Finite Group Databasebeta

← #6578596

Subgroups — #6578596

Order 512.

2075 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3117C212yesmaximal
2#36116C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic, complemented
4#33109C211yesmaximal, characteristic, complemented
5#33010C212yesmaximal, complemented
6#26910C214yesmaximal, complemented
7#26604C214yesmaximal, complemented
8#26465C212yesmaximal, complemented
9#25722C211yesmaximal, characteristic, complemented
10#25365C212yesmaximal, complemented
11#24508C212yesmaximal, complemented
12#24209C211yesmaximal, characteristic
13#23937C212yesmaximal
14#23187C212yesmaximal, complemented
15#23002C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic, complemented
17C4×#481C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×Q8C2212yes—
21C2×#494C2212yes—
22#3117—216no—
23C2×#484C2212yes—
24C2×#481C2211yescharacteristic
25C2×#479C2214yes—
26C2×#464C2214yes—
27C2×#461C2212yes—
28C2×#451C2212yes—
29C2×#451C2212yes—
30C2×#448C2212yes—
31C2×#441C2214yes—
32C2×#440C2214yes—
33C2×#432C2212yes—
34#2857C2211yescharacteristic
35#2855C2212yes—
36#2552C2211yescharacteristic
37#2547C2211yescharacteristic
38#2496—21no—
39#2486—21no—
40#2478—22no—
41#2453C2212yes—
42#2447C2212yes—
43#2419—22no—
44#2365—21no—
45#2365—22no—
46#2365—21no—
47#2365C2211yescharacteristic
48#2365C2212yes—
49#2365C2214yes—
50#2365C2212yes—
51D4×C4⋊C4—21no—
52D4×C4⋊C4C2211yescharacteristic
53#2344—21no—
54C4×#121C2211yescharacteristic
55C2×#391C2212yes—
56C2×#389—21no—
57C2×#387—22no—
58C2×C4×C4⋊C4—21no—
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61#2066C2214yes—
62#2056C2214yes—
63#2043C2212yes—
64#2026C2214yes—
65#2016C2214yes—
66#2015C2214yes—
67#2013C2212yes—
68#1981C2212yes—
69#1981C2212yes—
70#1971C2214yes—
71#1956C2212yes—
72#1947C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1928C2214yes—
76#1926C2212yes—
77#1919C2212yes—
78#1917C2214yes—
79#1878C2214yes—
80#1878C2214yes—
81#1878C2214yes—
82#1877C2214yes—
83#1874C2211yescharacteristic
84#1872C2212yes—
85#1863C2214yes—
86#1863C2214yes—
87#1853C2211yescharacteristic
88#1824C2211yescharacteristic
89#1817C2212yes—
90#1797C2211yescharacteristic
91#1787C2212yes—
92C4×#106C2212yes—
93#1770C2212yes—
94C4×#102C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2312yes—
97C22×Q16C2314yes—
98C22×Q16C2314yes—
99C22×SD16C2314yes—
100C2×C4⋊Q8—28no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106#529—28no—
107#529—44no—
108C2×#126—22no—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121C2311yescharacteristic
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—28no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2314yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D411yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—22no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#494—28no—
139#484—28no—
140#481D418yes—
141#479—216no—
142#464—216no—
143#461—28no—
144#451—28no—
145#451—44no—
146#448—28no—
147#441—216no—
148#440—216no—
149#432—28no—
150C2×#107C2314yes—
151C2×#106C2312yes—
152C2×#106C2312yes—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2311yescharacteristic
166C2×C22⋊C8C2311yescharacteristic
167C2×#97C2314yes—
168C2×C4×C8C2312yes—
169#400—22no—
170#398—22no—
171#398—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#391—22no—
176#391—22no—
177#391C2312yes—
178#391C2314yes—
179#391C2312yes—
180#391C2312yes—
181#391C2314yes—
182#390C2311yescharacteristic
183#389—21no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—24no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389C2312yes—
192#389C2312yes—
193#389C2311yescharacteristic
194#388—21no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—22no—
203#386—21no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384—21no—
208#384C2311yescharacteristic
209#381—22no—
210#381—22no—
211#380—21no—
212#380—21no—
213#380—21no—
214#380—21no—
215#380—21no—
216#380C2311yescharacteristic
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—24no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D412yes—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2311yescharacteristic
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2311yescharacteristic
232C4×C22⋊C4—21no—
233C2×#95—22no—
234C2×#95—21no—
235C2×#95—22no—
236C2×#95D412yes—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245#336C2312yes—
246C22×Q8—28no—
247C22×Q8—22no—
248C22×Q8—24no—
249C22×Q8—24no—
250C22×Q8—44no—
251C22×Q8C2412yes—
