Finite Group Databasebeta

← #6489375

Subgroups — #6489375

Order 512.

1668 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic, complemented
2#34211C211yesmaximal, characteristic
3#32964C212yesmaximal, complemented
4#32962C212yesmaximal
5#30090C212yesmaximal, complemented
6#30089C212yesmaximal, complemented
7#28727C211yesmaximal, characteristic, complemented
8#28727C211yesmaximal, characteristic, complemented
9#28520C211yesmaximal, characteristic, complemented
10#28519C211yesmaximal, characteristic, complemented
11#28212C212yesmaximal, complemented
12#28211C212yesmaximal, complemented
13#27955C212yesmaximal, complemented
14#27596C212yesmaximal, complemented
15#27594C212yesmaximal
16#25864C212yesmaximal
17#22224C212yesmaximal, complemented
18#22223C212yesmaximal, complemented
19#20783C211yesmaximal, characteristic, complemented
20C22×#124C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#489C2212yes—
24C2×#489C2212yes—
25C2×#477C2212yescomplemented
26C2×#475C2212yescomplemented
27C2×#476C2212yes—
28C2×#474C2212yes—
29#2868C2211yescharacteristic, complemented
30#2862C2211yescharacteristic, complemented
31#2857C2212yes—
32#2647C2212yescomplemented
33#2628C2212yes—
34#2511—22no—
35#2492—22no—
36#2451—22no—
37#2423—22no—
38#2367—22no—
39#2367—22no—
40#2342—22no—
41#2342—22no—
42#2331C2212yescomplemented
43#2326C2211yescharacteristic
44#2326C2211yescharacteristic
45#2326C2211yescharacteristic
46#2326C2211yescharacteristic
47#2314C2212yes—
48#2310C2211yescharacteristic
49#2309C2211yescharacteristic
50C4×C4⋊Q8C2212yescomplemented
51C4×#124C2211yescharacteristic
52C4×#124C2211yescharacteristic
53C4×#124C2211yescharacteristic
54C4×#124C2211yescharacteristic
55#2285C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2096C2212yescomplemented
58#2095C2212yescomplemented
59#2094C2212yes—
60#2093C2212yes—
61#2084C2214yescomplemented
62#2083C2214yescomplemented
63#2074C2212yescomplemented
64#2074C2212yescomplemented
65#2073C2212yes—
66#2073C2212yes—
67#2042C2214yes—
68#2042C2214yes—
69#1993C2212yescomplemented
70#1993C2212yescomplemented
71#1959C2212yes—
72#1959C2212yes—
73#1928C2214yescomplemented
74#1927C2214yescomplemented
75#1926C2214yes—
76#1925C2214yes—
77#1865C2214yescomplemented
78#1864C2214yes—
79#1863C2214yescomplemented
80#1862C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1847C2212yescomplemented
84#1847C2212yescomplemented
85#1840C2212yescomplemented
86#1832C2212yescomplemented
87#1818C2212yes—
88#1805C2212yescomplemented
89#1805C2212yescomplemented
90#1804C2212yes—
91#1804C2212yes—
92#1798C2212yes—
93#1787C2214yes—
94#1786C2214yes—
95#1767C2211yescharacteristic, complemented
96#1767C2211yescharacteristic, complemented
97#1749C2212yes—
98#1745C2212yescomplemented
99C23×Q8C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×#124—24no—
105C2×#124—24no—
106C2×#124—22no—
107C2×#124C2311yescharacteristic
108C2×#124C2311yescharacteristic
109C2×#124C2311yescharacteristic
110C2×#124C2311yescharacteristic
111C2×C22⋊Q8—22no—
112#519—44no—
113#518—44no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C2×#117—21no—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C22⋊C4—22no—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C42C2311yescharacteristic
137#489—28no—
138#489—28no—
139#477—44no—
140#476—44no—
141#475—44no—
142#474—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C22⋊C8C2314yes—
158C2×#97C2312yes—
159C2×#97C2312yes—
160#400—22no—
161#400—24no—
162#399—22no—
163#398—24no—
164#395—22no—
165#393—22no—
166#391—22no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391—24no—
172#391C2311yescharacteristic
173#391C2311yescharacteristic
174#391C2311yescharacteristic
175#391C2311yescharacteristic
176#390—22no—
177#390C2311yescharacteristic
178#390C2311yescharacteristic
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#389—22no—
182#389—22no—
183#389C2314yes—
184#389C2314yes—
185#388—24no—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—24no—
195#387—24no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#384—22no—
201#384C2312yes—
202#384C2312yes—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382C2312yes—
206#382C2312yes—
207#382C2311yescharacteristic
208#382C2311yescharacteristic
209#381—22no—
210#380—22no—
211#380—22no—
212#380C2314yes—
213#379—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2314yes—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225#376C2311yescharacteristic
226#376C2311yescharacteristic
227C43C2311yescharacteristic
228C43C2311yescharacteristic
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2314yes—
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8C2411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—24no—
252C22×D4—42no—
253C22×D4C2411yescharacteristic
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4C2×D412yes—
