Finite Group Databasebeta

← #6816555

Subgroups — #6816555

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3384C211yesmaximal, characteristic, complemented
2C2×#3354C211yesmaximal, characteristic
3#38958C211yesmaximal, characteristic, complemented
4#38955C211yesmaximal, characteristic, complemented
5#33313C211yesmaximal, characteristic, complemented
6#31090C214yesmaximal, complemented
7#31088C214yesmaximal, complemented
8#29865C211yesmaximal, characteristic, complemented
9#29865C211yesmaximal, characteristic, complemented
10#28713C211yesmaximal, characteristic, complemented
11#28684C211yesmaximal, characteristic, complemented
12#26079C214yesmaximal, complemented
13#25839C211yesmaximal, characteristic, complemented
14#25838C211yesmaximal, characteristic, complemented
15#25704C211yesmaximal, characteristic, complemented
16#25703C211yesmaximal, characteristic, complemented
17#25048C214yesmaximal, complemented
18#21631C211yesmaximal, characteristic, complemented
19#21275C211yesmaximal, characteristic, complemented
20C2×#555C2211yescharacteristic
21C2×#554C2211yescharacteristic
22C2×D4×Q8C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3384—28no—
25#3354—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#481C2212yes—
32C2×#482C2212yes—
33C2×#478C2212yes—
34C2×#476C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#476C2212yes—
37C2×#474C2211yescharacteristic
38C2×#474C2211yescharacteristic
39C2×#469C2212yes—
40C2×#446C2211yescharacteristic
41C2×#442C2212yes—
42C2×C4×D8C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2700—24no—
48#2687—24no—
49#2621—21no—
50#2620—21no—
51#2567C2211yescharacteristic
52#2563C2211yescharacteristic
53#2547C2211yescharacteristic
54#2546C2211yescharacteristic
55#2503—21no—
56#2499—21no—
57#2442C2214yes—
58#2439C2214yes—
59#2436—21no—
60#2432—21no—
61#2420C2214yes—
62#2420C2214yes—
63#2311—21no—
64#2310—21no—
65C4×#124C2211yescharacteristic
66#2293C2211yescharacteristic
67#2293C2211yescharacteristic
68#2088C2214yes—
69#2085C2214yes—
70#2085C2214yes—
71#2077C2214yes—
72#2073C2214yes—
73#2073C2214yes—
74#2059C2214yes—
75#2057C2214yes—
76#2033C2214yes—
77#2033C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1970C2212yes—
84#1954C2211yescharacteristic
85#1943C2214yes—
86#1942C2212yes—
87#1934C2212yes—
88#1934C2212yes—
89#1923C2214yes—
90#1862C2214yes—
91#1862C2214yes—
92#1854C2214yes—
93#1852C2214yes—
94#1846C2214yes—
95#1846C2214yes—
96#1831C2211yescharacteristic
97#1809C2214yes—
98C4×#107C2211yescharacteristic
99C4×#102C2211yescharacteristic
100C4×#102C2211yescharacteristic
101#1753C2214yes—
102C42×C8C2211yescharacteristic
103C22×#41—21no—
104C23×Q8—21no—
105C22×D8C2311yescharacteristic
106C22×M16C2312yes—
107#555—24no—
108#555—42no—
109#554—24no—
110#554—42no—
111D4×Q8—24no—
112D4×Q8—42no—
113#547—24no—
114#547—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#125C2312yes—
122C2×#124—21no—
123C2×#124—22no—
124C2×#124D411yescharacteristic
125C2×#123—22no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—22no—
131C2×#121—21no—
132C2×#121C2312yes—
133C2×#121C2312yes—
134C2×C4×Q8C2311yescharacteristic
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2311yescharacteristic
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C2×#117—21no—
141C2×#117—21no—
142C2×#117—21no—
143C2×#117C2312yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2312yes—
146#501—28no—
147#500—42no—
148#500—42no—
149#498—24no—
150#498—24no—
151#482—28no—
152#481—28no—
153#478—28no—
154#476—24no—
155#476—24no—
156#476—28no—
157#474—24no—
158#474—24no—
159#469—28no—
160#446—24no—
161#442—28no—
162C4×D8D418yes—
163#436—24no—
164#428—28no—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2314yes—
169C2×#107C2314yes—
170C2×#107C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2311yescharacteristic
180C2×#102C2312yes—
181C2×#102C2312yes—
182C2×#102C2312yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2311yescharacteristic
187C2×#102C2311yescharacteristic
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2314yes—
190C2×#97C2312yes—
191C2×C4×C8C2312yes—
192C2×C4×C8C2312yes—
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195#400—24no—
196#398—22no—
197#398—24no—
198#397—22no—
199#397—24no—
200#397—24no—
201#396—24no—
202#395—22no—
203#394—24no—
204#394—24no—
205#393—22no—
206#393—24no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—22no—
213#391—21no—
214#391—21no—
215#391—22no—
216#391—24no—
217#391D411yescharacteristic
218#390—21no—
219#390D411yescharacteristic
220#390C2312yes—
221#390C2312yes—
222#389C2314yes—
223#389C2314yes—
224#389C2312yes—
225#389C2312yes—
226#388—22no—
227#388—22no—
228#388D414yes—
229#387—24no—
230#386—24no—
231#386—24no—
232#385—24no—
233#384—21no—
234#382—21no—
235#382D411yescharacteristic
236#382C2314yes—
237#382C2314yes—
238#381—24no—
239#381—24no—
240#380—24no—
241#380C2314yes—
