Finite Group Databasebeta

← #6388928

Subgroups — #6388928

Order 512.

2282 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#463C211yesmaximal, characteristic
2#34240C211yesmaximal, characteristic, complemented
3#34126C211yesmaximal, characteristic, complemented
4#23104C214yesmaximal, complemented
5#23005C214yesmaximal, complemented
6#22567C212yesmaximal, complemented
7#22566C212yesmaximal, complemented
8#22437C212yesmaximal, complemented
9#22436C212yesmaximal
10#22155C214yesmaximal, complemented
11#21960C211yesmaximal, characteristic, complemented
12#21806C211yesmaximal, characteristic, complemented
13#21507C214yesmaximal, complemented
14#20733C211yesmaximal, characteristic, complemented
15#20547C211yesmaximal, characteristic, complemented
16C23×SD16C2211yescharacteristic
17C23×D8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#463C4116yescomplemented
21C2×#463C2212yes—
22C2×#463C2212yes—
23C2×#463C2212yes—
24C2×#463C2212yes—
25C2×#442C2214yes—
26C2×#440C2214yescomplemented
27C2×#431C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#102C2211yescharacteristic
31#2372—22no—
32#2367C2214yescomplemented
33#2357C2212yescomplemented
34#2353—22no—
35#2349C2212yescomplemented
36#2345C2212yes—
37#2342—22no—
38#2342—22no—
39Q8×C22⋊C4C2212yes—
40D4×C22⋊C4C2212yes—
41D4×C22⋊C4C2212yes—
42#2337—22no—
43#2312—21no—
44C4×#124C2214yescomplemented
45#2293C2212yescomplemented
46#2290—22no—
47#2283C2211yescharacteristic
48#2283C2211yescharacteristic
49#2280—21no—
50#2273—21no—
51C42×D4C2212yescomplemented
52C4×#117C2211yescharacteristic
53C2×C4×C22⋊C4—21no—
54#1983C2214yescomplemented
55#1973C2214yescomplemented
56#1972C2214yes—
57#1958C2214yescomplemented
58#1949C2214yescomplemented
59#1949C2214yescomplemented
60#1936C2214yescomplemented
61#1935C2214yescomplemented
62#1927C2214yescomplemented
63#1925C2214yescomplemented
64#1880C2214yescomplemented
65#1879C2214yes—
66#1879C2214yes—
67#1877C2214yescomplemented
68#1865C2214yescomplemented
69#1862C2214yescomplemented
70#1824C2212yes—
71#1808C2212yes—
72#1807C2212yes—
73#1804C2212yescomplemented
74#1804C2212yescomplemented
75#1787C2212yes—
76#1786C2212yes—
77#1786C2212yes—
78#1770C2214yescomplemented
79#1767C2214yescomplemented
80#1763C2214yescomplemented
81#1762C2214yescomplemented
82C4×#102C2212yescomplemented
83C4×#102C2212yescomplemented
84#1752C2212yes—
85#1752C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16—44no—
91C22×SD16C2×C418yescomplemented
92C22×SD16C2312yes—
93C22×SD16C2312yes—
94C22×D8—44no—
95C22×D8C2×C418yes—
96C22×D8C2312yes—
97C22×D8C2312yes—
98C23×C8C2311yescharacteristic
99C23×C8C2311yescharacteristic
100C2×#124—21no—
101C2×#124—21no—
102C2×#124—24no—
103C2×#124C2×C418yescomplemented
104C2×#124C2312yes—
105C2×#124C2312yes—
106C2×C4×Q8C2314yes—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—24no—
110C2×C4×D4C2×C418yescomplemented
111C2×C4×D4C2314yes—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2314yes—
117C2×#117—22no—
118C2×#117D412yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—22no—
124C22×C22⋊C4—22no—
125C22×C42—22no—
126C22×C42C2311yescharacteristic
127#463C2×C4116yescomplemented
128#463C2×C4116yescomplemented
129#463C2×C4116yescomplemented
130#463C2×C4116yescomplemented
131#442—48no—
132#440—48no—
133#431—24no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2×C418yescomplemented
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×#103C2×C418yescomplemented
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#102C2×C418yescomplemented
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2314yes—
