Finite Group Databasebeta

← #6816591

Subgroups — #6816591

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3381C211yesmaximal, characteristic, complemented
2C2×#3353C211yesmaximal, characteristic
3#39744C211yesmaximal, characteristic, complemented
4#38955C211yesmaximal, characteristic
5#33314C211yesmaximal, characteristic, complemented
6#31106C214yesmaximal, complemented
7#31105C214yesmaximal, complemented
8#29867C211yesmaximal, characteristic, complemented
9#29864C211yesmaximal, characteristic, complemented
10#28713C211yesmaximal, characteristic, complemented
11#28675C211yesmaximal, characteristic, complemented
12#26073C214yesmaximal, complemented
13#25840C211yesmaximal, characteristic, complemented
14#25835C211yesmaximal, characteristic, complemented
15#25703C211yesmaximal, characteristic
16#25701C211yesmaximal, characteristic, complemented
17#25037C214yesmaximal, complemented
18#21634C211yesmaximal, characteristic, complemented
19#21279C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#547C2211yescharacteristic
24#3381—28no—
25#3353—28no—
26C2×#500C2211yescharacteristic
27C2×#500C2211yescharacteristic
28C2×#499C2212yes—
29C2×#498C2211yescharacteristic
30C2×#498C2211yescharacteristic
31C2×#484C2212yes—
32C2×#482C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#475C2211yescharacteristic
36C2×#478C2212yes—
37C2×#474C2211yescharacteristic
38C2×#474C2211yescharacteristic
39C2×#468C2212yes—
40C2×C8×Q8C2211yescharacteristic
41C2×C4×SD16C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2868C2211yescharacteristic
46#2862C2211yescharacteristic
47#2721—24no—
48#2687—24no—
49#2658—21no—
50#2621—21no—
51#2563C2211yescharacteristic
52#2547C2211yescharacteristic
53#2546—21no—
54#2546C2211yescharacteristic
55#2546C2211yescharacteristic
56#2545—21no—
57#2503—21no—
58#2442C2214yes—
59#2441C2214yes—
60#2432—21no—
61#2423C2214yes—
62#2420C2214yes—
63#2327—21no—
64#2311—21no—
65C4×#124C2211yescharacteristic
66#2295C2211yescharacteristic
67#2293C2211yescharacteristic
68#2086C2214yes—
69#2085C2214yes—
70#2083C2214yes—
71#2077C2214yes—
72#2074C2214yes—
73#2071C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2035C2214yes—
77#2033C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1988C2211yescharacteristic
81#1987C2211yescharacteristic
82#1984C2211yescharacteristic
83#1973C2212yes—
84#1954C2211yescharacteristic
85#1944C2212yes—
86#1942C2214yes—
87#1936C2212yes—
88#1935C2212yes—
89#1924C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1854C2214yes—
93#1852C2214yes—
94#1847C2214yes—
95#1847C2214yes—
96#1831C2211yescharacteristic
97#1809C2214yes—
98C4×#107C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C4×#102C2211yescharacteristic
101#1753C2214yes—
102C4×#97C2211yescharacteristic
103C22×#41—21no—
104C22×SD16C2311yescharacteristic
105C22×M16C2312yes—
106#557—24no—
107#557—42no—
108#556—24no—
109#556—42no—
110#554—24no—
111#554—42no—
112#547—24no—
113#547—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×#126—22no—
124C2×#125—22no—
125C2×#125C2312yes—
126C2×#125C2312yes—
127C2×#124—21no—
128C2×#124—21no—
129C2×#124D411yescharacteristic
130C2×#123—22no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—22no—
134C2×C22⋊Q8C2312yes—
135C2×#121C2312yes—
136C2×C4×Q8—21no—
137C2×C4×Q8—21no—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2311yescharacteristic
141C2×C4×D4C2312yes—
142C2×C4×D4C2311yescharacteristic
143C2×C4×D4C2311yescharacteristic
144C2×#117—21no—
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4C2312yes—
148#500—24no—
149#500—42no—
150#499—28no—
151#498—24no—
152#498—42no—
153#484—28no—
154#482—28no—
155#479—28no—
156#478—28no—
157#477—24no—
158#475—24no—
159#474—24no—
160#474—24no—
161#468—28no—
162C8×Q8—24no—
163C4×SD16—28no—
164C4×SD16D418yes—
165#436—24no—
166#428—28no—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2314yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2314yes—
178C2×#106C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2311yescharacteristic
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2311yescharacteristic
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2314yes—
192C2×#97C2312yes—
193C2×#97C2312yes—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2312yes—
196C2×C4×C8C2311yescharacteristic
197#400—24no—
198#399—24no—
199#399—24no—
200#398—22no—
201#398—22no—
202#398—24no—
203#398—24no—
204#397—22no—
205#397—24no—
206#396—24no—
207#394—24no—
208#393—22no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391—21no—
215#391D411yescharacteristic
216#391C2312yes—
217#390—21no—
218#390D411yescharacteristic
219#390C2312yes—
220#389—24no—
221#389C2314yes—
222#389C2314yes—
223#389C2312yes—
224#389C2312yes—
225#388—24no—
226#388—22no—
227#388D414yes—
228#387—22no—
229#387—24no—
230#387—24no—
231#386—24no—
232#385—24no—
233#384—22no—
234#384C2314yes—
235#382—21no—
236#382D411yescharacteristic
237#382C2314yes—
238#381—24no—
239#380—24no—
240#380C2314yes—
