Finite Group Databasebeta

← #6470684

Subgroups — #6470684

Order 512.

2938 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
1C2×#3186C212yesmaximal, complemented
2#35001C211yesmaximal, characteristic, complemented
3C2×#2361C211yesmaximal, characteristic, complemented
4#33106C212yesmaximal, complemented
5#32902C212yesmaximal, complemented
6#29754C212yesmaximal, complemented
7#29712C212yesmaximal, complemented
8#25200C218yesmaximal, complemented
9#25076C218yesmaximal, complemented
10#21930C211yesmaximal, characteristic, complemented
11C2×#1874C212yesmaximal, complemented
12C2×#534—21no—
13C2×#534—21no—
14C22×#121C2212yes—
15C22×#121C2211yescharacteristic
16C22×C4×D4—21no—
17C22×C4×D4C2211yescharacteristic, complemented
18C23×C4⋊C4C2211yescharacteristic
19#3186—216no—
20C2×#480C2214yescomplemented
21C2×#480C2214yescomplemented
22C2×#466C2218yes—
23C2×#459C2214yescomplemented
24C2×#447C2214yescomplemented
25C2×#447C2214yescomplemented
26C22×#102C2214yescomplemented
27C22×C22⋊C8C2212yes—
28#2856C2212yes—
29#2856C2212yes—
30#2656—21no—
31#2656—21no—
32#2539—24no—
33#2501C2214yescomplemented
34#2433C2214yescomplemented
35#2361—24no—
36#2361C2218yescomplemented
37C2×#390C2211yescharacteristic, complemented
38C2×#389C2211yescharacteristic, complemented
39C2×#388—21no—
40C2×#386—21no—
41C2×#385—21no—
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43C2×C4×C22⋊C4C2211yescharacteristic
44#2083C2214yescomplemented
45#2071C2214yescomplemented
46#2067C2218yescomplemented
47#2054C2218yescomplemented
48#2041C2214yescomplemented
49#2011C2214yescomplemented
50#1982C2214yescomplemented
51#1947C2214yescomplemented
52#1925C2218yescomplemented
53#1879C2218yescomplemented
54#1874C2214yescomplemented
55#1874C2214yescomplemented
56#1874C2218yescomplemented
57#1850C2218yescomplemented
58#1802C2214yescomplemented
59#1797C2212yescomplemented
60#1797C2212yescomplemented
61C2×#340C2214yescomplemented
62C23×D4—21no—
63C23×D4—22no—
64C23×D4C2312yes—
65C23×D4C2311yescharacteristic
66C24×C4—21no—
67C24×C4—21no—
68C24×C4C2311yescharacteristic
69C22×D8C2318yes—
70C23×C8C2314yes—
71#534—48no—
72#534—48no—
73C2×#127C2312yes—
74C2×#127C2312yes—
75C2×#123—24no—
76C2×#123—24no—
77C2×#121—22no—
78C2×#121—24no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—22no—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121D418yescomplemented
88C2×#121C2314yes—
89C2×#121C2314yes—
90C2×#121C2314yes—
91C2×#120—24no—
92C2×#120—24no—
93C2×C4×D4—28no—
94C2×C4×D4—22no—
95C2×C4×D4—44no—
96C2×C4×D4—48no—
97C2×C4×D4C2314yes—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—22no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4D411yescharacteristic
113C22×C22⋊C4D411yescharacteristic
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#480—216no—
118#480—216no—
119#466—232no—
120#459—48no—
121#447—216no—
122#447—216no—
123C2×#107C2318yes—
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#102—48no—
127C2×#102C2318yes—
128C2×#102C2318yes—
129C2×#102C2318yes—
130C2×#102C2318yes—
131C2×C22⋊C8—28no—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134#399—24no—
135#399—24no—
136#392—24no—
137#392—24no—
138#390—22no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390C2314yes—
144#390C2314yes—
145#390C2314yes—
146#389—22no—
147#389C2314yes—
148#388—28no—
149#386—28no—
150#385—28no—
151#384C2312yes—
152#384C2312yes—
153C4×C4⋊C4—22no—
154C4×C4⋊C4C2314yes—
155C4×C22⋊C4D414yes—
156C4×C22⋊C4C2314yes—
157C2×#95—21no—
158C2×#95—21no—
159C2×#95D411yescharacteristic
160C2×#95D411yescharacteristic
161#340C2318yes—
162#340C2318yes—
163#336C2314yes—
164#336C2314yes—
165C25—21no—
166C25—22no—
167C25C2×D411yescharacteristic
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—48no—
188C22×D4—48no—
189C22×D4—48no—
190C22×D4C2×D414yes—
191C22×D4C2414yes—
192C22×D4C2412yes—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—82no—
206C23×C4—82no—
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D412yes—
211C23×C4C2×D414yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4#4114yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C2×D8—216no—
218C2×D8—48no—
219C2×D8—48no—
220C22×C8—28no—
221C22×C8—44no—
222C22×C8—44no—
223C22×C8C2418yes—
224#127—44no—
225#127—44no—
226#123—416no—
227#123—416no—
228#121—216no—
229#121—28no—
230#121—28no—
231#121—28no—
232#121—28no—
233#121—28no—
234#121—48no—
235#121—48no—
236#121—48no—
237#121—48no—
238#121—48no—
239#121—48no—
240#121C2×D4116yes—
241#120—416no—
242#120—416no—
243C4×D4—28no—
244C4×D4—44no—
245C4×D4—48no—
246C4×D4—432no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—82no—
288C2×C22⋊C4D818yescomplemented
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D414yes—
291C2×C42—24no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2412yes—
298C2×C42C2412yes—
299#107—216no—
300C4⋊C8—44no—
301C4⋊C8—44no—
302#102—216no—
303#102—216no—
304#102—216no—
305#102—48no—
306C22⋊C8—28no—