252C22×Q8C2414yes—
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4C2×D412yes—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2411yescharacteristic
265C2×Q16—28no—
266C2×Q16—28no—
267C2×Q16—28no—
268C2×Q16—44no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×SD16—28no—
272C2×SD16—28no—
273C2×SD16—44no—
274C22×C8C2414yes—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—28no—
280C4⋊Q8—24no—
281C4⋊Q8—24no—
282C4⋊Q8#4118yes—
283#126—48no—
284#125—24no—
285#125—42no—
286#124—24no—
287#124—42no—
288#123—48no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291#121—22no—
292#121C2×D414yes—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—24no—
296C4×Q8—44no—
297C4×Q8—44no—
298C4×Q8—42no—
299C4×Q8—48no—
300C4×Q8—48no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—42no—
304C4×D4—48no—
305C4×D4—44no—
306#117—48no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—48no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—48no—
366C2×C42—44no—
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375#107—28no—
376#106—42no—
377#106C2×D418yes—
378C4⋊C8—28no—
379C4⋊C8—24no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—44no—
385#103—44no—
386#103—44no—
387#102—42no—
388#102C2×D418yes—
389C22⋊C8—24no—
390C22⋊C8—22no—
391C22⋊C8C2×D414yes—
392#97—28no—
393C4×C8—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415C24—21no—
416C24C22×D411yescharacteristic
417C2×Q8—28no—
418C2×Q8—24no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—416no—
428C2×Q8—416no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8C2×#4118yes—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4C22×D412yes—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—41no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—82no—
475C22×C4#13714yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4H(5,𝔽2)12yes—
487C22×C4C2511yescharacteristic
488Q16—416no—
489Q16—88no—
490SD16—416no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—24no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8C22×D414yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—42no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4C22×D414yes—
534C4⋊C4C22×D412yes—
535C22⋊C4—24no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—81no—
544C42—22no—
545C42—22no—
546C42—24no—
547C42—24no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—48no—
554C42—48no—
555C42—44no—
556C42—44no—
557C42—42no—
558C42—44no—
559C42C2×#4114yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23D4212yes—
572C23C2×#13712yes—
573C23#57812yes—
574C23C23×D411yescharacteristic
575Q8—28no—
576Q8—28no—
577Q8—416no—
578Q8—44no—
579Q8—44no—
580Q8—44no—
581Q8—44no—
582D4—24no—
583D4—48no—
584D4—42no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—416no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—84no—
638C2×C4—88no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4#52512yes—
642C2×C4#52512yes—
643C2×C4D4212yes—
644C2×C4C2×#13712yes—
645C2×C4C2×#13712yes—
646C2×C4#57512yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4112yes—
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4#58512yes—
651C8—24no—
652C8—44no—
653C8—42no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22#327614yes—
668C22C2×#52511yescharacteristic
669C22C2×D4211yescharacteristic
670C22#348711yescharacteristic
671C22C22×#13711yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57811yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—48no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—42no—
684C4—44no—
685C4—41no—
686C4—82no—
687C4—81no—
688C4—82no—
689C4—81no—
690C4—82no—
691C4#311714yescomplemented
692C4D4×#4112yes—
693C4#351312yes—
694C2—24no—
695C2—82no—
696C2#3610211yescharacteristic
697C2C2×#311711yescharacteristic
698C2#6058311yescharacteristic
699C2C2×#327611yescharacteristic
700C2#6106711yescharacteristic
701C2#6189311yescharacteristic
702C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2365C2semidirect
4. #33109C2semidirect
5. #33010C2semidirect
6. #26910C2semidirect
7. #26604C2semidirect
8. #26465C2semidirect
9. #25722C2semidirect
10. #25365C2semidirect
11. #24508C2semidirect
14. #23187C2semidirect
15. #23002C2semidirect
16. #22603C2semidirect
691. C4#3117semidirect

Maximal subgroups

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

subgroupindex
C2×#31172
#361162
C2×#23652
#331092
#330102
#269102
#266042
#264652
#257222
#253652
#245082
#242092
#239372
#231872
#230022
#226032
C4×#4812