Finite Group Databasebeta

← #6500355

Subgroups — #6500355

Order 512.

2750 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic, complemented
2C2×#3007C211yesmaximal, characteristic
3#35305C211yesmaximal, characteristic
4#34380C211yesmaximal, characteristic, complemented
5#28131C214yesmaximal, complemented
6#25084C214yesmaximal, complemented
7#25082C214yesmaximal, complemented
8#24489C214yesmaximal, complemented
9#24369C214yesmaximal, complemented
10#24221C214yesmaximal, complemented
11#21933C211yesmaximal, characteristic, complemented
12#21765C211yesmaximal, characteristic, complemented
13C2×#1788C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#137C2212yes—
16C2×#536C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19#3189—28no—
20C2×#484C2212yescomplemented
21C2×#484C2212yescomplemented
22C2×#467C2214yes—
23C2×#464C2214yescomplemented
24C2×#462C2214yescomplemented
25#3007—28no—
26C2×#452C2214yes—
27C2×#451C2212yes—
28C2×#451C2212yescomplemented
29C2×#451C2212yescomplemented
30C2×#450C2212yes—
31C2×#417C2212yes—
32C22×#103C2212yes—
33C22×#102C2212yes—
34C2×#407C2211yescharacteristic
35C22×C22⋊C8C2211yescharacteristic
36#2848—22no—
37#2616—21no—
38#2616—21no—
39#2435C2212yes—
40#2434C2212yes—
41#2432C2214yes—
42#2406—22no—
43#2405—22no—
44#2378—22no—
45#2360C2214yes—
46#2344C2212yescomplemented
47#2342—22no—
48Q8×C22⋊C4C2212yescomplemented
49#2329—22no—
50#2323C2214yescomplemented
51#2316—21no—
52#2313—21no—
53#2311C2212yes—
54#2280C2211yescharacteristic
55C2×#380C2211yescharacteristic
56#2068C2214yescomplemented
57#2067C2214yescomplemented
58#2056C2214yescomplemented
59#2055C2214yescomplemented
60#2043C2214yescomplemented
61#2042C2214yescomplemented
62#2016C2214yescomplemented
63#2015C2214yescomplemented
64#1983C2214yescomplemented
65#1981C2214yescomplemented
66#1928C2214yescomplemented
67#1925C2214yescomplemented
68#1894C2214yescomplemented
69#1880C2214yescomplemented
70#1878C2214yescomplemented
71#1872C2214yes—
72#1851C2214yescomplemented
73#1850C2214yes—
74#1823C2212yescomplemented
75#1798C2214yescomplemented
76#1788C2214yes—
77#1788C2214yes—
78#1787C2212yescomplemented
79C22×#41—21no—
80C22×#41—21no—
81C22×#41—24no—
82C22×#41—24no—
83C22×#41—44no—
84C22×#41—44no—
85C22×#41C2312yes—
86C22×#41C2312yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×D4—21no—
91C23×D4C2311yescharacteristic
92C24×C4—21no—
93C24×C4C2311yescharacteristic
94C2×#137—28no—
95C2×#137—44no—
96C2×#137—44no—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100C22×M16C2312yes—
101C23×C8C2312yes—
102#536—44no—
103#536—44no—
104C2×C4⋊Q8C2314yes—
105C2×#124C2314yes—
106C2×#123—24no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8D418yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C4×Q8C2314yes—
122C2×#117—22no—
123C2×#117C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132#484—28no—
133#484—28no—
134#467—216no—
135#464—48no—
136#462—48no—
137#452—216no—
138#451—28no—
139#451—28no—
140#451—28no—
141#450—28no—
142C2×#107C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145#417—28no—
146C2×#103—44no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102—44no—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#407—24no—
159C2×C22⋊C8—24no—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164#395—22no—
165#395—22no—
166#394—24no—
167#394—24no—
168#394—22no—
169#394—22no—
170#391—24no—
171#391C2314yes—
172#391C2314yes—
173#390C2314yes—
174#389—24no—
175#389—24no—
176#389C2314yes—
177#388—24no—
178#388—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#386—24no—
183#385—24no—
184#385—24no—
185#385—24no—
186#385—24no—
187#381—22no—
188#381D412yes—
189#380—21no—
