Finite Group Databasebeta

← #6490805

Subgroups — #6490805

Order 512.

1657 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35276C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#33120C212yesmaximal, complemented
4#33118C212yesmaximal, complemented
5#30064C212yesmaximal, complemented
6#30059C212yesmaximal, complemented
7#29262C212yesmaximal, complemented
8#29254C212yesmaximal, complemented
9#29216C212yesmaximal, complemented
10#28761C212yesmaximal, complemented
11#27913C211yesmaximal, characteristic, complemented
12#27909C211yesmaximal, characteristic, complemented
13#25911C211yesmaximal, characteristic, complemented
14#25906C211yesmaximal, characteristic, complemented
15#24073C212yesmaximal, complemented
16#24069C212yesmaximal, complemented
17#22232C212yesmaximal, complemented
18#22228C212yesmaximal, complemented
19#20784C211yesmaximal, characteristic, complemented
20C2×#542C2211yescharacteristic
21C2×#540C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#490C2212yes—
24C2×#488C2212yes—
25C2×#484C2212yes—
26C2×#481C2212yes—
27C2×#480C2212yes—
28C2×#480C2212yes—
29#2861C2211yescharacteristic
30#2858C2211yescharacteristic
31#2856C2211yescharacteristic
32#2850C2211yescharacteristic
33#2660C2211yescharacteristic
34#2657C2211yescharacteristic
35#2620C2211yescharacteristic
36#2619C2211yescharacteristic
37#2555—22no—
38#2554—22no—
39#2464—22no—
40#2461—22no—
41#2360—22no—
42#2360—22no—
43#2355—22no—
44Q8×C4⋊C4—22no—
45#2330C2212yes—
46#2330C2212yes—
47#2328C2211yescharacteristic
48#2326C2211yescharacteristic
49#2317C2212yes—
50C4×C4⋊Q8C2212yes—
51C4×C4⋊Q8C2212yes—
52C4×#127C2211yescharacteristic
53C4×#124C2211yescharacteristic
54C2×#384C2211yescharacteristic
55#2274C2212yes—
56C4×#117C2211yescharacteristic
57#2096C2212yes—
58#2093C2212yes—
59#2093C2212yes—
60#2093C2212yes—
61#2059C2214yes—
62#2057C2214yes—
63#2042C2214yes—
64#2041C2214yes—
65#2024C2212yes—
66#2023C2212yes—
67#2023C2212yes—
68#2023C2212yes—
69#1992C2212yes—
70#1991C2212yes—
71#1983C2214yes—
72#1982C2214yes—
73#1959C2212yes—
74#1957C2212yes—
75#1949C2214yes—
76#1947C2214yes—
77#1928C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1875C2212yes—
82#1875C2212yes—
83#1875C2212yes—
84#1873C2212yes—
85#1864C2214yes—
86#1863C2214yes—
87#1862C2214yes—
88#1862C2214yes—
89#1847C2212yes—
90#1845C2212yes—
91#1839C2212yes—
92#1831C2212yes—
93#1818C2212yes—
94#1798C2212yes—
95#1768C2212yes—
96#1749C2211yescharacteristic
97C4×C22⋊C8C2211yescharacteristic
98#1746C2212yes—
99#542—44no—
100#540—44no—
101#533—28no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#127C2311yescharacteristic
105C2×#127C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#126—21no—
108C2×#125C2312yes—
109C2×#125C2311yescharacteristic
110C2×#124—21no—
111C2×#124—22no—
112C2×#124C2311yescharacteristic
113C2×#124C2311yescharacteristic
114C2×#124C2311yescharacteristic
115C2×#123—21no—
116C2×#123—21no—
117C2×#123—21no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2311yescharacteristic
121C2×#121—22no—
122C2×#121—21no—
123C2×#121—21no—
124C2×#121C2311yescharacteristic
125C2×#121C2311yescharacteristic
126C2×#121C2311yescharacteristic
127C2×C4×Q8—22no—
128C2×C4×Q8—22no—
129C2×C4×Q8—22no—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×D4—22no—
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2312yes—
140C22×C42C2311yescharacteristic
141#490—28no—
142#488—28no—
143#484—44no—
144#481—44no—
145#480—44no—
146#480—44no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163C2×C4×C8C2312yes—
164#398—22no—
165#396—22no—
166#395—22no—
167#394—22no—
168#394—22no—
169#392—22no—
170#392—22no—
171#392—22no—
172#391—22no—
173#391—22no—
174#391C2311yescharacteristic
175#391C2311yescharacteristic
176#390—22no—
177#390—22no—
178#390—22no—
179#390C2311yescharacteristic
180#390C2311yescharacteristic
181#390C2311yescharacteristic
182#390C2311yescharacteristic
183#390C2311yescharacteristic
184#390C2311yescharacteristic
185#389—22no—
186#389—22no—
187#389—24no—
188#389—24no—
189#389C2314yes—
190#389C2314yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#387—24no—
197#387—24no—
198#386—22no—
199#386—22no—
200#386—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#384—22no—
205#384—22no—
206#384C2311yescharacteristic
207#384C2312yes—
208#384C2311yescharacteristic
209#384C2312yes—
210#382—22no—
211#382C2311yescharacteristic
212#382C2311yescharacteristic
213C4×C4⋊C4—24no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—24no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224#376C2311yescharacteristic
