28no—
265C2×Q16—44no—
266C2×SD16—28no—
267C2×SD16—28no—
268C2×SD16—44no—
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8C2414yes—
276C22×C8C2414yes—
277C22⋊Q8—28no—
278C22⋊Q8—28no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8C2×D418yes—
282C22⋊Q8C2×D418yes—
283C4×Q8—48no—
284C4×Q8—48no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—44no—
290#117—44no—
291#117—48no—
292#117—44no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—28no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—88no—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D414yes—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—44no—
340C2×C42C2×D414yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343#107—216no—
344C4⋊C8—48no—
345#103—216no—
346#103—28no—
347#103—28no—
348#103—44no—
349#102—28no—
350#102—28no—
351#102—44no—
352C22⋊C8—24no—
353C22⋊C8—24no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356#95—28no—
357#95—28no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—24no—
363#95—24no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—22no—
369C24—42no—
370C24—44no—
371C24—42no—
372C24—42no—
373C24—44no—
374C24#12114yes—
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C24C22×D411yescharacteristic
378#41—216no—
379#41—216no—
380#41—48no—
381#41—48no—
382#41—48no—
383#41—48no—
384#41—432no—
385#41—432no—
386#41—48no—
387#41—48no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—216no—
393C2×Q8—22no—
394C2×Q8—24no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—48no—
406C2×Q8C22×D414yes—
407C2×Q8C2×#4114yes—
408C2×D4—24no—
409C2×D4—28no—
410C2×D4—28no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—42no—
419C2×D4—432no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4C2×#4114yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—432no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—82no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—816no—
469C22×C4#12114yes—
470C22×C4#12114yes—
471C22×C4#12114yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C4C2511yescharacteristic
486Q16—416no—
487SD16—416no—
488M16—48no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—48no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—44no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—84no—
512C4⋊C4C22×D414yes—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—44no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—82no—
524C42—24no—
525C42—24no—
526C42—44no—
527C42—48no—
528C42—42no—
529C42—42no—
530C42—84no—
531C23—22no—
532C23—22no—
533C23—21no—
534C23—21no—
535C23—22no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—28no—
540C23—28no—
541C23—22no—
542C23—24no—
543C23—44no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23—416no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—84no—
553C23—88no—
554C23—84no—
555C23C2×#12112yes—
556C23C2×#12112yes—
557C23C2×#12112yes—
558C23#52512yes—
559C23#53412yes—
560C23#57814yes—
561C23C23×D411yescharacteristic
562Q8—28no—
563Q8—24no—
564Q8—24no—
565Q8—48no—
566Q8—42no—
567Q8—44no—
568Q8—42no—
569Q8—48no—
570D4—24no—
571D4—24no—
572D4—48no—
573D4—416no—
574D4—416no—
575D4—42no—
576D4—42no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—416no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—88no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4C2×#12112yes—
620C2×C4C2×#12112yes—
621C2×C4C2×#12112yes—
622C2×C4#52512yes—
623C2×C4#53412yes—
624C2×C4#54618yes—
625C2×C4#55118yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4112yes—
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×H(5,𝔽2)12yes—
630C8—44no—
631C8—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—416no—
644C22—42no—
645C22—42no—
646C22—44no—
647C22—48no—
648C22—816no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22#253414yes—
653C22#304414yes—
654C22C22×#12111yescharacteristic
655C22C2×#52511yescharacteristic
656C22C2×#53411yescharacteristic
657C22C2×#57812yes—
658C22#358512yes—
659C4—22no—
660C4—22no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—44no—
666C4—84no—
667C4—82no—
668C4—82no—
669C4#253414yes—
670C4C2×#54612yes—
671C4C2×#55112yes—
672C2—24no—
673C2—24no—
674C2—42no—
675C2—84no—
676C2C2×#253411yescharacteristic
677C2C2×#304411yescharacteristic
678C2#6042611yescharacteristic
679C2#6058912yes—
680C2#6076412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#484C2semidirect
3. #34203C2semidirect
5. #28131C2semidirect
6. #28123C2semidirect
7. #28121C2semidirect
8. #23611C2semidirect
9. #23545C2semidirect
10. #21067C2semidirect
11. #20930C2semidirect
12. #20700C2semidirect
13. #20628C2semidirect
18. C2×#519C22semidirect
23. C2×#484C22semidirect
28. C22×#107C22semidirect
29. C2×#424C22semidirect
33. C22×#103C22semidirect
40. #2323C22semidirect
41. #2316C22semidirect
47. #2310C22semidirect
54. #2274C22semidirect
55. #2068C22semidirect
56. #2067C22semidirect
57. #2056C22semidirect
58. #2055C22semidirect
59. #1928C22semidirect
60. #1925C22semidirect
61. #1880C22semidirect
62. #1878C22semidirect
64. #1823C22semidirect
65. #1798C22semidirect
70. #1736C22semidirect
71. #1736C22semidirect
72. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#30072
#342032
C2×#23112
#281312
#281232
#281212
#236112
#235452
#210672
#209302
#207002
#206282
#205972