Finite Group Databasebeta

← #6502286

Subgroups — #6502286

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2448C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#29244C212yesmaximal, complemented
4#27583C211yesmaximal, characteristic, complemented
5#27582C211yesmaximal, characteristic
6#27193C214yesmaximal, complemented
7#26695C214yesmaximal, complemented
8#26591C214yesmaximal, complemented
9#26169C214yesmaximal, complemented
10#24217C212yesmaximal
11#23934C211yesmaximal, characteristic, complemented
12#23933C211yesmaximal, characteristic
13#22043C212yesmaximal
14#21912C211yesmaximal, characteristic, complemented
15C4×#451C212yesmaximal, complemented
16C2×#533C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#464C2214yes—
20C2×#451C2212yes—
21C2×#450C2212yes—
22C2×C4×Q16C2214yescomplemented
23#2448—22no—
24#2448—22no—
25#2448C2214yes—
26#2448C2214yes—
27C2×#399—21no—
28C2×#397—21no—
29#2370—22no—
30#2369C2212yescomplemented
31#2365C2212yescomplemented
32#2355—22no—
33#2355C2212yescomplemented
34Q8×C4⋊C4C2212yescomplemented
35#2329—21no—
36#2329—21no—
37#2317—21no—
38C4×C22⋊Q8C2212yes—
39C4×C22⋊Q8C2212yes—
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#388—21no—
45C2×#388—21no—
46#2285C2211yescharacteristic
47#2274—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2096C2212yescomplemented
50#2093C2212yes—
51#2086C2214yescomplemented
52#2072C2212yescomplemented
53#2071C2212yes—
54#2066C2214yescomplemented
55#2042C2214yes—
56#2024C2212yescomplemented
57#2023C2212yes—
58#2016C2212yescomplemented
59#2015C2214yes—
60#2013C2212yes—
61#1983C2214yes—
62#1968C2214yes—
63#1934C2214yes—
64#1927C2214yescomplemented
65#1926C2214yes—
66#1925C2214yes—
67#1920C2214yescomplemented
68#1916C2214yescomplemented
69#1916C2214yescomplemented
70#1880C2214yes—
71#1880C2214yes—
72#1878C2214yescomplemented
73#1878C2214yescomplemented
74#1872C2212yes—
75#1872C2212yes—
76#1863C2214yescomplemented
77#1863C2214yescomplemented
78#1853C2212yes—
79#1852C2212yes—
80#1817C2211yescharacteristic
81#1817C2211yescharacteristic
82#1797C2212yes—
83#1787C2212yes—
84#1786C2212yes—
85C4×#103C2214yescomplemented
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C22×Q16C2314yes—
90C22×SD16C2314yes—
91#533—28no—
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125—21no—
95C2×#124—22no—
96C2×#124—44no—
97C2×#124C2314yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—22no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2314yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×#117—21no—
108C2×#117—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#464—48no—
126#451—28no—
127#450—28no—
128C4×Q16—48no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C4×C8C2314yes—
143#399—24no—
144#399—24no—
145#397—24no—
146#397—24no—
147#391—21no—
148#391—21no—
149#391—22no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#391C2312yes—
154#391C2312yes—
155#390—21no—
156#390—21no—
157#390—22no—
158#390C2312yes—
159#390C2314yes—
160#390C2312yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389C2314yes—
167#389C2312yes—
168#389C2312yes—
169#388—24no—
170#388—24no—
171#388—24no—
172#388—24no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#385—22no—
180#385—22no—
181#384—22no—
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—22no—
185#382—21no—
186#380—21no—
187#380C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95D411yescharacteristic
208C2×#95D411yescharacteristic
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2314yes—
219C25—21no—
220C22×Q8—24no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8C2412yes—
226C22×Q8C2412yes—
227C22×D4—21no—
228C22×D4—24no—
229C22×D4—42no—
230C22×D4—42no—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—44no—
245C2×SD16—28no—
246C2×SD16—44no—
247C2×SD16—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252#125—28no—
253#124—28no—
254#124—44no—
255#124—48no—
256C22⋊Q8—28no—
257C22⋊Q8—24no—
258C22⋊Q8#4118yes—
