Finite Group Databasebeta

← #6686332

Subgroups — #6686332

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3364C211yesmaximal, characteristic, complemented
2C2×#3357C211yesmaximal, characteristic
3#39173C211yesmaximal, characteristic, complemented
4#36185C211yesmaximal, characteristic
5#33201C211yesmaximal, characteristic, complemented
6#30637C214yesmaximal, complemented
7#30610C214yesmaximal, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29830C211yesmaximal, characteristic, complemented
10#28676C211yesmaximal, characteristic, complemented
11#28606C211yesmaximal, characteristic, complemented
12#26561C214yesmaximal, complemented
13#26367C214yesmaximal, complemented
14#25836C211yesmaximal, characteristic, complemented
15#25700C211yesmaximal, characteristic
16#25646C211yesmaximal, characteristic, complemented
17#25573C211yesmaximal, characteristic, complemented
18#22210C211yesmaximal, characteristic, complemented
19#21114C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#556C2211yescharacteristic
22C2×#555C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3364—28no—
25#3357—28no—
26C2×#500C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#488C2212yes—
29C2×#487C2211yescharacteristic
30C2×#487C2211yescharacteristic
31C2×#485C2212yes—
32C2×#480C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#476C2211yescharacteristic
36C2×#474C2211yescharacteristic
37C2×#468C2212yes—
38C2×#464C2212yes—
39C2×#462C2211yescharacteristic
40C2×C8×Q8C2211yescharacteristic
41C2×C4×Q16C2212yes—
42C2×C4×SD16C2211yescharacteristic
43C2×#436C2211yescharacteristic
44C2×#428C2212yes—
45#2862C2211yescharacteristic
46#2862C2211yescharacteristic
47#2850—21no—
48#2690—24no—
49#2642—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2565—21no—
53#2547—21no—
54#2547C2211yescharacteristic
55#2542C2211yescharacteristic
56#2521—24no—
57#2500—21no—
58#2499—21no—
59#2442C2214yes—
60#2427C2214yes—
61#2421C2214yes—
62#2362C2214yes—
63#2326—21no—
64C4×#124C2211yescharacteristic
65#2295C2211yescharacteristic
66#2285—21no—
67C42×D4C2211yescharacteristic
68#2097C2214yes—
69#2097C2214yes—
70#2087C2214yes—
71#2068C2214yes—
72#2059C2214yes—
73#2056C2214yes—
74#2042C2214yes—
75#2036C2214yes—
76#2034C2214yes—
77#2001C2214yes—
78#1988C2211yescharacteristic
79#1987C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1954C2211yescharacteristic
84#1949C2214yes—
85#1942C2214yes—
86#1928C2212yes—
87#1923C2214yes—
88#1919C2212yes—
89#1894C2214yes—
90#1877C2212yes—
91#1872C2214yes—
92#1865C2214yes—
93#1865C2214yes—
94#1862C2212yes—
95#1854C2214yes—
96#1851C2214yes—
97#1842C2211yescharacteristic
98#1804C2214yes—
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×Q16C2312yes—
104C22×SD16C2311yescharacteristic
105C22×M16C2312yes—
106#557—24no—
107#557—42no—
108#556—24no—
109#556—42no—
110#555—24no—
111#555—42no—
112#554—24no—
113#554—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×#126—22no—
120C2×#125—21no—
121C2×#125—22no—
122C2×#125C2312yes—
123C2×#124—21no—
124C2×#124—22no—
125C2×#124D411yescharacteristic
126C2×#124C2312yes—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8C2312yes—
130C2×C22⋊Q8C2314yes—
131C2×#121—21no—
132C2×#121C2312yes—
133C2×C4×Q8—21no—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×D4—21no—
139C2×C4×D4C2312yes—
140C2×C4×D4C2311yescharacteristic
141C2×C4×D4C2311yescharacteristic
142C2×#117—21no—
143C2×#117—21no—
144C2×#117C2312yes—
145C22×C4⋊C4C2312yes—
146C22×C42—21no—
147#500—24no—
148#498—42no—
149#488—28no—
150#487—24no—
151#487—42no—
152#485—28no—
153#480—28no—
154#479—28no—
155#477—24no—
156#476—24no—
157#474—24no—
158#468—28no—
159#464—28no—
160#462—24no—
161C8×Q8—24no—
162C4×Q16—28no—
163C4×SD16D418yes—
164#436—24no—
165#428—28no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#106C2312yes—
172C2×#106C2314yes—
173C2×#106C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2314yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2311yescharacteristic
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×C22⋊C8C2312yes—
191C2×C22⋊C8C2314yes—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2312yes—
194C2×C4×C8C2311yescharacteristic
195#400—24no—
196#400—24no—
197#399—22no—
198#398—24no—
199#397—24no—
200#397—22no—
201#397—24no—
202#395—22no—
203#395—22no—
204#394—24no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—22no—
211#391—22no—
212#391D411yescharacteristic
213#391C2312yes—
214#391C2314yes—
215#390—21no—
216#390D411yescharacteristic
217#389C2314yes—
218#389C2314yes—
219#389C2312yes—
220#388—24no—
221#388—24no—
222#388—24no—
223#388D414yes—
224#387—22no—
225#387—24no—
226#387—24no—
227#386—24no—
228#385—24no—
229#384—21no—
230#384—21no—
231#384—21no—
232#382—21no—
233#382—21no—
234#382D411yescharacteristic
235#382C2314yes—
236#380—24no—
