Finite Group Databasebeta

← #6388968

Subgroups — #6388968

Order 512.

2298 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3038C211yesmaximal, characteristic
2#34364C211yesmaximal, characteristic, complemented
3#34193C211yesmaximal, characteristic, complemented
4#23099C212yesmaximal, complemented
5#23093C212yesmaximal, complemented
6#23003C212yesmaximal, complemented
7#22997C212yesmaximal, complemented
8#22580C214yesmaximal, complemented
9#22443C214yesmaximal, complemented
10#22158C214yesmaximal, complemented
11#21963C211yesmaximal, characteristic, complemented
12#21803C211yesmaximal, characteristic, complemented
13#21515C214yesmaximal, complemented
14#20736C211yesmaximal, characteristic, complemented
15#20548C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C22×#127C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#3038C4116yescomplemented
20C2×#460C2212yes—
21C2×#460C2212yes—
22C2×#459C2212yes—
23C2×#459C2212yes—
24C2×#440C2214yescomplemented
25C2×C4×SD16C2214yescomplemented
26C2×#431C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×#418C2212yes—
29#2380—22no—
30#2366C2212yes—
31#2366C2212yes—
32#2354—22no—
33#2354C2212yescomplemented
34Q8×C4⋊C4C2212yescomplemented
35#2347—22no—
36#2347C2212yes—
37#2341C2212yes—
38D4×C22⋊C4—22no—
39D4×C22⋊C4C2214yes—
40#2328—21no—
41#2328—21no—
42#2328C2214yes—
43#2319—22no—
44#2319C2212yescomplemented
45#2308—22no—
46#2285—21no—
47#2285C2211yescharacteristic
48#2283—21no—
49#2283C2211yescharacteristic
50#2283C2211yescharacteristic
51#2276C2212yescomplemented
52#1982C2214yes—
53#1972C2214yescomplemented
54#1970C2214yescomplemented
55#1957C2212yes—
56#1957C2212yes—
57#1949C2214yescomplemented
58#1948C2212yescomplemented
59#1947C2212yescomplemented
60#1936C2214yescomplemented
61#1934C2214yescomplemented
62#1926C2214yescomplemented
63#1925C2214yescomplemented
64#1880C2214yescomplemented
65#1880C2214yescomplemented
66#1879C2214yescomplemented
67#1879C2214yescomplemented
68#1864C2214yescomplemented
69#1862C2214yescomplemented
70#1825C2212yes—
71#1809C2214yes—
72#1805C2212yescomplemented
73#1804C2212yescomplemented
74#1788C2212yes—
75#1786C2214yes—
76#1770C2212yescomplemented
77C8×C4⋊C4C2212yescomplemented
78#1768C2214yescomplemented
79#1762C2214yescomplemented
80C4×#103C2211yescharacteristic, complemented
81C4×#103C2211yescharacteristic, complemented
82C4×#102C2211yescharacteristic, complemented
83C4×#102C2211yescharacteristic, complemented
84C4×#102C2214yescomplemented
85#1753C2214yes—
86C22×#41C2311yescharacteristic
87C23×D4—21no—
88C23×D4—21no—
89C23×D4C2312yes—
90C2×AGL(1,ℤ/8)—48no—
91C2×AGL(1,ℤ/8)C2×C4116yescomplemented
92C22×SD16C2314yes—
93C22×D8C2314yes—
94C22×M16C2312yes—
95C2×#127—21no—
96C2×#127—21no—
97C2×#127—24no—
98C2×#127C2×C418yes—
99C2×#127C2312yes—
100C2×#127C2312yes—
101C4×#41—24no—
102C4×#41C2×C418yescomplemented
103C2×C4×Q8C2314yes—
104C2×C4×Q8C2311yescharacteristic
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2314yes—
113C2×C4×D4C2314yes—
114C2×#117—21no—
115C2×#117—22no—
116C2×#117—21no—
117C2×#117C2311yescharacteristic
118C22×C22⋊C4—24no—
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121#460C2×C4116yescomplemented
122#460C2×C4116yescomplemented
123#459C2×C4116yescomplemented
124#459C2×C4116yescomplemented
125#440—48no—
126C4×SD16—48no—
127#431—24no—
128C2×#107C2314yes—
129C2×#106C2314yes—
130#423C2×C418yescomplemented
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134#418C2×C4116yescomplemented
135C2×#103C2312yes—
136C2×#103C2312yes—
137C2×#102C2312yes—
138C2×#102C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2312yes—
146C2×#97C2314yes—
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2314yes—
150#390—22no—
151#390—22no—
152#390—24no—
153#390C2314yes—
154#390C2314yes—
155#390C2314yes—
156#389—24no—
157#389C2314yes—
158#388—24no—
