Finite Group Databasebeta

← #6502468

Subgroups — #6502468

Order 512.

1918 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#28120C212yesmaximal
4#27610C211yesmaximal, characteristic, complemented
5#27582C211yesmaximal, characteristic
6#27439C214yesmaximal, complemented
7#27019C214yesmaximal, complemented
8#26919C214yesmaximal, complemented
9#26397C214yesmaximal, complemented
10#24302C212yesmaximal, complemented
11#24063C211yesmaximal, characteristic, complemented
12#23933C211yesmaximal, characteristic
13#22084C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21453C212yesmaximal
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×D4C2211yescharacteristic
20C2×#483C2212yes—
21C2×#477C2214yescomplemented
22C2×#450C2212yes—
23C2×#440C2214yes—
24#2452—22no—
25#2452—22no—
26#2452C2214yes—
27#2452C2214yes—
28C2×#396—21no—
29C2×#393—21no—
30#2370—22no—
31#2369C2212yescomplemented
32#2365C2212yescomplemented
33#2355—22no—
34#2355C2212yescomplemented
35Q8×C4⋊C4C2212yescomplemented
36#2329—21no—
37#2329—21no—
38#2317—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46#2285C2211yescharacteristic
47#2274—21no—
48C2×C4×C4⋊C4C2211yescharacteristic
49#2099C2212yescomplemented
50#2093C2212yes—
51#2083C2212yescomplemented
52#2071C2212yes—
53#2065C2214yes—
54#2058C2212yescomplemented
55#2057C2214yescomplemented
56#2055C2214yes—
57#2043C2212yescomplemented
58#2024C2214yescomplemented
59#2023C2212yes—
60#2013C2212yes—
61#1987C2214yescomplemented
62#1944C2214yescomplemented
63#1935C2214yescomplemented
64#1926C2214yes—
65#1925C2214yes—
66#1920C2214yescomplemented
67#1918C2214yescomplemented
68#1917C2214yescomplemented
69#1917C2214yescomplemented
70#1880C2214yes—
71#1880C2214yes—
72#1877C2214yes—
73#1872C2212yes—
74#1872C2212yes—
75#1865C2214yescomplemented
76#1863C2214yescomplemented
77#1862C2214yes—
78#1852C2212yes—
79#1852C2212yes—
80#1823C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1786C2212yes—
85#1762C2214yes—
86#1749C2212yes—
87C23×Q8—21no—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91#533—28no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2314yes—
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—48no—
102C2×C22⋊Q8C2312yes—
103C2×C4×Q8—21no—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4—44no—
109C2×C4×D4C2314yes—
110C2×#117—21no—
111C2×#117—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#483—28no—
129#477—48no—
130#450—28no—
131#440—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×#97C2314yes—
145#396—24no—
146#396—24no—
147#393—24no—
148#393—24no—
149#391—22no—
150#391—22no—
151#391C2312yes—
152#391C2312yes—
153#390—21no—
154#390—21no—
155#390—22no—
156#390C2314yes—
157#390C2312yes—
158#390C2312yes—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2314yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#385—22no—
178#385—22no—
179#385—24no—
180#385—24no—
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×#95D411yescharacteristic
207C2×#95D411yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2314yes—
215#336C2312yes—
216#336C2312yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C25—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—82no—
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—21no—
229C22×D4—24no—
230C22×D4—42no—
231C22×D4C2412yes—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—24no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—82no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×SD16—28no—
247C2×SD16—28no—
248C2×SD16—44no—
249C22×C8C2414yes—
250C22×C8C2414yes—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—28no—
254C4⋊Q8—44no—
255C4⋊Q8—48no—
256#125—28no—
257C22⋊Q8—28no—
258C22⋊Q8—24no—
259C22⋊Q8—416no—
260C22⋊Q8—416no—
261C22⋊Q8#4118yes—
262C4×Q8—28no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—44no—
266C4×Q8—44no—
267C4×D4—28no—
268C4×D4—44no—
269C4×D4—48no—
270#117—28no—
271#117—44no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—82no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341#107—28no—
342#107—44no—
343#106—28no—
344C4⋊C8—44no—
345#103—28no—
346#103—28no—
347#103—44no—
348#102—28no—
349#102—28no—
350#102—44no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353#97—44no—
354#95—22no—
355#95—22no—
356#95—24no—
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#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24—24no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24C22×D411yescharacteristic
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—84no—
389C2×Q8—816no—
390C2×Q8C2×#4114yes—
391C2×D4—24no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—416no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D42−1+414yes—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
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—41no—
416C22×C4—41no—
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—48no—
426C22×C4—48no—
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—84no—
436C22×C4—816no—
437C22×C4AGL(1,ℤ/8)14yes—
438C22×C4#13714yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D412yes—
448C22×C4C2×#4114yes—
449C22×C4C2511yescharacteristic
450SD16—416no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8—24no—
455C2×C8—24no—
456C2×C8—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—28no—
461C4⋊C4—24no—
462C4⋊C4—28no—
463C4⋊C4—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C4⋊C4—81no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4C2×#4114yes—
488C4⋊C4H(5,𝔽2)14yes—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C42—28no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—24no—
504C42—24no—
505C42—42no—
506C42—42no—
507C42—44no—
508C42—82no—
509C42—82no—
510C42—82no—
511C42—82no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—24no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—84no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23#52512yes—
531C23C2×AGL(1,ℤ/8)12yes—
532C23C2×#13712yes—
533C23C23×D411yescharacteristic
534Q8—28no—
535Q8—44no—
536Q8—84no—
537Q8—816no—
538D4—28no—
539D4—48no—
540D4—44no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—44no—
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—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—816no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4#52512yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×#13714yes—
600C2×C4C22×#4112yes—
601C2×C4C2×H(5,𝔽2)12yes—
602C2×C4C2×2−1+412yes—
603C8—44no—
604C8—44no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—816no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22#311214yes—
620C22C2×#52511yescharacteristic
621C22#348411yescharacteristic
622C22#348611yescharacteristic
623C22C22×AGL(1,ℤ/8)11yescharacteristic
624C22C22×#13711yescharacteristic
625C22#358511yescharacteristic
626C22#358611yescharacteristic
627C4—22no—
628C4—22no—
629C4—22no—
630C4—24no—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—41no—
637C4—41no—
638C4—44no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4#352014yes—
644C2—24no—
645C2—84no—
646C2#3535711yescharacteristic
647C2C2×#311211yescharacteristic
648C2#6056711yescharacteristic
649C2#6157311yescharacteristic
650C2#6158211yescharacteristic
651C2#6182611yescharacteristic
652C2#6182711yescharacteristic
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
4. #27610C2semidirect
6. #27439C2semidirect
7. #27019C2semidirect
8. #26919C2semidirect
9. #26397C2semidirect
10. #24302C2semidirect
11. #24063C2semidirect
13. #22084C2semidirect
14. #21916C2semidirect
21. C2×#477C22semidirect
31. #2369C22semidirect
32. #2365C22semidirect
34. #2355C22semidirect
35. Q8×C4⋊C4C22semidirect
49. #2099C22semidirect
51. #2083C22semidirect
54. #2058C22semidirect
55. #2057C22semidirect
57. #2043C22semidirect
58. #2024C22semidirect
61. #1987C22semidirect
62. #1944C22semidirect
63. #1935C22semidirect
66. #1920C22semidirect
67. #1918C22semidirect
68. #1917C22semidirect
69. #1917C22semidirect
75. #1865C22semidirect
76. #1863C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
#342132
#281202
#276102
#275822
#274392
#270192
#269192
#263972
#243022
#240632
#239332
#220842
#219162
#214532