Finite Group Databasebeta

← #6498096

Subgroups — #6498096

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic
3#33150C212yesmaximal
4#32960C212yesmaximal
5#29985C212yesmaximal, complemented
6#29871C212yesmaximal, complemented
7#29577C214yesmaximal, complemented
8#29395C212yesmaximal, complemented
9#29391C212yesmaximal, complemented
10#29297C211yesmaximal, characteristic
11#29296C211yesmaximal, characteristic
12#29105C211yesmaximal, characteristic
13#29105C211yesmaximal, characteristic
14#27617C212yesmaximal
15#27589C212yesmaximal
16#22115C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#479C2214yes—
20#2864C2212yes—
21#2863—21no—
22#2863—21no—
23#2862C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2855—21no—
28#2855—21no—
29#2851C2212yes—
30C4×C4⋊Q8C2211yescharacteristic
31C4×C4⋊Q8C2211yescharacteristic
32C4×#125C2212yes—
33C4×C22⋊Q8—21no—
34C4×C22⋊Q8—21no—
35#2295—21no—
36#2295—21no—
37#2295—21no—
38#2295—21no—
39#2295—21no—
40#2295—21no—
41#2295C2212yes—
42#2295C2212yes—
43#2295C2212yes—
44#2295C2212yes—
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×#387—21no—
51#2283C2212yes—
52C2×C4×C4⋊C4—21no—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×#376—21no—
55C2×C43C2211yescharacteristic
56#2101C2214yes—
57#2097C2214yes—
58#2085C2214yes—
59#2084C2212yes—
60#2083C2212yes—
61#2074C2214yes—
62#2072C2212yes—
63#2072C2212yes—
64#2058C2212yes—
65#2057C2212yes—
66#2024C2212yes—
67#2024C2212yes—
68#1989C2214yes—
69#1949C2214yes—
70#1942C2214yes—
71#1941C2214yes—
72#1933C2214yes—
73#1933C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1920C2214yes—
77#1919C2214yes—
78#1918C2214yes—
79#1918C2214yes—
80#1865C2214yes—
81#1863C2214yes—
82#1853C2212yes—
83#1853C2212yes—
84#1852C2212yes—
85#1852C2212yes—
86#1839C2212yes—
87#1831C2212yes—
88#1824C2212yes—
89#1824C2212yes—
90#1817C2212yes—
91#1805C2214yes—
92#1797C2212yes—
93#1787C2212yes—
94#1787C2212yes—
95C4×C4⋊C8C2212yes—
96C4×C22⋊C8C2211yescharacteristic
97C4×C22⋊C8C2211yescharacteristic
98#1744C2212yes—
99C23×Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C4×Q8—28no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42—21no—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#479—216no—
128C2×#107C2314yes—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—21no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#385—22no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#383—22no—
198#382C2312yes—
199#382C2312yes—
200#382C2312yes—
201#381—22no—
202#380—21no—
203#380—21no—
204#380C2312yes—
205#380C2312yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—22no—
221#376—22no—
222#376—22no—
223#376—22no—
224#376—22no—
225#376—22no—
226C2×#95—21no—
227C2×#95—21no—
228C2×#95—21no—
229C2×#95—21no—
230C2×#95—21no—
231C2×#95—21no—
232C2×#95—21no—
233C43D411yescharacteristic
234C43D411yescharacteristic
235C43D411yescharacteristic
236C43D411yescharacteristic
237C43C2311yescharacteristic
238C43C2311yescharacteristic
239C43C2311yescharacteristic
240C43C2311yescharacteristic
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2312yes—
251#336C2312yes—
252#336C2314yes—
253#336C2314yes—
254C22×Q8—24no—
255C22×Q8—21no—
256C22×Q8—21no—
257C22×Q8—21no—
258C22×Q8—24no—
259C22×Q8—42no—
260C22×Q8C2412yes—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C22×C8C2414yes—
273C22×C8C2414yes—
274C4⋊Q8—42no—
275C4⋊Q8—42no—
276#125—28no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×Q8#4118yes—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42—44no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—28no—
367#107—44no—
368#106—28no—
369#106—44no—
370C4⋊C8—28no—
371#103—28no—
372#103—28no—
373#103—44no—
374#103—44no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377C4×C8—42no—
378C4×C8—42no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407C24C22×D411yescharacteristic
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—416no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8C2×#4114yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4#13514yes—
461C22×C4#13714yes—
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C42−1+414yes—
476C22×C4C2511yescharacteristic
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8—24no—
482C2×C8—24no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4C2×#4114yes—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C42—28no—
515C42—28no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—44no—
544C42—44no—
545C42—44no—
546C42—44no—
547C42C2×#4114yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23#53612yes—
553C23C2×#13512yes—
554C23C2×#13712yes—
555C23C23×D411yescharacteristic
556Q8—28no—
557Q8—48no—
558Q8—44no—
559Q8—84no—
560Q8—84no—
561C2×C4—28no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—28no—
581C2×C4—21no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
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—84no—
625C2×C4—84no—
626C2×C4#52512yes—
627C2×C4#52512yes—
628C2×C4#53612yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13512yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13712yes—
633C2×C4#57512yes—
634C2×C4#57512yes—
635C2×C4C22×#4112yes—
636C2×C4C2×2−1+412yes—
637C2×C4#58512yes—
638C8—44no—
639C8—44no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22#323114yes—
647C22C2×#52511yescharacteristic
648C22C2×#53611yescharacteristic
649C22#344911yescharacteristic
650C22C22×#13511yescharacteristic
651C22C22×#13711yescharacteristic
652C22C2×#57511yescharacteristic
653C22C2×#57511yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—22no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—44no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—44no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4#349514yes—
676C2—24no—
677C2#3532511yescharacteristic
678C2#6053611yescharacteristic
679C2#6056011yescharacteristic
680C2C2×#323111yescharacteristic
681C2#6086611yescharacteristic
682C2#6094511yescharacteristic
683C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29985C2semidirect
6. #29871C2semidirect
7. #29577C2semidirect
8. #29395C2semidirect
9. #29391C2semidirect
16. #22115C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C2×#22952
#331502
#329602
#299852
#298712
#295772
#293952
#293912
#292972
#292962
#291052
#276172
#275892
#221152
#207772