159#388—24no—
160#388—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#386—24no—
165#386—24no—
166#386—24no—
167#386—24no—
168#386—24no—
169#384—22no—
170#384C2312yes—
171#383—21no—
172#383—22no—
173#383D412yes—
174#382—21no—
175#382D411yescharacteristic
176#382D411yescharacteristic
177#382C2312yes—
178#381—21no—
179#381—22no—
180#381D412yes—
181#380—21no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380D412yes—
186#380C2312yes—
187#380C2312yes—
188#379—24no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—24no—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2314yes—
196C4×C4⋊C4C2311yescharacteristic
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4C2312yes—
202#376—21no—
203#376—22no—
204#376C2312yes—
205#340C2312yes—
206#340C2312yes—
207#340C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25—22no—
214C2×#41—22no—
215C2×#41—24no—
216C2×#41—42no—
217C2×#41C22×C414yes—
218C22×Q8C2411yescharacteristic
219C22×D4—21no—
220C22×D4—28no—
221C22×D4—28no—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—24no—
228C22×D4—44no—
229C22×D4—44no—
230C22×D4—82no—
231C22×D4—82no—
232C22×D4—84no—
233C22×D4C22×C418yes—
234C22×D4C2414yes—
235C22×D4C2411yescharacteristic
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244AGL(1,ℤ/8)—432no—
245C2×SD16—28no—
246C2×SD16—28no—
247C2×SD16C22×C4116yes—
248C2×D8—44no—
249C2×D8—44no—
250C2×D8C22×C4116yes—
251C2×M16—44no—
252C2×M16C22×C418yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2414yes—
256#127—28no—
257#127—28no—
258#127—24no—
259#127—24no—
260#127C22×C418yes—
261#127C22×C418yes—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—28no—
265C4×Q8—44no—
266C4×Q8C22×C414yes—
267C4×Q8C22×C414yes—
268C4×D4—28no—
269C4×D4—28no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—44no—
275C4×D4—44no—
276C4×D4—84no—
277C4×D4—84no—
278C4×D4C22×C414yes—
279C4×D4C22×C414yes—
280#117—22no—
281#117—28no—
282#117—84no—
283#117—82no—
284#117C22×C414yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2411yescharacteristic
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
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—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—88no—
312C2×C22⋊C4C2×D414yes—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—28no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42C22×C414yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42#4114yes—
326C2×C42C2414yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—44no—
330#106—44no—
331C4⋊C8—24no—
332C4⋊C8C22×C418yes—
333C4⋊C8C22×C418yes—
334#103—24no—
335#103—24no—
336#102—44no—
337#102—44no—
338#102—42no—
339#102—42no—
340#102—44no—
341#102—44no—
342#102C22×C4116yes—
343#102C22×C4116yes—
344C22⋊C8—24no—
345#97—44no—
346C4×C8—22no—
347C4×C8—22no—
348C4×C8—44no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—24no—
356#95—21no—
357#95—21no—
358#95C2×D412yes—
359#95C2×D412yes—
360#95C2×D414yes—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D414yes—
364C24—21no—
365C24—24no—
366C24—28no—
367C24—48no—
368C24—84no—
369C24—82no—
370C24—82no—
371C24C22×D411yescharacteristic
372#41—28no—
373#41—44no—
374#41—48no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8C23×C412yes—
378C2×Q8H(5,𝔽2)14yes—
379C2×D4—24no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—21no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—88no—
399C2×D4—816no—
400C2×D4—84no—
401C2×D4—82no—
402C2×D4—82no—
403C2×D4—84no—
404C2×D4—82no—
405C2×D4—82no—
406C2×D4—816no—
407C2×D4C23×C418yes—
408C2×D4C23×C412yes—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—82no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4C4×D414yes—
434C22×C4C4×D414yes—
435C22×C4C23×C412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445SD16—232no—
446D8—416no—
