Finite Group Databasebeta

← #6502463

Subgroups — #6502463

Order 512.

1926 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2451C211yesmaximal, characteristic, complemented
2#34206C211yesmaximal, characteristic
3#28115C212yesmaximal, complemented
4#27614C211yesmaximal, characteristic, complemented
5#27586C211yesmaximal, characteristic, complemented
6#27445C214yesmaximal, complemented
7#27017C214yesmaximal, complemented
8#26912C214yesmaximal, complemented
9#26384C214yesmaximal, complemented
10#24298C212yesmaximal, complemented
11#24061C211yesmaximal, characteristic
12#23931C211yesmaximal, characteristic
13#22082C212yesmaximal, complemented
14#21916C211yesmaximal, characteristic, complemented
15#21454C212yesmaximal, complemented
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#520—21no—
19C22×C4×Q8C2211yescharacteristic
20C2×#481C2212yes—
21C2×#474C2214yes—
22C2×#448C2212yes—
23C2×#440C2214yes—
24#2451—22no—
25#2451—22no—
26#2451C2214yes—
27#2451C2214yes—
28C2×#398—21no—
29C2×#395—21no—
30#2367—22no—
31#2366C2212yes—
32#2365C2212yes—
33#2354C2212yes—
34#2352—22no—
35D4×C4⋊C4C2212yes—
36#2325—21no—
37#2325—21no—
38#2311—21no—
39C4×#121C2212yes—
40C4×#121C2212yes—
41C2×#391—21no—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#387—21no—
47#2285C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2212yes—
50#2095C2212yes—
51#2083C2212yes—
52#2072C2212yes—
53#2066C2214yes—
54#2058C2212yes—
55#2057C2214yes—
56#2055C2214yes—
57#2041C2212yes—
58#2024C2212yes—
59#2023C2214yes—
60#2014C2212yes—
61#1988C2214yes—
62#1944C2214yes—
63#1936C2214yes—
64#1928C2214yes—
65#1927C2214yes—
66#1921C2214yes—
67#1919C2214yes—
68#1917C2214yes—
69#1917C2214yes—
70#1880C2214yes—
71#1877C2214yes—
72#1877C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1864C2214yes—
76#1863C2214yes—
77#1862C2214yes—
78#1852C2212yes—
79#1852C2212yes—
80#1824C2212yes—
81#1817C2211yescharacteristic
82#1817C2211yescharacteristic
83#1797C2212yes—
84#1787C2212yes—
85#1763C2214yes—
86#1749C2212yes—
87C22×#41—21no—
88C23×Q8—21no—
89C23×Q8C2311yescharacteristic
90C22×SD16C2314yes—
91#532—28no—
92C2×C4⋊Q8—22no—
93C2×C4⋊Q8—44no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2314yes—
96C2×#127C2311yescharacteristic
97C2×#124—21no—
98C2×#121—21no—
99C2×#121C2312yes—
100#520—48no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—44no—
104C2×C4×Q8C2314yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—21no—
109C2×C4×D4C2312yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—22no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4D411yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C42—21no—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#481—28no—
129#474—48no—
130#448—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#398—24no—
146#398—24no—
147#395—24no—
148#395—24no—
149#391—22no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—24no—
154#391—24no—
155#391C2314yes—
156#391C2312yes—
157#391C2312yes—
158#390—22no—
159#390C2312yes—
160#390C2312yes—
161#389—22no—
162#389—21no—
163#389—21no—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#389C2312yes—
168#389C2314yes—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—24no—
174#387—24no—
175#387—22no—
176#387—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#385—22no—
182#385—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380C2312yes—
190C4×C4⋊C4—21no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
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—
220C2×#41—28no—
221C2×#41—44no—
222C2×#41—44no—
223C22×Q8—24no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8—24no—
227C22×Q8—24no—
228C22×Q8—42no—
229C22×Q8—44no—
230C22×Q8—44no—
231C22×Q8—82no—
232C22×Q8C2412yes—
233C22×D4—22no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×SD16—28no—
252C2×SD16—28no—
253C2×SD16—44no—
254C22×C8C2414yes—
255C22×C8C2414yes—
256C4⋊Q8—24no—
257C4⋊Q8—28no—
258C4⋊Q8—44no—
259C4⋊Q8—48no—
260#127—24no—
261#124—28no—
262#121—28no—
263#121—24no—
264#121#4118yes—
265C4×Q8—28no—
266C4×Q8—44no—
267C4×Q8—48no—
268C4×Q8—48no—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—48no—
275C4×D4—48no—
276C4×D4—48no—
277#117—48no—
278#117—44no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339#107—28no—
340#107—44no—
341#106—28no—
342C4⋊C8—44no—
343#103—28no—
344#103—28no—
345#103—44no—
346#102—28no—
347#102—28no—
348#102—44no—
349C22⋊C8—24no—
350C22⋊C8—24no—
351#97—44no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
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—22no—
372C24C22×D411yescharacteristic
373#41—432no—
374C2×Q8—24no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—48no—
380C2×Q8—416no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—48no—
385C2×Q8—42no—
386C2×Q8—84no—
387C2×Q8—816no—
388C2×Q82−1+414yes—
389C2×D4—24no—
390C2×D4—28no—
391C2×D4—24no—
392C2×D4—22no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4C2×#4114yes—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
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—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—44no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4#13714yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C2×#4114yes—
460C22×C4C2511yescharacteristic
461SD16—416no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4C2×#4114yes—
494C4⋊C4H(5,𝔽2)14yes—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—24no—
509C42—24no—
510C42—42no—
511C42—42no—
512C42—48no—
513C42—44no—
514C42—44no—
515C42—44no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23#52512yes—
528C23C2×AGL(1,ℤ/8)12yes—
529C23C2×#13712yes—
530C23C23×D411yescharacteristic
531Q8—28no—
532Q8—48no—
533Q8—48no—
534Q8—44no—
535Q8—816no—
536D4—28no—
537D4—48no—
538D4—416no—
539D4—44no—
540D4—84no—
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—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—44no—
570C2×C4—416no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
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—42no—
592C2×C4—42no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—164no—
598C2×C4—164no—
599C2×C4#52512yes—
600C2×C4C2×AGL(1,ℤ/8)14yes—
601C2×C4C2×#13712yes—
602C2×C4C2×#13712yes—
603C2×C4C22×#4112yes—
604C2×C4C2×H(5,𝔽2)12yes—
605C2×C4C2×2−1+412yes—
606C8—44no—
607C8—44no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22#311014yes—
622C22C2×#52511yescharacteristic
623C22#348411yescharacteristic
624C22#348611yescharacteristic
625C22C22×AGL(1,ℤ/8)11yescharacteristic
626C22C22×#13711yescharacteristic
627C22#358511yescharacteristic
628C22#358611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#352014yes—
646C2—24no—
647C2—84no—
648C2#3535711yescharacteristic
649C2C2×#311011yescharacteristic
650C2#6056711yescharacteristic
651C2#6156911yescharacteristic
652C2#6158411yescharacteristic
653C2#6182811yescharacteristic
654C2#6183011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. C2×#2451C2semidirect
3. #28115C2semidirect
4. #27614C2semidirect
5. #27586C2semidirect
6. #27445C2semidirect
7. #27017C2semidirect
8. #26912C2semidirect
9. #26384C2semidirect
10. #24298C2semidirect
13. #22082C2semidirect
14. #21916C2semidirect
15. #21454C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24512
#342062
#281152
#276142
#275862
#274452
#270172
#269122
#263842
#242982
#240612
#239312
#220822
#219162
#214542