Finite Group Databasebeta

← #6502667

Subgroups — #6502667

Order 512.

2030 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2C22×#470C211yesmaximal, characteristic
3#37453C211yesmaximal, characteristic
4#34370C211yesmaximal, characteristic
5#28588C214yesmaximal, complemented
6#25229C214yesmaximal, complemented
7#25095C214yesmaximal, complemented
8#24764C214yesmaximal, complemented
9#24579C212yesmaximal, complemented
10#24529C212yesmaximal, complemented
11#24482C212yesmaximal, complemented
12#24360C212yesmaximal, complemented
13#21762C211yesmaximal, characteristic
14#20972C211yesmaximal, characteristic
15#20836C211yesmaximal, characteristic
16C23×Q16C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21#3189—28no—
22C2×#484C2212yes—
23C2×#484C2212yes—
24C2×#470—28no—
25C2×#470C2214yes—
26C2×#470C2214yes—
27C2×#467C2212yes—
28C2×#467C2212yes—
29C2×#462C2214yescomplemented
30C2×#451C2212yes—
31C2×#451C2212yes—
32C22×#106C2211yescharacteristic
33C2×#424C2212yes—
34C2×#419C2212yes—
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38#2817—22no—
39#2627—22no—
40#2627—22no—
41#2621C2214yes—
42#2503C2212yes—
43#2503C2212yes—
44#2483—21no—
45#2483—21no—
46#2434C2212yes—
47#2434C2212yes—
48#2378—22no—
49#2362C2214yes—
50#2345—22no—
51#2345C2212yes—
52Q8×C22⋊C4C2212yes—
53#2332—22no—
54#2323C2214yes—
55#2312—21no—
56#2312—21no—
57#2280C2211yescharacteristic
58#2280C2211yescharacteristic
59#2085C2214yescomplemented
60#2074C2214yescomplemented
61#2068C2214yescomplemented
62#2066C2214yescomplemented
63#2056C2214yescomplemented
64#2056C2214yescomplemented
65#2043C2214yescomplemented
66#2016C2214yescomplemented
67#1981C2214yescomplemented
68#1949C2214yescomplemented
69#1928C2214yescomplemented
70#1927C2214yescomplemented
71#1894C2214yes—
72#1892C2212yes—
73#1892C2212yes—
74#1892C2212yes—
75#1892C2212yes—
76#1878C2214yescomplemented
77#1878C2214yescomplemented
78#1873C2214yes—
79#1851C2214yes—
80#1850C2212yes—
81#1850C2212yes—
82#1823C2212yes—
83#1805C2214yescomplemented
84#1787C2212yes—
85#1736C2212yes—
86#1729C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×Q16—24no—
92C22×Q16—44no—
93C22×Q16C2314yes—
94C22×Q16C2312yes—
95C22×Q16C2312yes—
96C22×M16—24no—
97C22×M16—44no—
98C22×M16C2312yes—
99C22×M16C2312yes—
100C23×C8C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103C23×C8C2311yescharacteristic
104C2×C4⋊Q8C2314yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8D418yes—
117C2×C22⋊Q8C2312yes—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C4×Q8C2314yes—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—21no—
135#484—28no—
136#484—28no—
137#470—216no—
138#470—216no—
139#467—28no—
140#467—28no—
141#462—48no—
142#451—28no—
143#451—28no—
144C2×#107C2312yes—
145C2×#107C2312yes—
146C2×#106—24no—
147C2×#106C2314yes—
148#424—44no—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2314yes—
151#419—44no—
152C2×#103—24no—
153C2×#103—24no—
154C2×#103C2312yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×C22⋊C8—24no—
165C2×C22⋊C8C2312yes—
166C2×C22⋊C8C2312yes—
167#400—24no—
168#400—24no—
169#398—24no—
170#398—24no—
171#397—24no—
172#397—24no—
173#395—24no—
174#395—24no—
175#391—24no—
176#391C2314yes—
177#391C2314yes—
178#391C2314yes—
179#389—24no—
180#389C2314yes—
181#388—24no—
182#388—24no—
183#388—24no—
184#388—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#382C2314yes—
193#381—21no—
194#381—22no—
195#381—22no—
196#381—22no—
197#381—22no—
198#381D412yes—
199#380—22no—
200#380—22no—
201#380D412yes—
202#380D412yes—
203#380C2312yes—
204#380C2312yes—
205#379—21no—
206#379D411yescharacteristic
207#379D411yescharacteristic
208C4×C4⋊C4—24no—
209C4×C4⋊C4C2314yes—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4C2312yes—
215C4×C22⋊C4C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25C2×D411yescharacteristic
223C22×Q8—21no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8—42no—
234C22×Q8—42no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×Q8C2412yes—
