Finite Group Databasebeta

← #6529807

Subgroups — #6529807

Order 512.

2146 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3109C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic, complemented
3#35178C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic, complemented
5#28276C214yesmaximal, complemented
6#28197C214yesmaximal, complemented
7#25485C214yesmaximal, complemented
8#25304C214yesmaximal, complemented
9#24607C212yesmaximal, complemented
10#24603C212yesmaximal, complemented
11#24506C212yesmaximal, complemented
12#24500C212yesmaximal, complemented
13#21986C211yesmaximal, characteristic, complemented
14#21865C211yesmaximal, characteristic, complemented
15#20738C211yesmaximal, characteristic, complemented
16C2×#536—21no—
17C2×#532C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3109—28no—
21C2×#481C2212yes—
22C2×#476C2212yes—
23C2×#459C2212yes—
24C2×#447C2212yes—
25#2983—28no—
26C2×#446C2214yescomplemented
27C2×#446C2214yescomplemented
28C2×#442C2214yes—
29C2×#432C2211yescharacteristic
30C2×#432C2212yes—
31C2×#432C2211yescharacteristic
32C2×#431C2212yes—
33#2854C2212yes—
34#2854C2212yes—
35#2618—21no—
36#2618—21no—
37#2580—22no—
38#2580—22no—
39#2580C2214yes—
40#2580C2214yes—
41#2548C2212yescomplemented
42#2546C2212yescomplemented
43#2545—22no—
44#2538C2212yescomplemented
45#2538C2212yescomplemented
46#2536—22no—
47C2×#399—21no—
48C2×#393—21no—
49#2311—21no—
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#385—21no—
55C2×#385—21no—
56#2285C2211yescharacteristic
57C2×#382C2211yescharacteristic
58#2088C2214yescomplemented
59#2088C2214yescomplemented
60#2067C2214yes—
61#2067C2214yescomplemented
62#2065C2214yes—
63#2065C2214yescomplemented
64#2054C2214yes—
65#2054C2214yes—
66#1993C2212yescomplemented
67#1972C2214yes—
68#1970C2214yes—
69#1961C2212yescomplemented
70#1949C2214yescomplemented
71#1943C2214yescomplemented
72#1941C2214yescomplemented
73#1936C2214yes—
74#1936C2214yescomplemented
75#1934C2214yes—
76#1934C2214yescomplemented
77#1874C2212yes—
78#1874C2212yes—
79#1853C2212yes—
80#1852C2212yes—
81#1847C2212yescomplemented
82#1836C2212yescomplemented
83#1824C2212yes—
84#1804C2214yescomplemented
85#1802C2212yes—
86#1802C2212yes—
87#1786C2212yes—
88#1762C2214yes—
89C22×#41—21no—
90C23×Q8D411yescharacteristic
91C23×D4C2311yescharacteristic
92C24×C4—21no—
93C22×D8C2314yes—
94#536—48no—
95#532—24no—
96#532—24no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×#127C2311yescharacteristic
100C2×#124—22no—
101C2×#124—21no—
102C2×#123—22no—
103C2×#123—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8D412yes—
111C2×C4×Q8D412yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4—24no—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×C4×D4C2312yes—
122C2×#117—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#481—28no—
140#476—28no—
141#459—28no—
142#447—28no—
143#446—216no—
144#446—216no—
145#442—216no—
146#432—24no—
147#432—28no—
148#432D418yes—
149#431—28no—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2312yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2312yes—
165C2×#97C2314yes—
166C2×#97C2312yes—
167C2×#97C2312yes—
168C2×C4×C8C2314yes—
169#399—24no—
170#399—24no—
171#398—22no—
172#398—22no—
173#396—22no—
174#396—22no—
175#396—22no—
176#396—22no—
177#394—22no—
178#394—22no—
179#393—24no—
180#393—24no—
181#391—22no—
182#390—22no—
183#390—22no—
184#390—22no—
185#390C2312yes—
186#390C2312yes—
187#390C2314yes—
188#390C2314yes—
189#390C2312yes—
190#390C2312yes—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2314yes—
195#389C2312yes—
196#389C2314yes—
197#389C2312yes—
198#389C2312yes—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#386—22no—
204#386—22no—
205#385—24no—
206#385—24no—
207#385—24no—
208#385—24no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#382—21no—
212#382—21no—
213#382—21no—
214#382C2312yes—
215#382C2312yes—
216#380—21no—
217#380—22no—
218#380—22no—
219#380C2312yes—
220C4×C4⋊C4—21no—
221C4×C4⋊C4C2312yes—
222C2×#95—21no—
223C2×#95—21no—
224C2×#95—21no—
225C2×#95—21no—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232C25—21no—
233C2×#41—44no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—24no—
237C22×Q8—24no—
238C22×Q8—21no—
239C22×Q8—42no—
240C22×Q8C2×D412yes—
241C22×Q8C2×D412yes—
242C22×D4—24no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4—22no—
249C22×D4—42no—
250C22×D4C2412yes—
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—44no—
259C23×C4—82no—
260C23×C4—82no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×D8—28no—
