Finite Group Databasebeta

← #6502652

Subgroups — #6502652

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3193C211yesmaximal, characteristic
2C22×#465C211yesmaximal, characteristic
3#37453C211yesmaximal, characteristic, complemented
4#34379C211yesmaximal, characteristic
5#28583C214yesmaximal, complemented
6#25212C214yesmaximal, complemented
7#25078C214yesmaximal, complemented
8#24766C214yesmaximal, complemented
9#24579C212yesmaximal, complemented
10#24527C212yesmaximal, complemented
11#24482C212yesmaximal, complemented
12#24358C212yesmaximal, complemented
13#21765C211yesmaximal, characteristic, complemented
14C2×#1892C211yesmaximal, characteristic, complemented
15#20834C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21#3193—28no—
22C2×#484C2212yes—
23C2×#480C2212yes—
24C2×#468C2212yes—
25C2×#468C2212yes—
26C2×#465—28no—
27C2×#465C2214yes—
28C2×#465C2214yes—
29C2×#461C2214yescomplemented
30C2×#448C2212yes—
31C2×#450C2212yes—
32C22×#106C2211yescharacteristic
33C2×#424C2212yescomplemented
34C22×#103C2211yescharacteristic
35C22×#103C2212yescomplemented
36C22×#102C2211yescharacteristic
37C2×#407C2211yescharacteristic
38#2817—22no—
39#2627—21no—
40#2627—21no—
41#2627—22no—
42#2621C2214yescomplemented
43#2503C2212yes—
44#2503C2212yescomplemented
45#2483—22no—
46#2434C2212yes—
47#2434C2212yescomplemented
48#2377—22no—
49#2361C2214yes—
50#2345C2212yes—
51D4×C22⋊C4—22no—
52D4×C22⋊C4C2212yes—
53#2325—22no—
54#2324C2214yes—
55#2315—21no—
56#2312—21no—
57#2280C2211yescharacteristic
58C2×#380C2211yescharacteristic
59#2085C2214yescomplemented
60#2073C2214yescomplemented
61#2068C2214yescomplemented
62#2066C2214yescomplemented
63#2055C2214yescomplemented
64#2055C2214yescomplemented
65#2043C2214yescomplemented
66#2013C2214yescomplemented
67#1981C2214yescomplemented
68#1949C2214yescomplemented
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1894C2212yescomplemented
72#1894C2212yescomplemented
73#1892C2212yes—
74#1892C2212yes—
75#1892C2214yescomplemented
76#1877C2214yescomplemented
77#1877C2214yescomplemented
78#1872C2214yescomplemented
79#1850C2212yes—
80#1850C2212yes—
81#1850C2214yescomplemented
82#1823C2212yes—
83#1804C2214yescomplemented
84#1787C2212yes—
85C2×#340C2212yescomplemented
86#1729C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×SD16—24no—
93C22×SD16—44no—
94C22×SD16C2314yes—
95C22×SD16C2312yes—
96C22×SD16C2312yes—
97C22×M16C2311yescharacteristic
98C22×M16C2311yescharacteristic
99C23×C8—24no—
100C23×C8—44no—
101C23×C8C2311yescharacteristic
102C23×C8C2312yes—
103C23×C8C2312yes—
104C23×C8C2311yescharacteristic
105C2×C4⋊Q8C2314yes—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×#121—21no—
120C2×#121—24no—
121C2×#121D418yes—
122C2×#121C2312yes—
123C2×#121C2312yes—
124C2×C4×D4—24no—
125C2×C4×D4C2314yes—
126C2×#117C2312yes—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—21no—
137C22×C22⋊C4—22no—
138#484—28no—
139#480—28no—
140#468—28no—
141#468—28no—
142#465—216no—
143#465—216no—
144#461—48no—
145#450—28no—
146#448—28no—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#106—24no—
150C2×#106C2314yes—
151#424—44no—
152C2×C4⋊C8C2314yes—
153C2×C4⋊C8C2314yes—
154C2×#103—24no—
155C2×#103—44no—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102—24no—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167#407—24no—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2312yes—
170#400—24no—
171#400—22no—
172#400—22no—
173#398—24no—
174#398—24no—
175#397—24no—
176#397—24no—
177#395—24no—
178#395—22no—
179#395—22no—
180#391C2314yes—
181#391C2314yes—
182#390—24no—
183#390C2314yes—
184#389C2314yes—
185#388—24no—
186#388—24no—
187#387—24no—
188#386—24no—
189#386—24no—
190#386—24no—
191#385—24no—
192#385—22no—
193#385—22no—
194#385—24no—
195#385—24no—
196#382C2314yes—
197#381—22no—
198#381—22no—
199#381—24no—
200#381D412yes—
201#380—21no—
202#380—21no—
203#380—22no—
204#380D411yescharacteristic
205#380D411yescharacteristic
206#380D412yes—
207#380C2312yes—
208#380C2312yes—
209#379—22no—
210#379D412yes—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4C2312yes—
214C4×C22⋊C4C2312yes—
215C2×#95—21no—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C25C2×D411yescharacteristic
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8C2412yes—
235C22×Q8C2412yes—
236C22×D4—22no—
237C22×D4—28no—
238C22×D4—22no—
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4—84no—
244C22×D4C2×D414yes—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D414yes—
259C23×C4#4114yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×SD16—28no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—28no—
267C2×SD16—44no—
268C2×SD16—416no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—42no—
274C2×M16—42no—
275C22×C8—22no—
276C22×C8—24no—
277C22×C8—22no—
278C22×C8—24no—
279C22×C8—24no—
280C22×C8—24no—
281C22×C8—24no—
282C22×C8—42no—
283C22×C8—42no—
284C22×C8—42no—