190#380—21no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380D411yescharacteristic
195#380D411yescharacteristic
196#380D412yes—
197#380C2312yes—
198#380C2312yes—
199#379—24no—
200#379D412yes—
201C4×C4⋊C4C2314yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—24no—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207#340C2314yes—
208#340C2314yes—
209#336C2314yes—
210#336C2314yes—
211C25—21no—
212C25C2×D411yescharacteristic
213C2×#41—24no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—24no—
220C2×#41—24no—
221C2×#41—42no—
222C2×#41—42no—
223C2×#41—42no—
224C2×#41—42no—
225C2×#41—48no—
226C2×#41—416no—
227C2×#41—48no—
228C2×#41—416no—
229C2×#41—432no—
230C2×#41—44no—
231C2×#41—44no—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—24no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8—42no—
245C22×Q8—42no—
246C22×Q8C2×D414yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×Q8C2412yes—
250C22×D4—24no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—42no—
257C22×D4—48no—
258C22×D4—44no—
259C22×D4C2412yes—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—42no—
269C23×C4—42no—
270C23×C4—48no—
271C23×C4—44no—
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D414yes—
276C23×C4#4114yes—
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280#137—432no—
281C2×Q16—28no—
282C2×Q16—28no—
283C2×Q16—28no—
284C2×Q16—44no—
285C2×Q16—44no—
286C2×Q16—44no—
287C2×SD16—28no—
288C2×SD16—28no—
289C2×SD16—44no—
290C2×M16—24no—
291C2×M16—42no—
292C2×M16—42no—
293C22×C8—24no—
294C22×C8—42no—
295C22×C8—42no—
296C22×C8C2414yes—
297C22×C8C2414yes—
298C4⋊Q8—28no—
299C4⋊Q8—44no—
300#124—28no—
301#124—44no—
302#123—416no—
303C22⋊Q8—28no—
304C22⋊Q8—28no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8—48no—
312C22⋊Q8—48no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8C2×D418yes—
316C22⋊Q8C2×D418yes—
317C4×Q8—44no—
318C4×Q8—44no—
319#117—42no—
320#117—42no—
321#117—84no—
322C2×C4⋊C4—28no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—42no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4C2×D414yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2414yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—28no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—82no—
359C2×C22⋊C4C2×D414yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D414yes—
362C2×C22⋊C4C2×D412yes—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42C2414yes—
366#107—28no—
367C4⋊C8—44no—
368C4⋊C8—44no—
369#103—28no—
370#103—28no—
371#103—28no—
372#103—28no—
373#103—44no—
374#103—44no—
375#103—44no—
376#102—28no—
377#102—28no—
378#102—44no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—24no—
395C24—22no—
396C24—41no—
397C24—42no—
398C24—41no—
399C24—44no—
400C24—42no—
401C24—44no—
402C24—81no—
403C24—81no—
404C24#12114yes—
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407C24C22×D411yescharacteristic
408#41—216no—
409#41—48no—
410#41—48no—
411#41—48no—
412#41—48no—
413#41—48no—
414#41—48no—
415#41—48no—
416#41—432no—
417#41—432no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—28no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—24no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—44no—
433C2×Q8—48no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—48no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—82no—
440C2×Q8—84no—
441C2×Q8—82no—
442C2×Q8—82no—
443C2×Q8—82no—
444C2×Q8—82no—
445C2×Q8C22×D414yes—
446C2×Q8H(5,𝔽2)14yes—
447C2×D4—28no—
448C2×D4—28no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—48no—