241#380C2314yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2311yescharacteristic
245C4×C22⋊C4—21no—
246C4×C22⋊C4C2314yes—
247#376—21no—
248C43C2311yescharacteristic
249#340C2312yes—
250#340C2312yes—
251#340C2314yes—
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—21no—
264C22×Q8—21no—
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×D4—22no—
268C22×D4C2×D411yescharacteristic
269C22×D4C2411yescharacteristic
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4C2412yes—
274C2×SD16—24no—
275C2×SD16C2×D414yes—
276C2×M16—24no—
277C2×M16—28no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2412yes—
282C22×C8C2411yescharacteristic
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—82no—
294C4⋊Q8—82no—
295C4⋊Q8—82no—
296#126—48no—
297#125—24no—
298#125—24no—
299#125—42no—
300#125—42no—
301#125—48no—
302#124—42no—
303#124—44no—
304#124—82no—
305#123—48no—
306C22⋊Q8—24no—
307C22⋊Q8—48no—
308C22⋊Q8—42no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311#121—24no—
312#121—42no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—24no—
317C4×Q8—42no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8C2×D414yes—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—42no—
326C4×D4C2×D414yes—
327#117—24no—
328#117—42no—
329#117—82no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—42no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C22⋊C4—21no—
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⋊C4C2×D412yes—
372C2×C22⋊C4C2×D414yes—
373C2×C22⋊C4C2×D414yes—
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—22no—
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—24no—
389#107—24no—
390#107—24no—
391#107—44no—
392#106—24no—
393#106—24no—
394#106—24no—
395#106—44no—
396#106C2×D414yes—
397C4⋊C8—24no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400#103—24no—
401#103—24no—
402#103—24no—
403#103C2×D414yes—
404#102—24no—
405#102—24no—
406#102—24no—
407#102C2×D414yes—
408C22⋊C8—24no—
409C22⋊C8—28no—
410#97—24no—
411#97—24no—
412C4×C8—24no—
413C4×C8—22no—
414C4×C8C2×D414yes—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—21no—
420#95—21no—
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—
430C24C22×D412yes—
431#41—816no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—22no—
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—42no—
447C2×Q8—42no—
448C2×Q8—82no—
449C2×Q8C22×D412yes—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—44no—
457C2×D4—44no—
458C2×D4—82no—
459C2×D4C22×D412yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489SD16—28no—
490M16—48no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8C22×D412yes—
499C2×C8C22×D412yes—
500C2×C82−1+414yes—
501C2×C82−1+414yes—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—48no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4C22×D412yes—
538C4⋊C4C22×D412yes—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—42no—
559C42—44no—
560C42—44no—
561C42—42no—
562C42—42no—
563C42C22×D412yes—
564C42H(5,𝔽2)12yes—
565C23—24no—
566C23—22no—
567C23—22no—
568C23—24no—
569C23—22no—
570C23—22no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23C23×D411yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—44no—
579Q8—42no—
580Q8—84no—
581Q8—84no—
582Q8—84no—
583Q8—162no—
584D4—22no—
585D4—22no—
586D4—44no—
587D4—42no—
588D4—88no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
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—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—88no—
632C2×C4—164no—
633C2×C4D4212yes—
634C2×C4D4212yes—
635C2×C4#57512yes—
636C2×C4#57612yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C2×H(5,𝔽2)11yescharacteristic
639C2×C4C2×2−1+412yes—
640C2×C4C2×2−1+412yes—
641C8—22no—
642C8—24no—
643C8—42no—
644C8—42no—
645C8#53614yes—
646C22—44no—
647C22—44no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—82no—
655C22—82no—
656C22C2×D4211yescharacteristic
657C22#349912yes—
658C22C2×#57511yescharacteristic
659C22C2×#57611yescharacteristic
660C22#358612yes—
661C4—22no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—42no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4#326814yes—
677C4C2×#53611yescharacteristic
678C4#349911yescharacteristic
679C4#355212yes—
680C2—82no—
681C2—82no—
682C2#3936011yescharacteristic
683C2C2×#326811yescharacteristic
684C2#6108311yescharacteristic
685C2#6251112yes—
686C2#6252012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
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×#3381C2semidirect
3. #39744C2semidirect
5. #33314C2semidirect
6. #31106C2semidirect
7. #31105C2semidirect
8. #29867C2semidirect
9. #29864C2semidirect
10. #28713C2semidirect
11. #28675C2semidirect
12. #26073C2semidirect
13. #25840C2semidirect
14. #25835C2semidirect
16. #25701C2semidirect
17. #25037C2semidirect
18. #21634C2semidirect
19. #21279C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33812
C2×#33532
#397442
#389552
#333142
#311062
#311052
#298672
#298642
#287132
#286752
#260732
#258402
#258352
#257032
#257012
#250372
#216342
#212792