154C2×#97C2314yes—
155C2×C4×C8C2312yes—
156#391—24no—
157#391C2314yes—
158#390C2314yes—
159#390C2314yes—
160#389C2314yes—
161#388—24no—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#388—24no—
167#387—24no—
168#386—24no—
169#386—24no—
170#385—24no—
171#385—24no—
172#383—21no—
173#383—21no—
174#382C2312yes—
175#382C2312yes—
176#381—21no—
177#381—21no—
178#381—22no—
179#380—22no—
180#380—22no—
181#380—22no—
182#380C2312yes—
183#380C2312yes—
184#379—22no—
185#379—22no—
186C4×C4⋊C4D412yes—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—21no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—24no—
196C4×C22⋊C4—24no—
197C4×C22⋊C4—24no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4—24no—
202C4×C22⋊C4D411yescharacteristic
203C4×C22⋊C4D411yescharacteristic
204C4×C22⋊C4C2312yes—
205#376—21no—
206#376—21no—
207#376D412yes—
208C2×#95—22no—
209C43C2312yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2314yes—
216C25—21no—
217C25—21no—
218C22×Q8—21no—
219C22×Q8—22no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C22×C414yes—
223C22×Q8C2412yes—
224C22×D4—24no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4—42no—
232C22×D4C22×C414yes—
233C22×D4C22×C414yes—
234C22×D4C2412yes—
235C22×D4C2411yescharacteristic
236C22×D4C2411yescharacteristic
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—28no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—84no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—28no—
251C2×SD16—44no—
252C2×SD16—416no—
253C2×SD16C22×C418yes—
254C2×SD16C22×C418yes—
255C2×D8—28no—
256C2×D8—44no—
257C2×D8—416no—
258C2×D8C22×C418yes—
259C2×D8C22×C418yes—
260C22×C8—42no—
261C22×C8—42no—
262C22×C8C22×C414yes—
263C22×C8C22×C414yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267#124—28no—
268#124—28no—
269#124—24no—
270#124—24no—
271#124C22×C418yes—
272#124C22×C418yes—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—44no—
283C4×D4—44no—
284C4×D4—44no—
285C4×D4C22×C414yes—
286C4×D4C22×C414yes—
287C4×D4C22×C414yes—
288C4×D4C22×C414yes—
289#117—44no—
290#117—84no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4C22×C414yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—84no—
329C2×C42C22×C414yes—
330C2×C42C2×D412yes—
331C2×C42C2×D414yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42#4114yes—
336C2×C42C2414yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339#107—44no—
340#106—44no—
341C4⋊C8—24no—
342C4⋊C8C22×C418yes—
343C4⋊C8C22×C418yes—
344#103—44no—
345#103—44no—
346#103C22×C418yes—
347#103C22×C418yes—
348#102—28no—
349#102—44no—
350#102—44no—
351#102—44no—
352#102C22×C418yes—
353#102C22×C418yes—
354C22⋊C8—24no—
355#97—44no—
356#97—44no—
357C4×C8—24no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—22no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D414yes—
373C24—21no—
374C24—22no—
375C24—21no—
376C24—24no—
377C24—24no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—82no—
382C24—82no—
383C24—82no—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—82no—
392C2×Q8—84no—
393C2×Q8C23×C414yes—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—48no—
409C2×D4—44no—
410C2×D4—84no—
411C2×D4—82no—
412C2×D4C23×C414yes—
413C2×D4C23×C412yes—
414C2×D4C23×C412yes—
415C2×D4H(5,𝔽2)14yes—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—84no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—84no—
440C22×C4—84no—
441C22×C4—816no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C4×D414yes—
445C22×C4C4×D414yes—
446C22×C4C23×C412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4112yes—
455C22×C4C2511yescharacteristic
456SD16—216no—