259C4×Q8—28no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—28no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—48no—
267#117—28no—
268#117—44no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—82no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—82no—
316C2×C22⋊C4—82no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—44no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336#107—44no—
337#106—44no—
338C4⋊C8—44no—
339#103—28no—
340#103—28no—
341#103—44no—
342#103—44no—
343#103—44no—
344#102—28no—
345#102—44no—
346#102—44no—
347C22⋊C8—24no—
348C22⋊C8—24no—
349C4×C8—44no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369C24—22no—
370C24—42no—
371C24—42no—
372C24—42no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24C22×D411yescharacteristic
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—416no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8C2×#4114yes—
392C2×Q8H(5,𝔽2)14yes—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—88no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4#13514yes—
436C22×C4#13714yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C2×#4114yes—
447C22×C4C2511yescharacteristic
448Q16—416no—
449SD16—88no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—28no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—48no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—82no—
476C4⋊C4—81no—
477C4⋊C4—82no—
478C4⋊C4—81no—
479C4⋊C4—84no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4C2×#4114yes—
483C22⋊C4—28no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4—82no—
491C22⋊C4—84no—
492C22⋊C4—82no—
493C42—28no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—24no—
500C42—42no—
501C42—42no—
502C42—48no—
503C42—44no—
504C42—82no—
505C42—82no—
506C42—82no—
507C42—82no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—84no—
520C23—84no—
521C23—84no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—84no—
526C23#52512yes—
527C23C2×#13512yes—
528C23C2×#13712yes—
529C23C23×D411yescharacteristic
530Q8—28no—
531Q8—28no—
532Q8—48no—
533Q8—44no—
534Q8—44no—
535Q8—84no—
536D4—44no—
537D4—44no—
538D4—84no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4#52512yes—
594C2×C4C2×#13512yes—
595C2×C4C2×#13512yes—
596C2×C4C2×#13714yes—
597C2×C4C22×#4112yes—
598C2×C4C2×H(5,𝔽2)12yes—
599C2×C4#58512yes—
600C8—44no—
601C8—44no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—168no—
616C22#310814yes—
617C22C2×#52511yescharacteristic
618C22#348411yescharacteristic
619C22#348711yescharacteristic
620C22C22×#13511yescharacteristic
621C22C22×#13711yescharacteristic
622C22#358511yescharacteristic
623C22#358611yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—24no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—44no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4#351314yes—
641C2—24no—
642C2—162no—
643C2#3535611yescharacteristic
644C2C2×#310811yescharacteristic
645C2#6056711yescharacteristic
646C2#6156611yescharacteristic
647C2#6157311yescharacteristic
648C2#6188211yescharacteristic
649C2#6189211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34213C2semidirect
3. #29244C2semidirect
4. #27583C2semidirect
6. #27193C2semidirect
7. #26695C2semidirect
8. #26591C2semidirect
9. #26169C2semidirect
11. #23934C2semidirect
14. #21912C2semidirect
15. C4×#451C2semidirect
22. C2×C4×Q16C22semidirect
30. #2369C22semidirect
31. #2365C22semidirect
33. #2355C22semidirect
34. Q8×C4⋊C4C22semidirect
49. #2096C22semidirect
51. #2086C22semidirect
52. #2072C22semidirect
54. #2066C22semidirect
56. #2024C22semidirect
58. #2016C22semidirect
64. #1927C22semidirect
67. #1920C22semidirect
68. #1916C22semidirect
69. #1916C22semidirect
72. #1878C22semidirect
73. #1878C22semidirect
76. #1863C22semidirect
77. #1863C22semidirect
85. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24482
#342132
#292442
#275832
#275822
#271932
#266952
#265912
#261692
#242172
#239342
#239332
#220432
#219122
C4×#4512