Finite Group Databasebeta

← #6519056

Subgroups — #6519056

Order 512.

1898 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35165C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic, complemented
3#27606C214yesmaximal, complemented
4#25022C214yesmaximal, complemented
5#25012C212yesmaximal, complemented
6#25010C212yesmaximal, complemented
7#24978C214yesmaximal, complemented
8#24968C212yesmaximal, complemented
9#24966C212yesmaximal, complemented
10#24013C214yesmaximal, complemented
11#21988C211yesmaximal, characteristic, complemented
12#21984C211yesmaximal, characteristic, complemented
13#21789C211yesmaximal, characteristic, complemented
14#20738C211yesmaximal, characteristic, complemented
15#20732C211yesmaximal, characteristic, complemented
16C2×#535C2212yes—
17C2×#532C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#481C2214yes—
20C2×#480C2214yes—
21C2×#432C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2847—22no—
24#2617—22no—
25#2547—21no—
26#2544—22no—
27#2541—21no—
28#2537C2212yes—
29#2537C2212yes—
30#2535C2214yes—
31#2379—22no—
32#2369—21no—
33#2366—21no—
34#2354C2212yescomplemented
35#2354C2212yescomplemented
36D4×C4⋊C4C2212yescomplemented
37D4×C4⋊C4C2212yescomplemented
38#2324—22no—
39#2324—22no—
40#2311C2214yes—
41#2311C2212yes—
42#2285C2211yescharacteristic
43#2285C2211yescharacteristic
44#2285C2211yescharacteristic
45#2077C2214yescomplemented
46#2077C2214yescomplemented
47#2076C2214yescomplemented
48#2075C2214yescomplemented
49#2034C2214yes—
50#2034C2214yes—
51#2033C2214yes—
52#2033C2214yes—
53#1955C2214yescomplemented
54#1954C2212yescomplemented
55#1954C2212yescomplemented
56#1949C2214yescomplemented
57#1948C2212yescomplemented
58#1947C2212yescomplemented
59#1875C2214yes—
60#1875C2214yes—
61#1854C2214yes—
62#1854C2214yes—
63#1853C2214yescomplemented
64#1852C2214yescomplemented
65#1847C2212yescomplemented
66#1845C2211yescharacteristic, complemented
67#1844C2211yescharacteristic, complemented
68#1836C2212yescomplemented
69#1834C2211yescharacteristic, complemented
70#1834C2211yescharacteristic, complemented
71#1825C2214yescomplemented
72#1825C2214yescomplemented
73#1824C2212yes—
74#1823C2212yes—
75#1802C2212yes—
76#1802C2212yes—
77#1802C2212yes—
78#1802C2212yes—
79#1786C2212yes—
80#1786C2212yes—
81#1780C2214yes—
82#1762C2214yes—
83C4×#102C2212yes—
84C4×#102C2212yes—
85C22×#41C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4—21no—
88#535—48no—
89#532—24no—
90#532—24no—
91C2×C4⋊Q8D411yescharacteristic
92C2×#127—22no—
93C2×#127D411yescharacteristic
94C2×#124—22no—
95C2×#124D412yes—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121D412yes—
99C2×#121D412yes—
100C2×#121C2314yes—
101C2×#121C2314yes—
102C2×#120—24no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—21no—
114C2×#117—21no—
115C2×#117C2314yes—
116C2×#117C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4—21no—
120C22×C42C2311yescharacteristic
121#481—48no—
122#480—48no—
123#432—24no—
124#431—24no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2312yes—
132C2×#102C2314yes—
133C2×#102C2312yes—
134C2×#102C2312yes—
135C2×#102C2312yes—
136C2×#102C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2314yes—
143C2×#97C2312yes—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146#397—24no—
147#394—24no—
148#394—24no—
149#392—24no—
150#391—24no—
151#391—22no—
152#390—24no—
153#390—22no—
154#390—22no—
155#390—22no—
156#389—22no—
157#389—22no—
158#388—24no—
159#388—24no—
160#386—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#386—24no—
165#385—24no—
166#385—24no—
167#384C2311yescharacteristic
168#384C2311yescharacteristic
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#384C2311yescharacteristic
172#384C2311yescharacteristic
173#382—21no—
174#382—21no—
175#382C2312yes—
176#380—21no—
177#380—21no—
178#380—21no—
179#380—21no—
180#380—22no—
181#380—22no—
182#380C2312yes—
183#380C2312yes—
184#380C2314yes—
185#380C2314yes—
186#380C2314yes—
187#380C2314yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196#340C2312yes—
197#340C2314yes—
198#340C2312yes—
199#340C2314yes—
200#340C2312yes—
201#340C2312yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206C25—21no—
207C2×#41—24no—
208C2×#41—24no—
209C2×#41—44no—
210C2×#41—44no—
211C22×Q8C2×D412yes—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—42no—
219C22×D4C2×D412yes—
220C22×D4C2×D412yes—
221C22×D4C2×D414yes—
222C22×D4C2×D412yes—
223C22×D4C2412yes—
224C23×C4—21no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—44no—
229C23×C4—82no—
230C23×C4—82no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2414yes—
