Finite Group Databasebeta

← #6809584

Subgroups — #6809584

Order 512.

1797 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3232C212yesmaximal
2#36116C211yesmaximal, characteristic
3C2×#2552C211yesmaximal, characteristic, complemented
4#33108C211yesmaximal, characteristic, complemented
5#30952C214yesmaximal, complemented
6#30811C214yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#28814C212yesmaximal, complemented
9#27910C211yesmaximal, characteristic, complemented
10#27768C212yesmaximal, complemented
11#27645C211yesmaximal, characteristic
12#26461C212yesmaximal, complemented
13#26041C212yesmaximal, complemented
14#24602C212yesmaximal, complemented
15#24067C212yesmaximal
16#22599C211yesmaximal, characteristic, complemented
17C4×#480C211yesmaximal, characteristic
18C2×#529C2212yes—
19C22×#125C2212yes—
20#3232—216no—
21C2×#499C2214yes—
22C2×#488C2214yes—
23C2×#484C2212yes—
24C2×#484C2212yes—
25C2×#481C2212yes—
26C2×#483C2212yes—
27C2×#480C2211yescharacteristic
28C2×#479C2214yes—
29C2×#478C2214yes—
30C2×#431C2212yes—
31#2857C2211yescharacteristic
32#2855C2212yes—
33#2552—21no—
34#2552—22no—
35#2552—21no—
36#2552C2211yescharacteristic
37#2552C2214yes—
38#2552C2212yes—
39#2552C2212yes—
40#2547C2211yescharacteristic
41#2496—21no—
42#2486—21no—
43#2478—22no—
44#2453C2212yes—
45#2447C2212yes—
46#2419—22no—
47C2×#398—22no—
48#2365C2211yescharacteristic
49D4×C4⋊C4—21no—
50D4×C4⋊C4C2211yescharacteristic
51#2344—21no—
52C4×#121C2211yescharacteristic
53C2×#389—21no—
54C2×#389C2212yes—
55C2×#384—21no—
56C2×#384C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59#2095C2214yes—
60#2087C2214yes—
61#2084C2212yes—
62#2071C2212yes—
63#2068C2214yes—
64#2059C2214yes—
65#2059C2214yes—
66#2059C2214yes—
67#2042C2214yes—
68#2026C2214yes—
69#2026C2214yes—
70#1981C2212yes—
71#1956C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1942C2214yes—
75#1939C2212yes—
76#1933C2214yes—
77#1927C2214yes—
78#1925C2212yes—
79#1918C2212yes—
80#1916C2214yes—
81#1874C2211yescharacteristic
82#1872C2212yes—
83#1865C2214yes—
84#1862C2214yes—
85#1852C2211yescharacteristic
86#1845C2212yes—
87#1824C2212yes—
88#1823C2211yescharacteristic
89#1817C2212yes—
90#1802C2212yes—
91#1797C2211yescharacteristic
92C4×#107C2212yes—
93C8×C4⋊C4C2212yes—
94C4×#102C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97#529—28no—
98#529—44no—
99C2×#126—22no—
100C2×#125—28no—
101C2×#125—22no—
102C2×#125—22no—
103C2×#125—44no—
104C2×#125C2314yes—
105C2×#125C2312yes—
106C2×#125C2314yes—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8C2312yes—
110C2×#121C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#499—216no—
132#488—216no—
133#484—28no—
134#484—44no—
135#483—28no—
136#481—28no—
137#480D418yes—
138#479—216no—
139#478—216no—
140#431—28no—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2314yes—
161#400—22no—
162#398—22no—
163#398—24no—
164#398—24no—
165#398—24no—
166#398—24no—
167#398—22no—
168#397—22no—
169#397—22no—
170#397—22no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#390C2311yescharacteristic
175#389—21no—
176#389—22no—
177#389—21no—
178#389—22no—
179#389—22no—
180#389—24no—
181#389—21no—
182#389—21no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#389C2314yes—
188#389C2314yes—
189#389C2311yescharacteristic
190#388—21no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#386—21no—
196#385—22no—
197#385—22no—
198#385—22no—
199#384—21no—
200#384—22no—
201#384—22no—
202#384—24no—
203#384—21no—
204#384—21no—
205#384C2312yes—
206#384C2312yes—
207#384C2311yescharacteristic
208#381—22no—
209#381—22no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380—21no—
214#380—21no—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—21no—
229C2×#95—21no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2312yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#336C2312yes—
240#336C2314yes—
241#336C2314yes—
242#336C2312yes—
