Finite Group Databasebeta

← #412405

Subgroups — #412405

Order 512.

1910 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#469C211yesmaximal, characteristic
2C2×#2034C212yesmaximal, complemented
3C2×#1972C211yesmaximal, characteristic, complemented
4C2×#1970C211yesmaximal, characteristic, complemented
5C2×#1844C211yesmaximal, characteristic, complemented
6C2×#1809C211yesmaximal, characteristic, complemented
7C2×#1753C211yesmaximal, characteristic, complemented
8#15130C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#11049C212yesmaximal, complemented
11#10863C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#121C2212yes—
15C2×#469—24no—
16C2×#469—24no—
17C2×#469—24no—
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2212yes—
21C22×#106C2211yescharacteristic
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#394—24no—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#384C2212yes—
32C2×#380C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34#2034—28no—
35#1972—24no—
36#1970—24no—
37#1844—24no—
38#1809—24no—
39#1753—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44#1439—24no—
45C4⋊#95C2212yes—
46C23×D4—22no—
47C23×D4C2312yes—
48C24×C4C2311yescharacteristic
49C22×D8—22no—
50C22×D8—22no—
51C22×D8—22no—
52C22×D8—22no—
53C22×D8—22no—
54C22×D8—22no—
55C22×D8D414yes—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16—22no—
61C22×M16—22no—
62C22×M16D414yes—
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×#121—24no—
68C2×#121—24no—
69C2×#121—24no—
70C2×#121—24no—
71C2×#121—42no—
72C2×#121—42no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—24no—
81C22×C22⋊C4D412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#469—232no—
85C2×#107—22no—
86C2×#107—22no—
87C2×#107—22no—
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—24no—
94C2×#106—24no—
95C2×#106—42no—
96C2×#106D414yes—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—42no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8D414yes—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8D414yescomplemented
115#394—416no—
116#390—24no—
117#390—24no—
118#390—24no—
119#390—42no—
120#384—24no—
121#384—42no—
122#380—24no—
123#380—24no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—24no—
127C2×#95—22no—
128C2×#95D412yes—
129#340—24no—
130#340—24no—
131#336—22no—
132#336—22no—
133C25—22no—
134C25C2×D411yescharacteristic
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—48no—
152C22×D4—84no—
153C22×D4C2×D414yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2411yescharacteristic
175C2×D8—28no—
176C2×D8—28no—
177C2×D8—28no—
178C2×D8—44no—
179C2×D8—44no—
180C2×D8—44no—
181C2×D8—44no—
182C2×M16—28no—
183C2×M16—28no—
184C2×M16—28no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8#4112yes—
214C22×C8#4112yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217#121—216no—
218#121—48no—
219#121—48no—
220#121—48no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—48no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—84no—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4—84no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C2×D412yes—
266C2×C42#4112yes—
267#107—44no—
268#106—28no—
269#106—48no—
270#106—44no—
271#102—216no—
272#102—48no—
273#102—48no—
274C22⋊C8—28no—
275#97—44no—
276C4×C8—28no—
277#95—24no—
278#95—48no—
279#95—44no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—44no—
296C24—44no—
297C24—44no—
298C24—44no—
299C24#12712yes—
300C24C22×D411yescharacteristic
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—416no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—816no—
331C2×D4—88no—
332C2×D4—88no—
333C2×D4—816no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4#12114yes—
392C22×C4#12712yes—
393C22×C4AGL(1,ℤ/8)18yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C42−1+411yescharacteristic
407C22×C42−1+411yescharacteristic
408D8—416no—
