Finite Group Databasebeta

← #6477732

Subgroups — #6477732

Order 512.

1984 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C2×#2924C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic
4C2×#2625C211yesmaximal, characteristic, complemented
5#33142C214yesmaximal, complemented
6#29787C214yesmaximal, complemented
7#24614C214yesmaximal, complemented
8#24608C212yesmaximal, complemented
9#24600C212yesmaximal, complemented
10#24516C214yesmaximal, complemented
11#24509C212yesmaximal, complemented
12#24503C212yesmaximal, complemented
13#21944C211yesmaximal, characteristic, complemented
14#21762C211yesmaximal, characteristic, complemented
15#21758C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3193—28no—
20C2×#484C2212yes—
21C2×#480C2212yes—
22C2×#468C2214yes—
23C2×#448C2212yes—
24C2×#450C2212yes—
25#2924—28no—
26C2×#433C2214yes—
27C2×#432C2212yescomplemented
28C2×#431C2212yescomplemented
29C2×#424C2212yescomplemented
30C2×#407C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C2×#405C2212yescomplemented
33#2854C2212yes—
34#2854C2212yes—
35#2625—21no—
36#2625—21no—
37#2625—22no—
38#2625C2214yes—
39#2625C2212yescomplemented
40#2625C2212yescomplemented
41#2624C2212yes—
42#2624C2212yes—
43#2504—21no—
44#2501—21no—
45#2500—22no—
46#2482—21no—
47#2482—21no—
48#2405—22no—
49C2×#400—21no—
50C2×#398—21no—
51C2×#397—22no—
52#2317C2212yes—
53#2309C2212yes—
54C2×#391C2212yescomplemented
55C2×#389C2212yescomplemented
56#2280C2211yescharacteristic
57C2×#380C2211yescharacteristic
58#2087C2214yescomplemented
59#2086C2214yescomplemented
60#2068C2214yescomplemented
61#2067C2214yescomplemented
62#2066C2214yescomplemented
63#2065C2214yescomplemented
64#2055C2214yescomplemented
65#2055C2214yescomplemented
66#1973C2214yescomplemented
67#1968C2214yescomplemented
68#1944C2214yescomplemented
69#1942C2214yescomplemented
70#1936C2214yescomplemented
71#1935C2214yescomplemented
72#1934C2214yescomplemented
73#1933C2214yescomplemented
74#1893C2214yes—
75#1892C2214yescomplemented
76#1874C2212yes—
77#1874C2212yes—
78#1873C2214yes—
79#1854C2214yes—
80#1853C2212yescomplemented
81#1852C2212yescomplemented
82#1823C2212yes—
83#1823C2211yescharacteristic, complemented
84#1823C2211yescharacteristic, complemented
85#1809C2214yescomplemented
86#1787C2211yescharacteristic, complemented
87#1787C2211yescharacteristic, complemented
88#1786C2212yes—
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C22×SD16C2314yes—
94C22×M16C2312yes—
95C23×C8C2312yes—
96C2×#123—22no—
97C2×#123—22no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—24no—
107C2×#121—24no—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#117—24no—
111C2×#117—24no—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C2×#117D412yes—
115C2×#117C2312yes—
116C2×#117C2312yes—
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4D411yescharacteristic
122C22×C4⋊C4D411yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4D412yes—
130C22×C42C2312yes—
131#484—28no—
132#480—28no—
133#468—216no—
134#450—28no—
135#448—28no—
136#433—216no—
137#432—28no—
138#431—28no—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142#424—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151#407D418yes—
152C2×C22⋊C8—24no—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158#405—28no—
159C2×#97C2314yes—
160C2×C4×C8C2314yes—
161#400—24no—
162#400—22no—
163#400—22no—
164#399—22no—
165#399—22no—
166#398—24no—
167#398—22no—
168#398—22no—
169#397—22no—
170#397—22no—
171#397—24no—
172#397—24no—
173#397—24no—
174#397—24no—
175#394—22no—
176#394—22no—
177#394—22no—
178#394—22no—
179#391—22no—
180#391—22no—
181#391C2314yes—
182#391C2314yes—
183#390—22no—
184#390—22no—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#390C2312yes—
189#389—22no—
190#389—22no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#389C2314yes—
196#389C2314yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#386—22no—
202#386—22no—
203#385—22no—
204#385—22no—
205#384—22no—
206#382—22no—
207#381—21no—
208#380—21no—
209#380—21no—
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213#380C2311yescharacteristic
214#380C2311yescharacteristic
215#379—21no—
216#379—22no—
217#379—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—21no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230C25—21no—
231C25C2×D411yescharacteristic
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—44no—
236C22×Q8C2412yes—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—22no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—82no—
