Finite Group Databasebeta

← #6644653

Subgroups — #6644653

Order 512.

1696 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#33005C212yesmaximal, complemented
4#32430C214yesmaximal, complemented
5#32346C214yesmaximal, complemented
6#31638C214yesmaximal, complemented
7#31522C214yesmaximal, complemented
8#29639C212yesmaximal, complemented
9#29105C211yesmaximal, characteristic, complemented
10#28305C211yesmaximal, characteristic
11#28115C212yesmaximal, complemented
12#27961C211yesmaximal, characteristic, complemented
13#27906C211yesmaximal, characteristic
14#27645C212yesmaximal, complemented
15#21912C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#481C2212yes—
20C2×#481C2212yes—
21C2×#481C2212yes—
22C2×#448C2212yes—
23#2857C2211yescharacteristic
24#2856C2211yescharacteristic
25#2854C2212yes—
26#2639—21no—
27#2618—21no—
28#2567—22no—
29#2567—22no—
30#2567C2214yes—
31#2567C2214yes—
32#2560—22no—
33#2558—22no—
34#2512C2212yes—
35#2506C2212yes—
36#2452C2212yes—
37#2445C2212yes—
38C2×#395—21no—
39C2×#395—21no—
40#2325—21no—
41C4×#121C2212yes—
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#389—21no—
45C2×#389—21no—
46C2×#384C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2100C2214yes—
50#2096C2212yes—
51#2094C2214yes—
52#2084C2214yes—
53#2072C2212yes—
54#2071C2214yes—
55#2066C2214yes—
56#2058C2214yes—
57#2058C2214yes—
58#2055C2214yes—
59#2024C2212yes—
60#2023C2214yes—
61#2023C2214yes—
62#2014C2212yes—
63#1990C2212yes—
64#1968C2214yes—
65#1957C2212yes—
66#1948C2214yes—
67#1933C2214yes—
68#1927C2214yes—
69#1916C2214yes—
70#1916C2214yes—
71#1880C2214yes—
72#1880C2214yes—
73#1877C2214yes—
74#1865C2214yes—
75#1865C2214yes—
76#1844C2212yes—
77#1834C2212yes—
78#1824C2212yes—
79#1824C2212yes—
80#1824C2212yes—
81#1817C2212yes—
82#1802C2214yes—
83#1797C2212yes—
84#1787C2212yes—
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C22×SD16C2314yes—
90#539—48no—
91C2×C4⋊Q8—22no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#126—22no—
97C2×#125—22no—
98C2×#123—22no—
99C2×#123—22no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—22no—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—22no—
106C2×C4×Q8—44no—
107C2×C4×Q8C2314yes—
108C2×C4×D4—22no—
109C2×C4×D4C2312yes—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4D411yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C42—21no—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#481—44no—
126#481—44no—
127#481—44no—
128#448—44no—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143#400—22no—
144#398—22no—
145#397—22no—
146#396—22no—
147#396—22no—
148#395—24no—
149#395—24no—
150#395—24no—
151#395—24no—
152#394—22no—
153#394—22no—
154#393—22no—
155#391—21no—
156#391—22no—
157#391—21no—
158#391—21no—
159#391C2314yes—
160#391C2312yes—
161#391C2312yes—
162#390—21no—
163#390—21no—
164#390C2312yes—
165#390C2312yes—
166#390C2312yes—
167#390C2312yes—
168#389—24no—
169#389—24no—
170#389—24no—
171#389—24no—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#385—22no—
186#384—22no—
187#384—21no—
188#384—21no—
189#384C2314yes—
190#384C2312yes—
191#384C2312yes—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#380—21no—
195#380C2312yes—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C22⋊C4—22no—
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#336C2314yes—
216#336C2314yes—
217#336C2312yes—
218#336C2312yes—
219C22×Q8—24no—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—24no—
223C22×Q8—24no—
224C22×Q8—42no—
225C22×Q8—44no—
226C22×Q8—82no—
227C22×Q8—82no—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C2×SD16—28no—
246C2×SD16—44no—
247C2×SD16—44no—
248C22×C8C2414yes—
249C22×C8C2414yes—
250C4⋊Q8—48no—
251C4⋊Q8—42no—
252C4⋊Q8—48no—
253#127—42no—
254#126—48no—
255#125—48no—
256#123—48no—
257#123—48no—
258C22⋊Q8—44no—
259C22⋊Q8—48no—
260#121—24no—
261#121—24no—
262#121—44no—
263#121—44no—