285C22×C8—416no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304#121—28no—
305#121—28no—
306#121—24no—
307#121—24no—
308#121C2×D418yes—
309#121C2×D418yes—
310C4×D4—44no—
311C4×D4—44no—
312C4×D4—88no—
313#117—42no—
314#117—42no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—44no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—84no—
357C2×C22⋊C4—82no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D414yes—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D414yes—
362C2×C42—24no—
363C2×C42C2414yes—
364#107—24no—
365#107—24no—
366#106—28no—
367C4⋊C8—28no—
368C4⋊C8—44no—
369#103—28no—
370#103—28no—
371#103—24no—
372#103—24no—
373#103—44no—
374#103—44no—
375#103—44no—
376#102—28no—
377#102—24no—
378#102—24no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381#95—22no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95C2×D412yes—
392#95C2×D414yes—
393#95C2×D412yes—
394C24—22no—
395C24—24no—
396C24—22no—
397C24—22no—
398C24—24no—
399C24—42no—
400C24—42no—
401C24—42no—
402C24—82no—
403C24—82no—
404C24#12114yes—
405C24C22×D411yescharacteristic
406C24C22×D411yescharacteristic
407C24C22×D411yescharacteristic
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—48no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—82no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8—82no—
425C2×Q8—84no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—48no—
435C2×D4—84no—
436C2×D4—82no—
437C2×D4—84no—
438C2×D4—816no—
439C2×D4C22×D414yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4#12114yes—
476C22×C4#12114yes—
477C22×C4#12114yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2511yescharacteristic
490SD16—416no—
491SD16—84no—
492SD16—84no—
493M16—82no—
494M16—82no—
495C2×C8—28no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—24no—
505C2×C8—48no—
506C2×C8—48no—
507C2×C8—44no—
508C2×C8H(5,𝔽2)14yes—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4C22×D414yes—
530C22⋊C4—24no—
531C22⋊C4—28no—
532C22⋊C4—24no—
533C22⋊C4—28no—
534C22⋊C4—28no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—82no—
545C42—24no—
546C42—24no—
547C42—84no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—22no—
554C23—28no—
555C23—28no—
556C23—21no—
557C23—22no—
558C23—21no—
559C23—24no—
560C23—44no—
561C23—48no—
562C23—48no—
563C23—44no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—48no—
568C23—84no—
569C23—84no—
570C23—164no—
571C23—164no—
572C23C2×#12112yes—
573C23C2×#12112yes—
574C23C2×#12112yes—
575C23#57714yes—
576C23#57712yes—
577C23#57812yes—
578C23C23×D411yescharacteristic
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—44no—
584Q8—44no—
585Q8—84no—
586Q8—84no—
587D4—24no—
588D4—24no—
589D4—416no—
590D4—44no—
591D4—84no—
592D4—88no—
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—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—168no—
634C2×C4—164no—
635C2×C4C2×#12112yes—
636C2×C4C2×#12112yes—
637C2×C4C2×#12112yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C2×H(5,𝔽2)12yes—
641C2×C4#58512yes—
642C8—24no—
643C8—42no—
644C8—42no—
645C8—42no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—416no—
658C22—42no—
659C22—42no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—84no—
664C22—84no—
665C22—168no—
666C22#304214yes—
667C22#304614yes—
668C22C22×#12111yescharacteristic
669C22#349612yes—
670C22C2×#57712yes—
671C22C2×#57711yescharacteristic
672C22C2×#57811yescharacteristic
673C4—22no—
674C4—22no—
675C4—22no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4#348411yescharacteristic
685C4#348511yescharacteristic
686C4#350512yes—
687C2—24no—
688C2—24no—
689C2—42no—
690C2—84no—
691C2#3535811yescharacteristic
692C2C2×#304211yescharacteristic
693C2C2×#304611yescharacteristic
694C2#6226812yes—
695C2#6228112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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
3. #37453C2semidirect
5. #28583C2semidirect
6. #25212C2semidirect
7. #25078C2semidirect
8. #24766C2semidirect
9. #24579C2semidirect
10. #24527C2semidirect
11. #24482C2semidirect
12. #24358C2semidirect
13. #21765C2semidirect
14. C2×#1892C2semidirect
15. #20834C2semidirect
29. C2×#461C22semidirect
33. C2×#424C22semidirect
35. C22×#103C22semidirect
42. #2621C22semidirect
44. #2503C22semidirect
47. #2434C22semidirect
59. #2085C22semidirect
60. #2073C22semidirect
61. #2068C22semidirect
62. #2066C22semidirect
63. #2055C22semidirect
64. #2055C22semidirect
65. #2043C22semidirect
66. #2013C22semidirect
67. #1981C22semidirect
68. #1949C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
71. #1894C22semidirect
72. #1894C22semidirect
75. #1892C22semidirect
76. #1877C22semidirect
77. #1877C22semidirect
78. #1872C22semidirect
81. #1850C22semidirect
83. #1804C22semidirect
85. C2×#340C22semidirect
86. #1729C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31932
C22×#4652
#374532
#343792
#285832
#252122
#250782
#247662
#245792
#245272
#244822
#243582
#217652
C2×#18922
#208342