Finite Group Databasebeta

← #6508748

Subgroups — #6508748

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2C22×#485C211yesmaximal, characteristic, complemented
3#35262C211yesmaximal, characteristic
4#34370C211yesmaximal, characteristic, complemented
5#28130C214yesmaximal, complemented
6#27657C214yesmaximal, complemented
7#25213C214yesmaximal, complemented
8#25203C214yesmaximal, complemented
9#24583C212yesmaximal, complemented
10#24578C212yesmaximal, complemented
11#24533C212yesmaximal, complemented
12#24528C212yesmaximal, complemented
13#21931C211yesmaximal, characteristic, complemented
14#21761C211yesmaximal, characteristic
15#20699C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#3187—28no—
20C2×#481C2212yes—
21C2×#485—28no—
22C2×#485C2214yes—
23C2×#485C2214yescomplemented
24C2×#483C2212yes—
25C2×#465C2212yes—
26C2×#465C2212yes—
27C2×#448C2212yes—
28C2×#450C2212yes—
29C22×#107C2211yescharacteristic, complemented
30C22×#106C2211yescharacteristic, complemented
31C22×#103C2211yescharacteristic, complemented
32C22×#103C2211yescharacteristic, complemented
33C22×C22⋊C8C2211yescharacteristic
34C22×C22⋊C8C2211yescharacteristic
35#2854—22no—
36#2624—22no—
37#2504C2212yes—
38#2501C2212yes—
39#2500C2214yes—
40#2482—22no—
41#2405—21no—
42#2405—21no—
43#2378—22no—
44#2362C2214yescomplemented
45#2345—22no—
46#2345C2212yescomplemented
47Q8×C22⋊C4C2212yescomplemented
48#2332—22no—
49#2323C2214yes—
50#2317C2212yes—
51#2312—21no—
52#2312—21no—
53#2309C2212yes—
54#2280C2211yescharacteristic
55#2280C2211yescharacteristic
56#2085C2214yescomplemented
57#2084C2214yescomplemented
58#2074C2214yescomplemented
59#2072C2214yescomplemented
60#2067C2214yescomplemented
61#2065C2214yescomplemented
62#2055C2214yescomplemented
63#2055C2214yescomplemented
64#1949C2214yescomplemented
65#1948C2214yescomplemented
66#1926C2214yescomplemented
67#1925C2214yescomplemented
68#1893C2212yes—
69#1893C2212yes—
70#1880C2214yescomplemented
71#1880C2214yescomplemented
72#1875C2214yes—
73#1850C2212yescomplemented
74#1850C2212yescomplemented
75#1850C2212yescomplemented
76#1850C2212yescomplemented
77#1825C2214yes—
78#1824C2212yes—
79#1824C2212yescomplemented
80#1823C2212yescomplemented
81#1805C2214yescomplemented
82#1803C2214yescomplemented
83#1798C2214yes—
84#1786C2212yes—
85#1735C2212yescomplemented
86#1735C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×SD16C2312yes—
92C22×SD16C2312yes—
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C2×#123—22no—
98C2×#123—22no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8D418yes—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×C4×Q8C2314yes—
112C2×#117—21no—
113C2×#117—21no—
114C2×#117—24no—
115C2×#117—24no—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129#485—216no—
130#485—216no—
131#483—28no—
132#481—28no—
133#465—28no—
134#465—28no—
135#450—28no—
136#448—28no—
137C2×#107—24no—
138C2×#107C2314yes—
139C2×#106—24no—
140C2×#106C2312yes—
141C2×#106C2312yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2312yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161#399—24no—
162#397—24no—
163#394—24no—
164#394—24no—
165#391—24no—
166#391C2314yes—
167#390—24no—
168#390C2314yes—
169#390C2314yes—
170#389—24no—
171#389—24no—
172#389C2314yes—
173#388—24no—
174#388—24no—
175#388—24no—
176#388—24no—
177#388—24no—
178#388—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#386—22no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384C2314yes—
187#382C2314yes—
188#381—21no—
189#381D412yes—
190#380—22no—
191#380—22no—
192#380D412yes—
193#380D412yes—
194#380C2312yes—
195#380C2312yes—
196#379—21no—
197#379—22no—
198#379—22no—
199#379D411yescharacteristic
200#379D411yescharacteristic
201C4×C4⋊C4—24no—
202C4×C4⋊C4C2314yes—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4C2312yes—
208C4×C22⋊C4C2312yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#336C2314yes—
216#336C2314yes—
217C25—21no—
218C25C2×D411yescharacteristic
219C22×Q8—21no—
220C22×Q8—22no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8C2×D414yes—
224C22×Q8C2412yes—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—22no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4—44no—
233C22×D4—84no—
234C22×D4C2412yes—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D414yes—
246C23×C4#4114yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—44no—
253C2×SD16—44no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—42no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8—42no—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266#123—84no—