261C23×C4C2411yescharacteristic
262C22×C8C2414yes—
263C22×C8C2414yes—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—24no—
267C4⋊Q8—24no—
268C4⋊Q8—84no—
269#124—216no—
270#124—22no—
271#124—22no—
272#124—22no—
273#124—22no—
274#124—22no—
275#124—22no—
276#124#4114yes—
277#124#4114yes—
278C22⋊Q8—84no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8—84no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—44no—
291C4×D4—44no—
292#117—24no—
293#117—24no—
294#117—44no—
295#117—44no—
296#117—42no—
297#117—42no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2414yes—
320C2×C22⋊C4—216no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4C2×D411yescharacteristic
339C2×C22⋊C4C2×D411yescharacteristic
340C2×C22⋊C4C2×D412yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2×D412yes—
354C2×C42C2×D412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361#107—44no—
362#107—44no—
363#106—44no—
364#106—44no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367#103—28no—
368#103—28no—
369#103—44no—
370#103—44no—
371#102—28no—
372#102—28no—
373#102—44no—
374#102—44no—
375C22⋊C8—28no—
376#97—24no—
377#97—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—24no—
390#95—22no—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394#95C2×D412yes—
395C24—24no—
396C24—21no—
397C24—21no—
398C24—42no—
399C24—42no—
400C24—82no—
401C24—82no—
402C24C22×D411yescharacteristic
403C2×Q8—24no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—42no—
418C2×Q8—84no—
419C2×Q8C2×#4112yes—
420C2×Q8C2×#4112yes—
421C2×D4—24no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4C2×#4112yes—
433C2×D4C2×#4112yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4C22×D412yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C42−1+414yes—
469C22×C4C2511yescharacteristic
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—24no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C22⋊C4—84no—
518C42—24no—
519C42—21no—
520C42—21no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—42no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—42no—
534C42—84no—
535C42C2×#4112yes—
536C42C2×#4112yes—
537C42H(5,𝔽2)14yes—
538C23—22no—
539C23—22no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—48no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—88no—
554C23—164no—
555C23—162no—
556C23—162no—
557C23#53612yes—
558C23C23×D411yescharacteristic
559Q8—24no—
560Q8—24no—
561Q8—48no—
562Q8—42no—
563Q8—42no—
564Q8—88no—
565Q8—164no—
566D4—24no—
567D4—24no—
568D4—48no—
569D4—42no—
570D4—42no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—48no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—84no—
614C2×C4—162no—
615C2×C4—162no—
616C2×C4—162no—
617C2×C4—162no—
618C2×C4#52514yes—
619C2×C4#53512yes—
620C2×C4#53512yes—
621C2×C4#53612yes—
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+412yes—
626C8—44no—
627C8—44no—
628C22—22no—
629C22—24no—
630C22—42no—
631C22—41no—
632C22—41no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—82no—
642C22—82no—
643C22—168no—
644C22C2×#52511yescharacteristic
645C22C2×#53511yescharacteristic
646C22C2×#53611yescharacteristic
647C22#358512yes—
648C22#358612yes—
649C4—24no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—42no—
654C4—41no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—84no—
661C4—81no—
662C4—81no—
663C4—84no—
664C4—81no—
665C4—81no—
666C4#352012yes—
667C4#352012yes—
668C2—42no—
669C2—82no—
670C2—82no—
671C2#3524011yescharacteristic
672C2#6056712yes—
673C2#6082511yescharacteristic
674C2#6082611yescharacteristic
675C2#6088512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35283C2semidirect
3. #32964C2semidirect
5. #30090C2semidirect
6. #30089C2semidirect
7. #28727C2semidirect
8. #28727C2semidirect
9. #28520C2semidirect
10. #28519C2semidirect
11. #28212C2semidirect
12. #28211C2semidirect
13. #27955C2semidirect
14. #27596C2semidirect
17. #22224C2semidirect
18. #22223C2semidirect
19. #20783C2semidirect
25. C2×#477C22semidirect
26. C2×#475C22semidirect
29. #2868C22semidirect
30. #2862C22semidirect
32. #2647C22semidirect
42. #2331C22semidirect
50. C4×C4⋊Q8C22semidirect
57. #2096C22semidirect
58. #2095C22semidirect
61. #2084C22semidirect
62. #2083C22semidirect
63. #2074C22semidirect
64. #2074C22semidirect
69. #1993C22semidirect
70. #1993C22semidirect
73. #1928C22semidirect
74. #1927C22semidirect
77. #1865C22semidirect
79. #1863C22semidirect
83. #1847C22semidirect
84. #1847C22semidirect
85. #1840C22semidirect
86. #1832C22semidirect
88. #1805C22semidirect
89. #1805C22semidirect
95. #1767C22semidirect
96. #1767C22semidirect
98. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#352832
#342112
#329642
#329622
#300902
#300892
#287272
#285202
#285192
#282122
#282112
#279552
#275962
#275942
#258642
#222242
#222232
#207832