269C2×D8—28no—
270C2×D8—44no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8—48no—
276#127—22no—
277#127—22no—
278#124—48no—
279#124—44no—
280#123—48no—
281#123—48no—
282C22⋊Q8—48no—
283C22⋊Q8—48no—
284#121—24no—
285#121—24no—
286#121—44no—
287C4×Q8—28no—
288C4×Q8—28no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×D4—24no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—44no—
302#117—44no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—82no—
325C2×C4⋊C4—82no—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
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—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—82no—
359C2×C22⋊C4—82no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D414yes—
370C2×C42#4114yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—44no—
376#106—28no—
377#106—44no—
378C4⋊C8—28no—
379C4⋊C8—28no—
380C4⋊C8—24no—
381C4⋊C8C2×D418yes—
382#102—28no—
383#102—28no—
384#102—28no—
385#102—44no—
386#102—44no—
387#102—44no—
388C22⋊C8—24no—
389C22⋊C8C2×D418yes—
390#97—28no—
391#97—24no—
392#97C2×D418yes—
393C4×C8—28no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406C24—24no—
407C24—22no—
408C24—22no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24C22×D411yescharacteristic
416#41—816no—
417C2×Q8—24no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—24no—
421C2×Q8—42no—
422C2×Q8—42no—
423C2×Q8—416no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—81no—
449C2×D4—81no—
450C2×D42−1+414yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—816no—
493C22×C4#12114yes—
494C22×C4#12114yes—
495C22×C4AGL(1,ℤ/8)14yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C2×#4112yes—
507C22×C4C2511yescharacteristic
508D8—416no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8C22×D418yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C42—24no—
561C42—24no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—44no—
567C42#12114yes—
568C42#12114yes—
569C42C22×D412yes—
570C42C22×D412yes—
571C42C2×#4112yes—
572C42C2×#4112yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—24no—
578C23—48no—
579C23—48no—
580C23—48no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—84no—
595C23—84no—
596C23—84no—
597C23—84no—
598C23C2×#12112yes—
599C23C2×AGL(1,ℤ/8)12yes—
600C23#57512yes—
601C23C23×D411yescharacteristic
602Q8—48no—
603Q8—48no—
604Q8—84no—
605D4—24no—
606D4—24no—
607D4—48no—
608D4—42no—
609D4—44no—
610D4—42no—
611D4—44no—
612D4—84no—
613D4—84no—
614D4—88no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4—88no—
663C2×C4—82no—
664C2×C4—82no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4C2×#12112yes—
669C2×C4C2×#12112yes—
670C2×C4C2×#12112yes—
671C2×C4C2×#12112yes—
672C2×C4C2×#12112yes—
673C2×C4C2×AGL(1,ℤ/8)14yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×2−1+412yes—
677C2×C4#58512yes—
678C8—28no—
679C8—44no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—816no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22#305514yes—
702C22C22×#12111yescharacteristic
703C22#348611yescharacteristic
704C22#348711yescharacteristic
705C22C22×AGL(1,ℤ/8)11yescharacteristic
706C22C2×#57511yescharacteristic
707C22#358511yescharacteristic
708C22#358511yescharacteristic
709C4—22no—
710C4—22no—
711C4—24no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4#348912yes—
725C4#349512yes—
726C2—24no—
727C2—84no—
728C2—162no—
729C2#3566111yescharacteristic
730C2C2×#305511yescharacteristic
731C2#6045411yescharacteristic
732C2#6182511yescharacteristic
733C2#6183111yescharacteristic
734C2#6189111yescharacteristic
735C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2983C2semidirect
3. #35178C2semidirect
4. C2×#2580C2semidirect
5. #28276C2semidirect
6. #28197C2semidirect
7. #25485C2semidirect
8. #25304C2semidirect
9. #24607C2semidirect
10. #24603C2semidirect
11. #24506C2semidirect
12. #24500C2semidirect
13. #21986C2semidirect
14. #21865C2semidirect
15. #20738C2semidirect
26. C2×#446C22semidirect
27. C2×#446C22semidirect
41. #2548C22semidirect
42. #2546C22semidirect
44. #2538C22semidirect
45. #2538C22semidirect
58. #2088C22semidirect
59. #2088C22semidirect
61. #2067C22semidirect
63. #2065C22semidirect
66. #1993C22semidirect
69. #1961C22semidirect
70. #1949C22semidirect
71. #1943C22semidirect
72. #1941C22semidirect
74. #1936C22semidirect
76. #1934C22semidirect
81. #1847C22semidirect
82. #1836C22semidirect
84. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
C2×#29832
#351782
C2×#25802
#282762
#281972
#254852
#253042
#246072
#246032
#245062
#245002
#219862
#218652
#207382