267#123—84no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270C22⋊Q8—24no—
271C22⋊Q8—24no—
272C22⋊Q8C2×D418yes—
273C22⋊Q8C2×D418yes—
274#121—24no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—44no—
279#121—44no—
280C4×Q8—44no—
281C4×Q8—44no—
282#117—24no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—42no—
287#117—42no—
288#117—44no—
289#117—44no—
290#117—42no—
291#117—42no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
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—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4C2×D412yes—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42C2414yes—
346#107—28no—
347#106—28no—
348#106—24no—
349#106—24no—
350C4⋊C8—44no—
351C4⋊C8—44no—
352#103—28no—
353#103—28no—
354#103—24no—
355#103—24no—
356#102—24no—
357#102—24no—
358#102—44no—
359#102—44no—
360C22⋊C8—24no—
361C22⋊C8—24no—
362C22⋊C8—24no—
363C22⋊C8—24no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95C2×D414yes—
373#95C2×D414yes—
374C24—24no—
375C24—22no—
376C24—42no—
377C24—42no—
378C24—44no—
379C24—42no—
380C24—82no—
381C24—82no—
382C24#12114yes—
383C24C22×D411yescharacteristic
384C24C22×D411yescharacteristic
385C24C22×D411yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—82no—
393C2×Q8—84no—
394C2×Q8C22×D414yes—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—82no—
406C2×D4—82no—
407C2×D4—84no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—816no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—48no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—44no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4#12114yes—
440C22×C4#12114yes—
441C22×C4#12114yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2511yescharacteristic
454SD16—84no—
455SD16—84no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—44no—
465C2×C8—44no—
466C2×C8—82no—
467C2×C8—82no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4C22×D414yes—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—24no—
509C42—24no—
510C42—44no—
511C42—44no—
512C42—84no—
513C42—82no—
514C42—84no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—88no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—168no—
536C23C2×#12112yes—
537C23C2×#12112yes—
538C23C2×#12112yes—
539C23#57714yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—44no—
544Q8—84no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—84no—
549D4—88no—
550D4—88no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4C2×#12112yes—
592C2×C4C2×#12112yes—
593C2×C4C2×#12112yes—
594C2×C4C23×D411yescharacteristic
595C2×C4C22×#4111yescharacteristic
596C2×C4#58512yes—
597C2×C4#58512yes—
598C8—42no—
599C8—42no—
600C8—42no—
601C8—42no—
602C22—22no—
603C22—28no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—44no—
612C22—44no—
613C22—88no—
614C22—82no—
615C22—82no—
616C22—84no—
617C22—84no—
618C22—82no—
619C22—168no—
620C22#304314yes—
621C22C22×#12111yescharacteristic
622C22#348512yes—
623C22C2×#57712yes—
624C22#358412yes—
625C4—22no—
626C4—22no—
627C4—22no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—44no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—84no—
636C4#348511yescharacteristic
637C4#348511yescharacteristic
638C4#349112yes—
639C2—24no—
640C2—42no—
641C2—42no—
642C2—162no—
643C2#3544711yescharacteristic
644C2C2×#304311yescharacteristic
645C2#6042211yescharacteristic
646C2#6177314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#485C2semidirect
4. #34370C2semidirect
5. #28130C2semidirect
6. #27657C2semidirect
7. #25213C2semidirect
8. #25203C2semidirect
9. #24583C2semidirect
10. #24578C2semidirect
11. #24533C2semidirect
12. #24528C2semidirect
13. #21931C2semidirect
15. #20699C2semidirect
23. C2×#485C22semidirect
29. C22×#107C22semidirect
30. C22×#106C22semidirect
31. C22×#103C22semidirect
32. C22×#103C22semidirect
44. #2362C22semidirect
46. #2345C22semidirect
47. Q8×C22⋊C4C22semidirect
56. #2085C22semidirect
57. #2084C22semidirect
58. #2074C22semidirect
59. #2072C22semidirect
60. #2067C22semidirect
61. #2065C22semidirect
62. #2055C22semidirect
63. #2055C22semidirect
64. #1949C22semidirect
65. #1948C22semidirect
66. #1926C22semidirect
67. #1925C22semidirect
70. #1880C22semidirect
71. #1880C22semidirect
73. #1850C22semidirect
74. #1850C22semidirect
75. #1850C22semidirect
76. #1850C22semidirect
79. #1824C22semidirect
80. #1823C22semidirect
81. #1805C22semidirect
82. #1803C22semidirect
85. #1735C22semidirect
86. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
C22×#4852
#352622
#343702
#281302
#276572
#252132
#252032
#245832
#245782
#245332
#245282
#219312
#217612
#206992