307C22⋊C8—28no—
308#95—24no—
309#95—24no—
310#95C2×D414yes—
311#95C2×D414yes—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—41no—
316C24—41no—
317C24—41no—
318C24—41no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—44no—
326C24—44no—
327C24—44no—
328C24—82no—
329C24—82no—
330C24—82no—
331C24—82no—
332C24#12114yes—
333C24C2×D814yes—
334C24AGL(1,ℤ/8)14yes—
335C24C22×D412yes—
336C24C22×D411yescharacteristic
337C2×D4—216no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—28no—
346C2×D4—48no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—432no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—88no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4—82no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—832no—
377C2×D4C22×D414yes—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—44no—
413C22×C4—42no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—816no—
421C22×C4—816no—
422C22×C4#12114yes—
423C22×C4#12118yes—
424C22×C4C2×D814yes—
425C22×C4C2×D814yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4112yes—
436C22×C4C2511yescharacteristic
437D8—816no—
438C2×C8—28no—
439C2×C8—28no—
440C2×C8—28no—
441C2×C8—88no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—48no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4C22×D414yes—
458C4⋊C4H(5,𝔽2)14yes—
459C22⋊C4—28no—
460C22⋊C4—216no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—84no—
470C22⋊C4C2×D8116yes—
471C42—24no—
472C42—24no—
473C42—42no—
474C42—42no—
475C42—48no—
476C23—21no—
477C23—22no—
478C23—21no—
479C23—22no—
480C23—21no—
481C23—22no—
482C23—21no—
483C23—22no—
484C23—24no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—48no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—48no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—84no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—88no—
512C23—88no—
513C23—88no—
514C23—164no—
515C23—164no—
516C23—164no—
517C23—164no—
518C23#459116yescomplemented
519C23C2×#12112yes—
520C23C2×#12114yes—
521C23C22×D814yes—
522C23C22×D812yes—
523C23C2×AGL(1,ℤ/8)14yes—
524C23C2×AGL(1,ℤ/8)12yes—
525C23C23×D411yescharacteristic
526D4—28no—
527D4—48no—
528D4—432no—
529D4—44no—
530D4—48no—
531D4—48no—
532D4—88no—
533D4—816no—
534D4—816no—
535D4—164no—
536D4—164no—
537D4—164no—
538D4—164no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—816no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—816no—
579C2×C4—84no—
580C2×C4—88no—
581C2×C4C2×#12114yes—
582C2×C4C2×#12112yes—
583C2×C4C22×D814yes—
584C2×C4C22×D814yes—
585C2×C4C23×D411yescharacteristic
586C2×C4C22×#4111yescharacteristic
587C2×C4C2×H(5,𝔽2)12yes—
588C2×C4#58512yes—
589C8—48no—
590C22—22no—
591C22—24no—
592C22—22no—
593C22—24no—
594C22—22no—
595C22—24no—
596C22—22no—
597C22—24no—
598C22—22no—
599C22—24no—
600C22—22no—
601C22—24no—
602C22—416no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—816no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—1616no—
619C22C2×#45914yes—
620C22C2×#45918yes—
621C22C22×#12111yescharacteristic
622C22#348411yescharacteristic
623C22#348511yescharacteristic
624C22C23×D811yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22#358511yescharacteristic
627C22#358611yescharacteristic
628C4—24no—
629C4—22no—
630C4—24no—
631C4—24no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—44no—
636C4—82no—
637C4—84no—
638C4#345912yes—
639C4#349612yes—
640C2—24no—
641C2—28no—
642C2—84no—
643C2—164no—
644C2#3498411yescharacteristic
645C2C22×#45911yescharacteristic
646C2#6042511yescharacteristic
647C2#6155911yescharacteristic
648C2#6157211yescharacteristic
649C2#6169411yescharacteristic
650C2#6170711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3186C2semidirect
2. #35001C2semidirect
3. C2×#2361C2semidirect
4. #33106C2semidirect
5. #32902C2semidirect
6. #29754C2semidirect
7. #29712C2semidirect
8. #25200C2semidirect
9. #25076C2semidirect
10. #21930C2semidirect
11. C2×#1874C2semidirect
17. C22×C4×D4C22semidirect
20. C2×#480C22semidirect
21. C2×#480C22semidirect
23. C2×#459C22semidirect
24. C2×#447C22semidirect
25. C2×#447C22semidirect
26. C22×#102C22semidirect
33. #2501C22semidirect
34. #2433C22semidirect
36. #2361C22semidirect
37. C2×#390C22semidirect
38. C2×#389C22semidirect
42. C2×C4×C4⋊C4C22semidirect
44. #2083C22semidirect
45. #2071C22semidirect
46. #2067C22semidirect
47. #2054C22semidirect
48. #2041C22semidirect
49. #2011C22semidirect
50. #1982C22semidirect
51. #1947C22semidirect
52. #1925C22semidirect
53. #1879C22semidirect
54. #1874C22semidirect
55. #1874C22semidirect
56. #1874C22semidirect
57. #1850C22semidirect
58. #1802C22semidirect
59. #1797C22semidirect
60. #1797C22semidirect
61. C2×#340C22semidirect
87. C2×#121D4semidirect
288. C2×C22⋊C4D8semidirect
518. C23#459semidirect

Maximal subgroups

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

subgroupindex
C2×#31862
#350012
C2×#23612
#331062
#329022
#297542
#297122
#252002
#250762
#219302
C2×#18742