455C2×D4—44no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—432no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4H(5,𝔽2)14yes—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—28no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—432no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—81no—
501C22×C4—81no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4#12114yes—
506C22×C4#12114yes—
507C22×C4#12114yes—
508C22×C4C22×D414yes—
509C22×C4C22×D414yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2×#4112yes—
519C22×C4C2511yescharacteristic
520Q16—416no—
521Q16—88no—
522SD16—416no—
523M16—48no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—84no—
531C4⋊C4—28no—
532C4⋊C4—28no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—82no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C4⋊C4C22×D414yes—
550C22⋊C4—28no—
551C22⋊C4—28no—
552C22⋊C4—28no—
553C22⋊C4—42no—
554C22⋊C4—44no—
555C22⋊C4—42no—
556C22⋊C4—48no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C22⋊C4—82no—
564C42—24no—
565C42—24no—
566C42—84no—
567C42—84no—
568C23—21no—
569C23—21no—
570C23—22no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—28no—
580C23—24no—
581C23—44no—
582C23—44no—
583C23—48no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—84no—
591C23—84no—
592C23—88no—
593C23—88no—
594C23—84no—
595C23—84no—
596C23C2×#12112yes—
597C23C2×#12112yes—
598C23C2×#12112yes—
599C23#57814yes—
600C23C23×D411yescharacteristic
601Q8—28no—
602Q8—24no—
603Q8—24no—
604Q8—48no—
605Q8—42no—
606Q8—42no—
607Q8—42no—
608Q8—42no—
609Q8—44no—
610Q8—42no—
611Q8—42no—
612Q8—84no—
613Q8—84no—
614D4—24no—
615D4—24no—
616D4—48no—
617D4—416no—
618D4—416no—
619D4—42no—
620D4—42no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—416no—
646C2×C4—416no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—164no—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#12112yes—
669C2×C4C23×D411yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)12yes—
672C2×C4C2×H(5,𝔽2)12yes—
673C8—44no—
674C8—44no—
675C22—22no—
676C22—22no—
677C22—24no—
678C22—24no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—88no—
693C22—816no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22#304414yes—
699C22C22×#12111yescharacteristic
700C22#348412yes—
701C22C2×#57812yes—
702C22#358511yescharacteristic
703C22#358611yescharacteristic
704C4—22no—
705C4—22no—
706C4—22no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—84no—
715C4#348412yes—
716C4#349011yescharacteristic
717C4#349011yescharacteristic
718C2—24no—
719C2—42no—
720C2—42no—
721C2—84no—
722C2#3534811yescharacteristic
723C2C2×#304411yescharacteristic
724C2#6042511yescharacteristic
725C2#6161912yes—
726C2#6162212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3189C2semidirect
4. #34380C2semidirect
5. #28131C2semidirect
6. #25084C2semidirect
7. #25082C2semidirect
8. #24489C2semidirect
9. #24369C2semidirect
10. #24221C2semidirect
11. #21933C2semidirect
12. #21765C2semidirect
20. C2×#484C22semidirect
21. C2×#484C22semidirect
23. C2×#464C22semidirect
24. C2×#462C22semidirect
28. C2×#451C22semidirect
29. C2×#451C22semidirect
46. #2344C22semidirect
48. Q8×C22⋊C4C22semidirect
50. #2323C22semidirect
56. #2068C22semidirect
57. #2067C22semidirect
58. #2056C22semidirect
59. #2055C22semidirect
60. #2043C22semidirect
61. #2042C22semidirect
62. #2016C22semidirect
63. #2015C22semidirect
64. #1983C22semidirect
65. #1981C22semidirect
66. #1928C22semidirect
67. #1925C22semidirect
68. #1894C22semidirect
69. #1880C22semidirect
70. #1878C22semidirect
72. #1851C22semidirect
74. #1823C22semidirect
75. #1798C22semidirect
78. #1787C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C2×#30072
#353052
#343802
#281312
#250842
#250822
#244892
#243692
#242212
#219332
#217652
C2×#17882