457SD16—48no—
458D8—216no—
459D8—48no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8C23×C414yes—
468C2×C8C23×C414yes—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—44no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4C23×C412yes—
480C4⋊C4C23×C412yes—
481C22⋊C4—24no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C42—24no—
500C42—24no—
501C42—42no—
502C42—44no—
503C42—42no—
504C42—44no—
505C42—82no—
506C42—84no—
507C42—84no—
508C42C4×D414yes—
509C42C4×D414yes—
510C42C23×C412yes—
511C42C23×C412yes—
512C42C2×#4112yes—
513C42C2×#4112yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—48no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—48no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—48no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—816no—
536C23—168no—
537C23C2×C4×D412yes—
538C23#57514yes—
539C23C23×D411yescharacteristic
540Q8—24no—
541Q8—24no—
542Q8—44no—
543Q8—84no—
544D4—24no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—24no—
549D4—24no—
550D4—48no—
551D4—42no—
552D4—44no—
553D4—84no—
554D4#51814yes—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—44no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—816no—
590C2×C4—168no—
591C2×C4C2×C4×D412yes—
592C2×C4C2×C4×D412yes—
593C2×C4C2×C4×D412yes—
594C2×C4C2×C4×D412yes—
595C2×C4C2×C4×D412yes—
596C2×C4C24×C411yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4#58512yes—
600C8—24no—
601C8—24no—
602C8—44no—
603C22—22no—
604C22—28no—
605C22—22no—
606C22—24no—
607C22—42no—
608C22—48no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—44no—
616C22—816no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—168no—
621C22#294314yes—
622C22C22×C4×D411yescharacteristic
623C22#348812yes—
624C22C2×#57512yes—
625C22#358512yes—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—42no—
632C4—42no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—84no—
637C4—84no—
638C4C2×#51811yescharacteristic
639C4#342811yescharacteristic
640C4#349412yes—
641C2—24no—
642C2—44no—
643C2—82no—
644C2—84no—
645C2#3420411yescharacteristic
646C2C2×#294311yescharacteristic
647C2#6016811yescharacteristic
648C2#6197114yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34240C2semidirect
3. #34126C2semidirect
4. #23104C2semidirect
5. #23005C2semidirect
6. #22567C2semidirect
7. #22566C2semidirect
8. #22437C2semidirect
10. #22155C2semidirect
11. #21960C2semidirect
12. #21806C2semidirect
13. #21507C2semidirect
14. #20733C2semidirect
15. #20547C2semidirect
20. C2×#463C4semidirect
26. C2×#440C22semidirect
32. #2367C22semidirect
33. #2357C22semidirect
35. #2349C22semidirect
44. C4×#124C22semidirect
45. #2293C22semidirect
51. C42×D4C22semidirect
54. #1983C22semidirect
55. #1973C22semidirect
57. #1958C22semidirect
58. #1949C22semidirect
59. #1949C22semidirect
60. #1936C22semidirect
61. #1935C22semidirect
62. #1927C22semidirect
63. #1925C22semidirect
64. #1880C22semidirect
67. #1877C22semidirect
68. #1865C22semidirect
69. #1862C22semidirect
73. #1804C22semidirect
74. #1804C22semidirect
78. #1770C22semidirect
79. #1767C22semidirect
80. #1763C22semidirect
81. #1762C22semidirect
82. C4×#102C22semidirect
83. C4×#102C22semidirect
91. C22×SD16C2×C4semidirect
103. C2×#124C2×C4semidirect
110. C2×C4×D4C2×C4semidirect
127. #463C2×C4semidirect
128. #463C2×C4semidirect
129. #463C2×C4semidirect
130. #463C2×C4semidirect
136. C2×C4⋊C8C2×C4semidirect
140. C2×#103C2×C4semidirect
145. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4632
#342402
#341262
#231042
#230052
#225672
#225662
#224372
#224362
#221552
#219602
#218062
#215072
#207332
#205472