225#376C2311yescharacteristic
226C43C2311yescharacteristic
227C43C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237C22×Q8—22no—
238C22×Q8—21no—
239C22×Q8C2411yescharacteristic
240C22×D4—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4C2411yescharacteristic
248C22×D4C2411yescharacteristic
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C2×D412yes—
255C23×C4C2411yescharacteristic
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—28no—
259C4⋊Q8—42no—
260#127—24no—
261#127—24no—
262#127—42no—
263#126—44no—
264#125—24no—
265#125—42no—
266#125#4118yes—
267#124—24no—
268#124—24no—
269#124—42no—
270#124—44no—
271#124—84no—
272#123—44no—
273#123—44no—
274#123—44no—
275C22⋊Q8—216no—
276C22⋊Q8—24no—
277C22⋊Q8—44no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121—44no—
282#121—44no—
283#121—84no—
284C4×Q8—216no—
285C4×Q8—42no—
286C4×Q8—84no—
287C4×Q8—84no—
288C4×D4—42no—
289C4×D4—42no—
290C4×D4—42no—
291C4×D4—84no—
292#117—24no—
293#117—28no—
294#117—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2414yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2414yes—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4C2×D411yescharacteristic
341C2×C22⋊C4C2×D411yescharacteristic
342C2×C22⋊C4C2×D412yes—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—21no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—44no—
360#107—44no—
361#107—44no—
362#106—44no—
363C4⋊C8—28no—
364#103—28no—
365#103—44no—
366#102—28no—
367#102—28no—
368#102—28no—
369#102—44no—
370#102—44no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374#97—24no—
375C4×C8—24no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D412yes—
387#95C2×D412yes—
388C24—21no—
389C24—21no—
390C24—21no—
391C24C22×D411yescharacteristic
392C2×Q8—28no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—42no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4—82no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—82no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4—84no—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D412yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4C2511yescharacteristic
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—84no—
483C4⋊C4—82no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—82no—
487C4⋊C4C2×#4118yes—
488C22⋊C4—28no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—24no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C42—28no—
510C42—24no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—84no—
521C42—84no—
522C42C2×#4114yes—
523C422−1+414yes—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—22no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23#53512yes—
544C23C23×D411yescharacteristic
545Q8—416no—
546Q8—44no—
547Q8—162no—
548D4—44no—
549D4—44no—
550D4—44no—
551D4—88no—
552D4—164no—
553D4—162no—
554D4—162no—
555D4—162no—
556D4—162no—
557D4—162no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—28no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—162no—
597C2×C4—162no—
598C2×C4—162no—
599C2×C4—164no—
600C2×C4#52514yes—
601C2×C4#53512yes—
602C2×C4#53614yes—
603C2×C4#57614yes—
604C2×C4C22×#4112yes—
605C2×C4C2×H(5,𝔽2)12yes—
606C2×C4C2×2−1+412yes—
607C8—44no—
608C8—44no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22C2×#52511yescharacteristic
630C22C2×#53511yescharacteristic
631C22C2×#53611yescharacteristic
632C22C2×#57612yes—
633C22#358612yes—
634C4—24no—
635C4—24no—
636C4—22no—
637C4—24no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—48no—
643C4—42no—
644C4—84no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4#352214yes—
651C2—42no—
652C2—161no—
653C2—161no—
654C2—161no—
655C2#3524911yescharacteristic
656C2#6058412yes—
657C2#6081711yescharacteristic
658C2#6082611yescharacteristic
659C2#6088812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35276C2
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
2. #34214C2semidirect
3. #33120C2semidirect
4. #33118C2semidirect
5. #30064C2semidirect
6. #30059C2semidirect
7. #29262C2semidirect
8. #29254C2semidirect
9. #29216C2semidirect
10. #28761C2semidirect
11. #27913C2semidirect
12. #27909C2semidirect
13. #25911C2semidirect
14. #25906C2semidirect
15. #24073C2semidirect
16. #24069C2semidirect
17. #22232C2semidirect
18. #22228C2semidirect
19. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#352762
#342142
#331202
#331182
#300642
#300592
#292622
#292542
#292162
#287612
#279132
#279092
#259112
#259062
#240732
#240692
#222322
#222282
#207842