243C22×Q8C2412yes—
244C22×D4—21no—
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260#126—48no—
261#125—28no—
262#125—28no—
263#125—24no—
264#125—44no—
265#125—44no—
266#125—42no—
267#125—48no—
268#125—48no—
269#124—24no—
270#124—42no—
271#123—48no—
272C22⋊Q8—24no—
273C22⋊Q8—24no—
274#121—22no—
275#121C2×D414yes—
276C4×Q8—24no—
277C4×Q8—42no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—42no—
281C4×D4—48no—
282C4×D4—44no—
283#117—48no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—48no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—84no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D414yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2412yes—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—28no—
360#107—24no—
361#107—44no—
362#107C2×D418yes—
363#106—28no—
364#106—28no—
365C4⋊C8—28no—
366C4⋊C8—24no—
367#103—28no—
368#103—28no—
369#103—28no—
370#103—44no—
371#102—28no—
372#102—24no—
373#102C2×D418yes—
374C22⋊C8—24no—
375C22⋊C8—22no—
376C22⋊C8C2×D414yes—
377C4×C8—28no—
378C4×C8—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—21no—
399C24C22×D411yescharacteristic
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4C22×D412yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—82no—
447C22×C4#13514yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4H(5,𝔽2)12yes—
459C22×C4C2511yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—24no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8C22×D414yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—42no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—81no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C4⋊C4—82no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C22×D414yes—
509C4⋊C4C22×D412yes—
510C4⋊C42−1+414yes—
511C22⋊C4—24no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—81no—
520C42—22no—
521C42—22no—
522C42—24no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—44no—
531C42—44no—
532C42—42no—
533C422−1+414yes—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23D4212yes—
546C23C2×#13512yes—
547C23#57812yes—
548C23C23×D411yescharacteristic
549Q8—44no—
550Q8—44no—
551D4—24no—
552D4—48no—
553D4—42no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—84no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4#53612yes—
611C2×C4#53612yes—
612C2×C4D4212yes—
613C2×C4C2×#13512yes—
614C2×C4C2×#13512yes—
615C2×C4#57512yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+412yes—
619C2×C4C2×2−1+412yes—
620C8—24no—
621C8—44no—
622C8—42no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22#328314yes—
637C22C2×#53611yescharacteristic
638C22C2×D4211yescharacteristic
639C22#348611yescharacteristic
640C22C22×#13511yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57811yescharacteristic
643C22#358511yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—42no—
648C4—42no—
649C4—44no—
650C4—42no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—41no—
655C4—81no—
656C4—84no—
657C4—82no—
658C4—81no—
659C4—82no—
660C4#323214yescomplemented
661C4#349912yes—
662C4#354412yes—
663C2—24no—
664C2—82no—
665C2#3933711yescharacteristic
666C2C2×#323211yescharacteristic
667C2#6087411yescharacteristic
668C2C2×#328311yescharacteristic
669C2#6107011yescharacteristic
670C2#6181911yescharacteristic
671C2#6184211yescharacteristic
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×#2552C2semidirect
4. #33108C2semidirect
5. #30952C2semidirect
6. #30811C2semidirect
7. #29643C2semidirect
8. #28814C2semidirect
9. #27910C2semidirect
10. #27768C2semidirect
12. #26461C2semidirect
13. #26041C2semidirect
14. #24602C2semidirect
16. #22599C2semidirect
660. C4#3232semidirect

Maximal subgroups

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

subgroupindex
C2×#32322
#361162
C2×#25522
#331082
#309522
#308112
#296432
#288142
#279102
#277682
#276452
#264612
#260412
#246022
#240672
#225992
C4×#4802