409M16—416no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8#12014yes—
431C2×C8#12114yes—
432C2×C8#12114yes—
433C2×C8#12114yes—
434C2×C8#12114yes—
435C2×C8#12314yes—
436C2×C8#12314yes—
437C2×C8#12414yes—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—816no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—816no—
454C22⋊C4—88no—
455C22⋊C4—816no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—44no—
470C23—44no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—44no—
477C23—44no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—48no—
484C23—48no—
485C23—44no—
486C23—84no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—84no—
496C23—82no—
497C23—84no—
498C23—82no—
499C23—84no—
500C23—84no—
501C23—82no—
502C23—84no—
503C23—84no—
504C23C2×#12111yescharacteristic
505C23C2×#12711yescharacteristic
506C23#53211yescharacteristic
507C23D4211yescharacteristic
508C23D4211yescharacteristic
509C23#54711yescharacteristic
510C23#54711yescharacteristic
511C23C2×AGL(1,ℤ/8)14yes—
512C23#57714yes—
513D4—44no—
514D4—44no—
515D4—44no—
516D4—44no—
517D4—832no—
518D4—82no—
519D4—82no—
520D4—82no—
521D4—82no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—84no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—84no—
557C2×C4—82no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4#46918yescomplemented
562C2×C4C2×#12011yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12311yescharacteristic
568C2×C4C2×#12311yescharacteristic
569C2×C4C2×#12411yescharacteristic
570C2×C4#53211yescharacteristic
571C2×C4#53211yescharacteristic
572C2×C4#53511yescharacteristic
573C2×C4#53611yescharacteristic
574C2×C4#53711yescharacteristic
575C2×C4#53711yescharacteristic
576C2×C4#54211yescharacteristic
577C2×C4#54211yescharacteristic
578C2×C4D4211yescharacteristic
579C2×C4D4211yescharacteristic
580C2×C4#54711yescharacteristic
581C2×C4#54711yescharacteristic
582C2×C4#54711yescharacteristic
583C2×C4#54711yescharacteristic
584C2×C4#54811yescharacteristic
585C2×C4#54811yescharacteristic
586C2×C4#55311yescharacteristic
587C2×C4#55311yescharacteristic
588C2×C4#55511yescharacteristic
589C2×C4#55511yescharacteristic
590C8—24no—
591C8—42no—
592C8—42no—
593C8#39418yescomplemented
594C22—42no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—84no—
604C22—88no—
605C22—84no—
606C22—84no—
607C22—88no—
608C22—162no—
609C22—162no—
610C22—162no—
611C22—164no—
612C22—162no—
613C22—164no—
614C22#241611yescharacteristic
615C22#242211yescharacteristic
616C22#243311yescharacteristic
617C22#249111yescharacteristic
618C22#250111yescharacteristic
619C22#253711yescharacteristic
620C22#285611yescharacteristic
621C22C2×#46912yes—
622C22#306012yes—
623C22#315112yes—
624C22#315412yes—
625C22#317312yes—
626C22#318412yes—
627C22#326514yes—
628C22#327214yes—
629C22#334514yes—
630C22#335414yes—
631C4—28no—
632C4—42no—
633C4—82no—
634C4—84no—
635C4—162no—
636C4—162no—
637C4C2×#39411yescharacteristic
638C4#241611yescharacteristic
639C4#242211yescharacteristic
640C4#243211yescharacteristic
641C4#249111yescharacteristic
642C4#250011yescharacteristic
643C4#253711yescharacteristic
644C4#262011yescharacteristic
645C2—82no—
646C2—164no—
647C2#1513011yescharacteristic
648C2#2478212yes—
649C2#2504612yes—
650C2#2579112yes—
651C2#2780412yes—
652C2#2794112yes—
653C2#2823112yes—
654C2#3317511yescharacteristic
655C2#3318111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#469C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2034C2semidirect
3. C2×#1972C2semidirect
4. C2×#1970C2semidirect
5. C2×#1844C2semidirect
6. C2×#1809C2semidirect
7. C2×#1753C2semidirect
8. #15130C2semidirect
9. #11121C2semidirect
10. #11049C2semidirect
11. #10863C2semidirect
114. C2×C4×C8D4semidirect
561. C2×C4#469semidirect
593. C8#394semidirect

Maximal subgroups

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

subgroupindex
C22×#4692
C2×#20342
C2×#19722
C2×#19702
C2×#18442
C2×#18092
C2×#17532
#151302
#111212
#110492
#108632