238C4⋊Q8—24no—
239#127—24no—
240#127—48no—
241#124—28no—
242#124—48no—
243#121—28no—
244#121—28no—
245#121—28no—
246#121—28no—
247#121—44no—
248#121—44no—
249#121—84no—
250#121—84no—
251#120—416no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—24no—
260C4×D4—44no—
261C4×D4—44no—
262C4×D4—84no—
263C4×D4—84no—
264#117—44no—
265#117—44no—
266#117—44no—
267#117—82no—
268#117—82no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42#4114yes—
318C2×C42#4114yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321#107—44no—
322#107—44no—
323#106—44no—
324#106—44no—
325C4⋊C8—24no—
326C4⋊C8—24no—
327#102—24no—
328#102—28no—
329#102—24no—
330#102—42no—
331#102—42no—
332#102—44no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335C22⋊C8—44no—
336C22⋊C8—44no—
337C22⋊C8—44no—
338C22⋊C8—44no—
339#97—24no—
340C4×C8—22no—
341C4×C8—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352C24—24no—
353C24—22no—
354C24—42no—
355C24—42no—
356C24—81no—
357C24—81no—
358C24—81no—
359C24—81no—
360C24C22×D414yes—
361C24C22×D411yescharacteristic
362#41—88no—
363#41—88no—
364C2×Q8—24no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×D4—24no—
368C2×D4—28no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—44no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—42no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4H(5,𝔽2)14yes—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—82no—
421C22×C4—82no—
422C22×C4—816no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4#12314yes—
429C22×C4#12314yes—
430C22×C4C22×D414yes—
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2511yescharacteristic
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—24no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—84no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—84no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C42—44no—
489C42—44no—
490C42—44no—
491C42—84no—
492C42#12314yes—
493C42#12314yes—
494C42C2×#4112yes—
495C42C2×#4112yes—
496C42C2×#4112yes—
497C42C2×#4112yes—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—48no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23C2×#12312yes—
526C23C23×D411yescharacteristic
527Q8—84no—
528D4—24no—
529D4—24no—
530D4—48no—
531D4—48no—
532D4—48no—
533D4—42no—
534D4—42no—
535D4—84no—
536D4—88no—
537D4—168no—
538D4—168no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—88no—
580C2×C4—84no—
581C2×C4—88no—
582C2×C4—168no—
583C2×C4C2×#12312yes—
584C2×C4C2×#12312yes—
585C2×C4C2×#12312yes—
586C2×C4C2×#12312yes—
587C2×C4C2×#12312yes—
588C2×C4#57612yes—
589C2×C4#57712yes—
590C2×C4C22×#4111yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4#58512yes—
594C8—42no—
595C8—42no—
596C8—44no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—816no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C22×#12311yescharacteristic
620C22#348411yescharacteristic
621C22#348511yescharacteristic
622C22C2×#57611yescharacteristic
623C22C2×#57711yescharacteristic
624C22#358511yescharacteristic
625C22#358611yescharacteristic
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4#309614yes—
639C4#349412yes—
640C4#349412yes—
641C2—42no—
642C2—84no—
643C2—84no—
644C2#3554011yescharacteristic
645C2C2×#309611yescharacteristic
646C2#6051711yescharacteristic
647C2#6167911yescharacteristic
648C2#6168011yescharacteristic
649C2#6180711yescharacteristic
650C2#6180811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35165C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34368C2semidirect
3. #27606C2semidirect
4. #25022C2semidirect
5. #25012C2semidirect
6. #25010C2semidirect
7. #24978C2semidirect
8. #24968C2semidirect
9. #24966C2semidirect
10. #24013C2semidirect
11. #21988C2semidirect
12. #21984C2semidirect
13. #21789C2semidirect
14. #20738C2semidirect
15. #20732C2semidirect
34. #2354C22semidirect
35. #2354C22semidirect
36. D4×C4⋊C4C22semidirect
37. D4×C4⋊C4C22semidirect
45. #2077C22semidirect
46. #2077C22semidirect
47. #2076C22semidirect
48. #2075C22semidirect
53. #1955C22semidirect
54. #1954C22semidirect
55. #1954C22semidirect
56. #1949C22semidirect
57. #1948C22semidirect
58. #1947C22semidirect
63. #1853C22semidirect
64. #1852C22semidirect
65. #1847C22semidirect
66. #1845C22semidirect
67. #1844C22semidirect
68. #1836C22semidirect
69. #1834C22semidirect
70. #1834C22semidirect
71. #1825C22semidirect
72. #1825C22semidirect

Maximal subgroups

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

subgroupindex
#351652
#343682
#276062
#250222
#250122
#250102
#249782
#249682
#249662
#240132
#219882
#219842
#217892
#207382
#207322