264C4×Q8—48no—
265C4×Q8—48no—
266C4×D4—44no—
267C4×D4—48no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4—82no—
298C2×C4⋊C4SD1618yescomplemented
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
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—22no—
320C2×C42—24no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—42no—
324C2×C42—42no—
325C2×C42—44no—
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#106—44no—
335#106—44no—
336#106—44no—
337C4⋊C8—44no—
338#103—28no—
339#103—44no—
340#103—44no—
341#102—44no—
342#102—44no—
343#102—44no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346C4×C8—42no—
347C4×C8—42no—
348#95—24no—
349#95—24no—
350#95—22no—
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#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364C24—22no—
365C24C22×D411yescharacteristic
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—24no—
370C2×Q8—416no—
371C2×Q8—42no—
372C2×Q8—48no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—84no—
376C2×Q8—84no—
377C2×Q8—81no—
378C2×Q8—81no—
379C2×Q8—816no—
380C2×Q82−1+414yes—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—24no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—81no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D4—81no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—44no—
430C22×C4—82no—
431C22×C4—82no—
432C22×C4C2×SD1614yes—
433C22×C4C2×SD1614yes—
434C22×C4C2×SD1614yes—
435C22×C4#13714yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4H(5,𝔽2)14yes—
446C22×C4C2511yescharacteristic
447SD16—88no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—84no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—81no—
483C4⋊C4C2×SD1618yes—
484C4⋊C4C2×SD1618yes—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C42—24no—
497C42—24no—
498C42—22no—
499C42—22no—
500C42—24no—
501C42—41no—
502C42—42no—
503C42—41no—
504C42—48no—
505C42—44no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23#53512yes—
518C23C22×SD1612yes—
519C23C2×#13712yes—
520C23C23×D411yescharacteristic
521Q8—28no—
522Q8—48no—
523Q8—44no—
524Q8—816no—
525Q8—162no—
526D4—44no—
527D4—44no—
528D4—88no—
529D4—162no—
530D4—162no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
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—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—164no—
584C2×C4#53512yes—
585C2×C4C22×SD1614yes—
586C2×C4C22×SD1614yes—
587C2×C4C22×SD1612yes—
588C2×C4C22×SD1612yes—
589C2×C4C2×#13712yes—
590C2×C4C2×#13712yes—
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4C2×2−1+412yes—
593C2×C4#58512yes—
594C8—44no—
595C8—44no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22#320614yes—
610C22C2×#53511yescharacteristic
611C22#348511yescharacteristic
612C22#348611yescharacteristic
613C22C23×SD1611yescharacteristic
614C22C22×#13711yescharacteristic
615C22#358511yescharacteristic
616C22#358511yescharacteristic
617C4—24no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—24no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—44no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4#353514yes—
634C2—24no—
635C2—162no—
636C2#3711811yescharacteristic
637C2C2×#320611yescharacteristic
638C2#6080611yescharacteristic
639C2#6169711yescharacteristic
640C2#6170911yescharacteristic
641C2#6182011yescharacteristic
642C2#6183011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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. C2×#2567C2semidirect
3. #33005C2semidirect
4. #32430C2semidirect
5. #32346C2semidirect
6. #31638C2semidirect
7. #31522C2semidirect
8. #29639C2semidirect
9. #29105C2semidirect
11. #28115C2semidirect
12. #27961C2semidirect
14. #27645C2semidirect
15. #21912C2semidirect
298. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#25672
#330052
#324302
#323462
#316382
#315222
#296392
#291052
#283052
#281152
#279612
#279062
#276452
#219122