447M16—48no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8C23×C418yes—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—82no—
472C4⋊C4—84no—
473C4⋊C4C23×C412yes—
474C4⋊C4C23×C412yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—24no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—84no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—82no—
489C42—24no—
490C42—24no—
491C42—24no—
492C42—24no—
493C42—24no—
494C42—48no—
495C42—42no—
496C42—42no—
497C42—82no—
498C42—84no—
499C42C4×D414yes—
500C42C4×D414yes—
501C42C23×C412yes—
502C42C23×C412yes—
503C42C2×#4112yes—
504C42C2×#4112yes—
505C23—22no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—28no—
511C23—416no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—42no—
518C23—41no—
519C23—44no—
520C23—44no—
521C23—41no—
522C23—88no—
523C23—88no—
524C23—168no—
525C23—168no—
526C23C2×C4×D412yes—
527C23C23×D411yescharacteristic
528Q8—22no—
529Q8—24no—
530Q8#51814yes—
531D4—22no—
532D4—28no—
533D4—28no—
534D4—22no—
535D4—48no—
536D4—416no—
537D4—44no—
538D4—44no—
539D4—42no—
540D4—88no—
541D4—88no—
542D4—88no—
543D4—88no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—84no—
579C2×C4—168no—
580C2×C4—168no—
581C2×C4C2×C4×D412yes—
582C2×C4C2×C4×D412yes—
583C2×C4C2×C4×D412yes—
584C2×C4C2×C4×D412yes—
585C2×C4C2×C4×D412yes—
586C2×C4#57612yes—
587C2×C4#57712yes—
588C2×C4C24×C411yescharacteristic
589C2×C4C22×#4111yescharacteristic
590C2×C4C2×H(5,𝔽2)12yes—
591C2×C4#58512yes—
592C8—28no—
593C8—42no—
594C8—42no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—48no—
602C22—42no—
603C22—48no—
604C22—42no—
605C22—48no—
606C22—84no—
607C22—82no—
608C22—84no—
609C22—84no—
610C22—82no—
611C22—84no—
612C22—1616no—
613C22C22×C4×D411yescharacteristic
614C22#348812yes—
615C22C2×#57611yescharacteristic
616C22C2×#57711yescharacteristic
617C22#358512yes—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—24no—
622C4—22no—
623C4—42no—
624C4—44no—
625C4—44no—
626C4—42no—
627C4—82no—
628C4—84no—
629C4—82no—
630C4#294614yes—
631C4C2×#51811yescharacteristic
632C4#342811yescharacteristic
633C4#349412yes—
634C2—42no—
635C2—82no—
636C2—88no—
637C2#3420411yescharacteristic
638C2C2×#294611yescharacteristic
639C2#6016811yescharacteristic
640C2#6203512yes—
641C2#6203712yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3038C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2283 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34364C2semidirect
3. #34193C2semidirect
4. #23099C2semidirect
5. #23093C2semidirect
6. #23003C2semidirect
7. #22997C2semidirect
8. #22580C2semidirect
9. #22443C2semidirect
10. #22158C2semidirect
11. #21963C2semidirect
12. #21803C2semidirect
13. #21515C2semidirect
14. #20736C2semidirect
15. #20548C2semidirect
19. #3038C4semidirect
24. C2×#440C22semidirect
25. C2×C4×SD16C22semidirect
33. #2354C22semidirect
34. Q8×C4⋊C4C22semidirect
44. #2319C22semidirect
51. #2276C22semidirect
53. #1972C22semidirect
54. #1970C22semidirect
57. #1949C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1936C22semidirect
61. #1934C22semidirect
62. #1926C22semidirect
63. #1925C22semidirect
64. #1880C22semidirect
65. #1880C22semidirect
66. #1879C22semidirect
67. #1879C22semidirect
68. #1864C22semidirect
69. #1862C22semidirect
72. #1805C22semidirect
73. #1804C22semidirect
76. #1770C22semidirect
77. C8×C4⋊C4C22semidirect
78. #1768C22semidirect
79. #1762C22semidirect
80. C4×#103C22semidirect
81. C4×#103C22semidirect
82. C4×#102C22semidirect
83. C4×#102C22semidirect
84. C4×#102C22semidirect
91. C2×AGL(1,ℤ/8)C2×C4semidirect
102. C4×#41C2×C4semidirect
121. #460C2×C4semidirect
122. #460C2×C4semidirect
123. #459C2×C4semidirect
124. #459C2×C4semidirect
130. #423C2×C4semidirect
134. #418C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30382
#343642
#341932
#230992
#230932
#230032
#229972
#225802
#224432
#221582
#219632
#218032
#215152
#207362
#205482