Finite Group Databasebeta

← #6396499

Subgroups — #6396499

Order 512.

1996 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3215C211yesmaximal, characteristic
2#34240C211yesmaximal, characteristic
3#34225C211yesmaximal, characteristic, complemented
4#23097C214yesmaximal, complemented
5#23006C214yesmaximal, complemented
6#22869C214yesmaximal, complemented
7#22864C214yesmaximal
8#22273C212yesmaximal, complemented
9#22224C212yesmaximal, complemented
10#21989C211yesmaximal, characteristic, complemented
11#21702C212yesmaximal, complemented
12C4×#489C212yesmaximal, complemented
13#21404C211yesmaximal, characteristic, complemented
14#21204C211yesmaximal, characteristic
15#20480C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C22×#124C2211yescharacteristic
18#3215C4116yescomplemented
19C2×#497C2212yes—
20C2×#489C2212yes—
21C2×#464C2214yes—
22C2×#451C2212yes—
23C2×#450C2212yes—
24C2×#441C2214yescomplemented
25C2×#440C2214yes—
26C2×#432C2211yescharacteristic
27C2×#419C2212yes—
28C2×#418C2212yes—
29C2×C4×M16C2211yescharacteristic
30#2372—21no—
31#2372—21no—
32#2370—22no—
33#2367C2212yes—
34#2367C2212yes—
35#2362C2214yes—
36#2356C2212yescomplemented
37#2351C2212yescomplemented
38#2347—21no—
39#2346—21no—
40#2342—22no—
41Q8×C22⋊C4C2212yes—
42D4×C22⋊C4C2212yes—
43#2337—22no—
44#2323—22no—
45#2320—21no—
46#2312—21no—
47C4×C4⋊Q8C2212yescomplemented
48C4×C4⋊Q8C2212yescomplemented
49C4×#124C2212yes—
50C4×#124C2212yes—
51#2285C2211yescharacteristic
52#2283C2211yescharacteristic
53#2276—21no—
54#2273—21no—
55C4×#117C2211yescharacteristic
56#1993C2211yescharacteristic, complemented
57#1993C2211yescharacteristic, complemented
58#1969C2214yescomplemented
59#1968C2214yes—
60#1960C2211yescharacteristic, complemented
61#1960C2211yescharacteristic, complemented
62#1959C2212yes—
63#1959C2212yes—
64#1958C2214yes—
65#1949C2214yescomplemented
66#1936C2214yes—
67#1935C2214yescomplemented
68#1927C2214yescomplemented
69#1925C2214yes—
70#1894C2214yes—
71#1880C2214yes—
72#1878C2214yescomplemented
73#1878C2214yescomplemented
74#1877C2214yes—
75#1864C2214yes—
76#1863C2214yescomplemented
77#1854C2214yes—
78#1847C2212yescomplemented
79#1836C2212yescomplemented
80#1770C2212yescomplemented
81C8×C4⋊C4C2212yescomplemented
82C4×#103C2214yescomplemented
83C4×#103C2214yescomplemented
84C4×#102C2214yes—
85#1753C2214yes—
86#1745C2212yescomplemented
87C4×#97C2212yescomplemented
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C22×Q16C2314yes—
91C22×SD16C2314yes—
92C22×M16C2312yes—
93#533—24no—
94#533C2×C418yescomplemented
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125—21no—
98C2×#124—21no—
99C2×#124—21no—
100C2×#124—24no—
101C2×#124C2×C418yes—
102C2×#124C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2314yes—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—21no—
115C2×#117D412yes—
116C2×#117C2312yes—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—22no—
120C22×C42C2311yescharacteristic
121#497C2×C4116yescomplemented
122#489C2×C4116yescomplemented
123#464C2×C4132yescomplemented
124#451C2×C4116yescomplemented
125#450C2×C4116yes—
126#441—48no—
127#440—48no—
128#432C2×C418yescomplemented
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2312yes—
132#419—44no—
133#418—44no—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C4×M16—24no—
144C2×#97C2311yescharacteristic
145C2×#97C2311yescharacteristic
146C2×#97C2314yes—
147C2×#97C2312yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#391—22no—
151#391—22no—
152#391—22no—
153#391C2314yes—
154#391C2312yes—
155#391C2312yes—
156#390C2312yes—
157#390C2312yes—
158#389—24no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2314yes—
163#388—22no—
164#388—22no—
165#388—24no—
166#388—24no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—24no—
174#387—24no—
175#386—22no—
176#386—22no—
177#385—22no—
178#385—22no—
179#384C2312yes—
180#383—21no—
181#383—22no—
182#382—22no—
183#382C2311yescharacteristic
184#382C2312yes—
185#382C2311yescharacteristic
186#381—21no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380C2312yes—
191#379—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—24no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2311yescharacteristic
204C4×C22⋊C4—21no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D411yescharacteristic
213C4×C22⋊C4D411yescharacteristic
214C4×C22⋊C4C2312yes—
215#376—21no—
216#376—22no—
217#376D411yescharacteristic
218#376D411yescharacteristic
219C43C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2314yes—
225#336C2314yes—
226C25—21no—
227C22×Q8—21no—
228C22×Q8—21no—
229C22×Q8—22no—
230C22×Q8—42no—
231C22×Q8—42no—
232C22×Q8C22×C414yes—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—42no—
238C22×D4—42no—
239C22×D4C22×C414yes—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—44no—
248C2×Q16—44no—
249C2×Q16C22×C4116yes—
250C2×SD16—44no—
251C2×SD16—44no—
252C2×SD16C22×C4116yes—
253C2×M16—24no—
254C2×M16—24no—
255C2×M16—42no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8C22×C414yes—
262C4⋊Q8C22×C414yes—
263#125—28no—
264#124—28no—
265#124—28no—
266#124—24no—