245C22×D4—82no—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4C2×D412yes—
253C23×C4C2×D412yes—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D414yes—
258C23×C4#4114yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C2×SD16—28no—
263C2×SD16—44no—
264C2×SD16—44no—
265C2×M16—44no—
266C2×M16C2×D418yes—
267C22×C8—24no—
268C22×C8—44no—
269C22×C8C2412yes—
270C22×C8C2412yes—
271C22×C8C2414yes—
272#123—84no—
273#123—84no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280#121—24no—
281#121—24no—
282#121—24no—
283#121—24no—
284#121—44no—
285#121—44no—
286#117—24no—
287#117—24no—
288#117—28no—
289#117—22no—
290#117—22no—
291#117—24no—
292#117—24no—
293#117—22no—
294#117—22no—
295#117—44no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2414yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2412yes—
359C2×C42C2412yes—
360#107—28no—
361#107—44no—
362#106—44no—
363C4⋊C8—28no—
364C4⋊C8—28no—
365#103—28no—
366#103—44no—
367#103—44no—
368#102—28no—
369#102—44no—
370#102—44no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—24no—
374C22⋊C8C2×D418yes—
375C22⋊C8C2×D418yes—
376#97—28no—
377C4×C8—28no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—42no—
397C24—42no—
398C24—44no—
399C24—42no—
400C24—81no—
401C24—81no—
402C24—82no—
403C24#12114yes—
404C24C22×D411yescharacteristic
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—84no—
414C2×Q8—84no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—82no—
428C2×D4—82no—
429C2×D4—816no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4#12114yes—
462C22×C4#12114yes—
463C22×C4#12114yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2511yescharacteristic
478SD16—88no—
479M16—48no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—48no—
487C2×C8C22×D418yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—24no—
516C22⋊C4—24no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—24no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—44no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—28no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—84no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—164no—
568C23—164no—
569C23C2×#12112yes—
570C23C2×#12112yes—
571C23C2×#12112yes—
572C23#57612yes—
573C23#57712yes—
574C23C23×D411yescharacteristic
575Q8—44no—
576Q8—42no—
577Q8—42no—
578Q8—84no—
579D4—44no—
580D4—44no—
581D4—84no—
582D4—88no—
583D4—88no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—84no—
627C2×C4—88no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4C2×#12112yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4#58514yes—
637C8—28no—
638C8—42no—
639C8—42no—
640C22—22no—
641C22—22no—
642C22—24no—
643C22—24no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—88no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—168no—
663C22#306214yes—
664C22C22×#12111yescharacteristic
665C22#344912yes—
666C22C2×#57611yescharacteristic
667C22C2×#57711yescharacteristic
668C22#358612yes—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4#348512yes—
683C4#349112yes—
684C2—24no—
685C2—42no—
686C2—42no—
687C2—162no—
688C2#3508911yescharacteristic
689C2C2×#306211yescharacteristic
690C2#6045511yescharacteristic
691C2#6097612yes—
692C2#6098212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
2. C2×#2924C2semidirect
4. C2×#2625C2semidirect
5. #33142C2semidirect
6. #29787C2semidirect
7. #24614C2semidirect
8. #24608C2semidirect
9. #24600C2semidirect
10. #24516C2semidirect
11. #24509C2semidirect
12. #24503C2semidirect
13. #21944C2semidirect
14. #21762C2semidirect
27. C2×#432C22semidirect
28. C2×#431C22semidirect
29. C2×#424C22semidirect
32. C2×#405C22semidirect
39. #2625C22semidirect
40. #2625C22semidirect
54. C2×#391C22semidirect
55. C2×#389C22semidirect
58. #2087C22semidirect
59. #2086C22semidirect
60. #2068C22semidirect
61. #2067C22semidirect
62. #2066C22semidirect
63. #2065C22semidirect
64. #2055C22semidirect
65. #2055C22semidirect
66. #1973C22semidirect
67. #1968C22semidirect
68. #1944C22semidirect
69. #1942C22semidirect
70. #1936C22semidirect
71. #1935C22semidirect
72. #1934C22semidirect
73. #1933C22semidirect
75. #1892C22semidirect
80. #1853C22semidirect
81. #1852C22semidirect
83. #1823C22semidirect
84. #1823C22semidirect
85. #1809C22semidirect
86. #1787C22semidirect
87. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C2×#29242
#352622
C2×#26252
#331422
#297872
#246142
#246082
#246002
#245162
#245092
#245032
#219442
#217622
#217582