240C22×Q8C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—42no—
248C23×C4—42no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D414yes—
253C23×C4#4114yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×Q16—28no—
258C2×Q16—24no—
259C2×Q16—24no—
260C2×Q16—28no—
261C2×Q16—44no—
262C2×Q16—416no—
263C2×Q16—44no—
264C2×Q16—44no—
265C2×M16—24no—
266C2×M16—24no—
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—42no—
270C2×M16—42no—
271C2×M16—416no—
272C22×C8—22no—
273C22×C8—24no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—42no—
278C22×C8—42no—
279C22×C8—42no—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C4⋊Q8—44no—
285C4⋊Q8—44no—
286C22⋊Q8—28no—
287C22⋊Q8—28no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8C2×D418yes—
303C22⋊Q8C2×D418yes—
304C4×Q8—44no—
305C4×Q8—44no—
306#117—42no—
307#117—42no—
308#117—42no—
309#117—42no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4C2×D414yes—
332C2×C4⋊C4C2×D414yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42C2414yes—
359#107—24no—
360#107—24no—
361#106—28no—
362C4⋊C8—28no—
363C4⋊C8—44no—
364#103—28no—
365#103—24no—
366#103—24no—
367#103—28no—
368#103—28no—
369#103—24no—
370#103—24no—
371#103—44no—
372#103—44no—
373#103—44no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95C2×D414yes—
389#95C2×D414yes—
390C24—24no—
391C24—42no—
392C24—42no—
393C24#12114yes—
394C24C22×D411yescharacteristic
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—42no—
415C2×Q8—42no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×Q8C22×D414yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4#12114yes—
456C22×C4#12114yes—
457C22×C4#12114yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2511yescharacteristic
470Q16—416no—
471Q16—84no—
472Q16—84no—
473M16—48no—
474M16—48no—
475C2×C8—28no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—24no—
485C2×C8—44no—
486C2×C8—82no—
487C2×C8—82no—
488C2×C8H(5,𝔽2)14yes—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4C22×D414yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C22⋊C4—82no—
526C42—24no—
527C42—24no—
528C42—84no—
529C42—82no—
530C42—84no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—44no—
541C23—44no—
542C23—88no—
543C23C2×#12112yes—
544C23C2×#12112yes—
545C23C2×#12112yes—
546C23#57814yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—24no—
550Q8—44no—
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—44no—
556Q8—44no—
557Q8—84no—
558Q8—84no—
559Q8—84no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4C2×#12112yes—
603C2×C4C2×#12112yes—
604C2×C4C2×#12112yes—
605C2×C4C23×D411yescharacteristic
606C2×C4C22×#4111yescharacteristic
607C2×C4C2×H(5,𝔽2)12yes—
608C2×C4#58512yes—
609C8—24no—
610C8—42no—
611C8—42no—
612C8—42no—
613C22—22no—
614C22—22no—
615C22—28no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—44no—
623C22—44no—
624C22—88no—
625C22#304414yes—
626C22C22×#12111yescharacteristic
627C22#349612yes—
628C22C2×#57812yes—
629C22#358612yes—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—84no—
642C4#348411yescharacteristic
643C4#348511yescharacteristic
644C4#350512yes—
645C2—24no—
646C2—42no—
647C2—42no—
648C2#3535811yescharacteristic
649C2C2×#304411yescharacteristic
650C2#6042711yescharacteristic
651C2#6231714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
5. #28588C2semidirect
6. #25229C2semidirect
7. #25095C2semidirect
8. #24764C2semidirect
9. #24579C2semidirect
10. #24529C2semidirect
11. #24482C2semidirect
12. #24360C2semidirect
29. C2×#462C22semidirect
59. #2085C22semidirect
60. #2074C22semidirect
61. #2068C22semidirect
62. #2066C22semidirect
63. #2056C22semidirect
64. #2056C22semidirect
65. #2043C22semidirect
66. #2016C22semidirect
67. #1981C22semidirect
68. #1949C22semidirect
69. #1928C22semidirect
70. #1927C22semidirect
76. #1878C22semidirect
77. #1878C22semidirect
83. #1805C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
C22×#4702
#374532
#343702
#285882
#252292
#250952
#247642
#245792
#245292
#244822
#243602
#217622
#209722
#208362