267#124—24no—
268#124C22×C418yes—
269#124C22×C418yes—
270C22⋊Q8—28no—
271C22⋊Q8—24no—
272C22⋊Q8C22×C418yes—
273C4×Q8—28no—
274C4×Q8—24no—
275C4×Q8—48no—
276C4×Q8—44no—
277C4×Q8—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8C22×C418yes—
281C4×D4—44no—
282C4×D4—44no—
283#117—28no—
284#117—44no—
285#117—42no—
286#117—42no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2414yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—22no—
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—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42C22×C414yes—
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42#4114yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#107—44no—
342#106—44no—
343C4⋊C8C22×C418yes—
344#103—28no—
345#103—28no—
346#103—44no—
347#103—44no—
348#103C22×C4116yes—
349#102—28no—
350#102—44no—
351#102C22×C4116yes—
352C22⋊C8C22×C418yes—
353#97—28no—
354#97—22no—
355#97—22no—
356#97C22×C418yes—
357C4×C8—24no—
358C4×C8—44no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—24no—
363#95—22no—
364#95—21no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—24no—
375#95C2×D411yescharacteristic
376#95C2×D411yescharacteristic
377#95C2×D411yescharacteristic
378#95C2×D411yescharacteristic
379#95C2×D412yes—
380#95C2×D412yes—
381C24—22no—
382C24—24no—
383C24—44no—
384C24—81no—
385C24—81no—
386C24—82no—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—44no—
397C2×Q8—48no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—84no—
401C2×Q8C23×C414yes—
402C2×Q8C23×C414yes—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—48no—
408C2×D4—84no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4C23×C414yes—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4C4×D414yes—
432C22×C4C4×D414yes—
433C22×C4C23×C412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445Q16—416no—
446SD16—416no—
447M16—48no—
448C2×C8—22no—
449C2×C8—24no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8C23×C418yes—
456C2×C8H(5,𝔽2)14yes—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—22no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—82no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4C23×C414yes—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—28no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—82no—
491C42—28no—
492C42—24no—
493C42—24no—
494C42—24no—
495C42—42no—
496C42—42no—
497C42—42no—
498C42—48no—
499C42—82no—
500C42—82no—
501C42—82no—
502C42—84no—
503C42—82no—
504C42C4×D414yes—
505C42C4×D414yes—
506C42C23×C412yes—
507C42C23×C412yes—
508C42C2×#4112yes—
509C42C2×#4112yes—
510C23—22no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—24no—
515C23—48no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—84no—
523C23—84no—
524C23—164no—
525C23—164no—
526C23C2×C4×D412yes—
527C23C23×D411yescharacteristic
528Q8—28no—
529Q8—28no—
530Q8—48no—
531Q8—44no—
532Q8—44no—
533Q8—84no—
534D4—28no—
535D4—44no—
536D4—84no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—84no—
576C2×C4—164no—
577C2×C4—164no—
578C2×C4C2×C4×D412yes—
579C2×C4C2×C4×D412yes—
580C2×C4C2×C4×D412yes—
581C2×C4C2×C4×D412yes—
582C2×C4C2×C4×D412yes—
583C2×C4#57514yes—
584C2×C4C24×C411yescharacteristic
585C2×C4C22×#4111yescharacteristic
586C2×C4C2×H(5,𝔽2)12yes—
587C2×C4#58512yes—
588C8—28no—
589C8—24no—
590C8—42no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—42no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—168no—
604C22C22×C4×D411yescharacteristic
605C22#349811yescharacteristic
606C22#350011yescharacteristic
607C22C2×#57512yes—
608C22#358612yes—
609C4—22no—
610C4—22no—
611C4—24no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—44no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4C4×H(5,𝔽2)12yes—
625C4#349412yes—
626C2—42no—
627C2—84no—
628C2#3427511yescharacteristic
629C2C4×#57511yescharacteristic
630C2#6016911yescharacteristic
631C2#6245612yes—
632C2#6258512yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
3. #34225C2semidirect
4. #23097C2semidirect
5. #23006C2semidirect
6. #22869C2semidirect
8. #22273C2semidirect
9. #22224C2semidirect
10. #21989C2semidirect
11. #21702C2semidirect
12. C4×#489C2semidirect
13. #21404C2semidirect
15. #20480C2semidirect
18. #3215C4semidirect
24. C2×#441C22semidirect
36. #2356C22semidirect
37. #2351C22semidirect
47. C4×C4⋊Q8C22semidirect
48. C4×C4⋊Q8C22semidirect
56. #1993C22semidirect
57. #1993C22semidirect
58. #1969C22semidirect
60. #1960C22semidirect
61. #1960C22semidirect
65. #1949C22semidirect
67. #1935C22semidirect
68. #1927C22semidirect
72. #1878C22semidirect
73. #1878C22semidirect
76. #1863C22semidirect
78. #1847C22semidirect
79. #1836C22semidirect
80. #1770C22semidirect
81. C8×C4⋊C4C22semidirect
82. C4×#103C22semidirect
83. C4×#103C22semidirect
86. #1745C22semidirect
87. C4×#97C22semidirect
94. #533C2×C4semidirect
121. #497C2×C4semidirect
122. #489C2×C4semidirect
123. #464C2×C4semidirect
124. #451C2×C4semidirect
128. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32152
#342402
#342252
#230972
#230062
#228692
#228642
#222732
#222242
#219892
#217022
C4×#4892
#214042
#212042
#204802