237#380—24no—
238#380C2314yes—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—24no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4C2314yes—
247#376—21no—
248C43C2311yescharacteristic
249#340C2314yes—
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8C2×D411yescharacteristic
262C22×Q8C2412yes—
263C22×Q8C2411yescharacteristic
264C22×D4—22no—
265C22×D4C2×D411yescharacteristic
266C22×D4C2411yescharacteristic
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4C2412yes—
271C2×Q16—24no—
272C2×Q16—44no—
273C2×SD16—42no—
274C2×SD16C2×D414yes—
275C2×M16—24no—
276C2×M16—44no—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—44no—
291#126—48no—
292#125—24no—
293#125—42no—
294#125—44no—
295#125—48no—
296#124—24no—
297#124—42no—
298#124—42no—
299#124—48no—
300#124—44no—
301C22⋊Q8—24no—
302C22⋊Q8—28no—
303C22⋊Q8—48no—
304C22⋊Q8—44no—
305C22⋊Q8—42no—
306C22⋊Q8—44no—
307#121—24no—
308#121—42no—
309#121—44no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—42no—
316C4×Q8—42no—
317C4×Q8—44no—
318C4×Q8C2×D414yes—
319C4×D4—24no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—42no—
324C4×D4—44no—
325C4×D4C2×D414yes—
326#117—24no—
327#117—42no—
328#117—44no—
329#117—44no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4C2×D412yes—
368C2×C22⋊C4C2×D414yes—
369C2×C22⋊C4C2×D414yes—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—44no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D412yes—
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384#107—24no—
385#107—24no—
386#107—24no—
387#107—24no—
388#107—24no—
389#106—24no—
390#106—44no—
391#106C2×D414yes—
392C4⋊C8—24no—
393C4⋊C8—22no—
394C4⋊C8—22no—
395C4⋊C8—44no—
396#103—24no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—44no—
402#103—44no—
403#103C2×D414yes—
404#102—24no—
405#102—24no—
406#102—42no—
407#102C2×D414yes—
408C22⋊C8—24no—
409C22⋊C8—44no—
410C4×C8—24no—
411C4×C8—22no—
412C4×C8C2×D414yes—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—21no—
417#95—24no—
418#95—22no—
419#95—21no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95C2×D414yes—
427#95C2×D414yes—
428C24—21no—
429C24C22×D412yes—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—82no—
446C2×Q8—82no—
447C2×Q8C22×D412yes—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—44no—
455C2×D4—84no—
456C2×D4C22×D412yes—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—82no—
477C22×C4—84no—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C2511yescharacteristic
486Q16—48no—
487SD16—44no—
488M16—48no—
489C2×C8—22no—
490C2×C8—22no—
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—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4C22×D412yes—
531C4⋊C4C22×D412yes—
532C4⋊C42−1+414yes—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—48no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—42no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42—44no—
557C42—42no—
558C42—42no—
559C42C22×D412yes—
560C42H(5,𝔽2)12yes—
561C23—24no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—22no—
566C23—22no—
567C23—42no—
568C23—42no—
569C23—41no—
570C23—41no—
571C23C23×D411yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—44no—
575Q8—42no—
576Q8—42no—
577Q8—42no—
578Q8—44no—
579Q8—84no—
580Q8—162no—
581Q8—162no—
582D4—22no—
583D4—22no—
584D4—44no—
585D4—42no—
586D4—164no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—88no—
628C2×C4—164no—
629C2×C4D4212yes—
630C2×C4D4212yes—
631C2×C4#57512yes—
632C2×C4#57812yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C2×H(5,𝔽2)11yescharacteristic
635C2×C4C2×2−1+412yes—
636C2×C4#58512yes—
637C8—22no—
638C8—22no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—82no—
651C22—82no—
652C22C2×D4211yescharacteristic
653C22#348912yes—
654C22C2×#57511yescharacteristic
655C22C2×#57811yescharacteristic
656C22#358412yes—
657C4—22no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4#326914yes—
673C4#344911yescharacteristic
674C4#349911yescharacteristic
675C4#355412yes—
676C2—82no—
677C2—82no—
678C2#3768011yescharacteristic
679C2C2×#326911yescharacteristic
680C2#6109011yescharacteristic
681C2#6215212yes—
682C2#6222012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
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×#3364C2semidirect
3. #39173C2semidirect
5. #33201C2semidirect
6. #30637C2semidirect
7. #30610C2semidirect
8. #30281C2semidirect
9. #29830C2semidirect
10. #28676C2semidirect
11. #28606C2semidirect
12. #26561C2semidirect
13. #26367C2semidirect
14. #25836C2semidirect
16. #25646C2semidirect
17. #25573C2semidirect
18. #22210C2semidirect
19. #21114C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33642
C2×#33572
#391732
#361852
#332012
#306372
#306102
#302812
#298302
#286762
#286062
#265612
#263672
#258362
#257002
#256462
#255732
#222102
#211142