242#380C2314yes—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4—21no—
247C4×C22⋊C4C2314yes—
248C43C2311yescharacteristic
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2312yes—
256#336C2314yes—
257#336C2314yes—
258C2×#41—44no—
259C2×#41—44no—
260C2×#41—82no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—44no—
267C22×Q8—44no—
268C22×Q8—82no—
269C22×Q8C2×D411yescharacteristic
270C22×D4—22no—
271C22×D4—22no—
272C22×D4C2×D411yescharacteristic
273C22×D4C2411yescharacteristic
274C22×D4C2411yescharacteristic
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—22no—
278C23×C4—22no—
279C23×C4—22no—
280C23×C4C2412yes—
281C2×D8—24no—
282C2×D8C2×D414yes—
283C2×M16—24no—
284C2×M16—28no—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2411yescharacteristic
288C22×C8C2412yes—
289C22×C8C2411yescharacteristic
290C4⋊Q8—24no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—42no—
300C4⋊Q8—82no—
301#125—24no—
302#125—42no—
303#124—42no—
304#124—48no—
305#124—82no—
306#123—48no—
307C22⋊Q8—48no—
308C22⋊Q8—48no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312#121—24no—
313#121—24no—
314#121—42no—
315#121—42no—
316#121—44no—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4C2×D414yes—
328C4×D4C2×D414yes—
329#117—24no—
330#117—42no—
331#117—44no—
332#117—82no—
333#117—82no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D414yes—
375C2×C22⋊C4C2×D414yes—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385#107—24no—
386#107—24no—
387#107—24no—
388#107—44no—
389#107—44no—
390#107C2×D414yes—
391#106—24no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—24no—
396#106—24no—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#102—24no—
401#102—24no—
402#102—24no—
403#102—24no—
404#102—24no—
405#102—24no—
406#102C2×D414yes—
407#102C2×D414yes—
408C22⋊C8—24no—
409C22⋊C8—28no—
410#97—24no—
411C4×C8—24no—
412C4×C8—24no—
413C4×C8—22no—
414C4×C8C2×D414yes—
415#95—22no—
416#95—22no—
417#95—21no—
418#95—21no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95C2×D414yes—
428#95C2×D414yes—
429C24—21no—
430C24—21no—
431C24C22×D412yes—
432#41—816no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—82no—
449C2×Q8—816no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—82no—
463C2×D4—84no—
464C2×D4C22×D412yes—
465C2×D4C22×D412yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—82no—
490C22×C4—84no—
491C22×C4—82no—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2511yescharacteristic
500D8—28no—
501M16—48no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—24no—
509C2×C8C22×D412yes—
510C2×C8C22×D412yes—
511C2×C82−1+414yes—
512C2×C82−1+414yes—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—24no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—84no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—84no—
542C4⋊C4C22×D412yes—
543C4⋊C4C22×D412yes—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
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—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—42no—
567C42—44no—
568C42—42no—
569C42—42no—
570C42C22×D412yes—
571C42H(5,𝔽2)12yes—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23C23×D411yescharacteristic
588Q8—44no—
589Q8—88no—
590Q8—84no—
591Q8—84no—
592Q8—162no—
593Q8—162no—
594D4—22no—
595D4—22no—
596D4—22no—
597D4—22no—
598D4—44no—
599D4—42no—
600D4—42no—
601D4—88no—
602D4—164no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—164no—
647C2×C4D4212yes—
648C2×C4D4212yes—
649C2×C4#57512yes—
650C2×C4#57612yes—
651C2×C4C23×D411yescharacteristic
652C2×C4C2×H(5,𝔽2)11yescharacteristic
653C2×C4C2×2−1+412yes—
654C2×C4C2×2−1+412yes—
655C8—22no—
656C8—24no—
657C8—42no—
658C8—42no—
659C8#53614yes—
660C22—44no—
661C22—44no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22C2×D4211yescharacteristic
676C22#349912yes—
677C22C2×#57511yescharacteristic
678C22C2×#57611yescharacteristic
679C22#358412yes—
680C4—22no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#327314yes—
695C4C2×#53611yescharacteristic
696C4#349911yescharacteristic
697C4#355212yes—
698C2—82no—
699C2—82no—
700C2—82no—
701C2#3936011yescharacteristic
702C2C2×#327311yescharacteristic
703C2#6108811yescharacteristic
704C2#6248512yes—
705C2#6249512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3354C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3384C2semidirect
3. #38958C2semidirect
4. #38955C2semidirect
5. #33313C2semidirect
6. #31090C2semidirect
7. #31088C2semidirect
8. #29865C2semidirect
9. #29865C2semidirect
10. #28713C2semidirect
11. #28684C2semidirect
12. #26079C2semidirect
13. #25839C2semidirect
14. #25838C2semidirect
15. #25704C2semidirect
16. #25703C2semidirect
17. #25048C2semidirect
18. #21631C2semidirect
19. #21275C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33842
C2×#33542
#389582
#389552
#333132
#310902
#310882
#298652
#287132
#286842
#260792
#258392
#258382
#257042
#257032
#250482
#216312
#212752