Finite Group Databasebeta

← #6800409

Subgroups — #6800409

Order 512.

1479 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36551C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic, complemented
3#33148C211yesmaximal, characteristic, complemented
4#32326C214yesmaximal, complemented
5#31470C214yesmaximal, complemented
6#29972C212yesmaximal, complemented
7#28798C212yesmaximal, complemented
8#27966C212yesmaximal, complemented
9#27960C211yesmaximal, characteristic, complemented
10#27735C212yesmaximal, complemented
11#27644C212yesmaximal
12#27644C211yesmaximal, characteristic
13#26512C212yesmaximal, complemented
14#26110C212yesmaximal, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22081C212yesmaximal
17C4×#480C211yesmaximal, characteristic
18C22×#125—21no—
19C2×#480C2212yes—
20C2×#480C2211yescharacteristic
21#2857C2211yescharacteristic
22#2851C2212yes—
23#2589—22no—
24#2563C2211yescharacteristic
25#2552—22no—
26#2552—21no—
27#2552—21no—
28#2552C2214yes—
29#2552C2212yes—
30#2552C2212yes—
31#2552C2211yescharacteristic
32#2509C2212yes—
33#2497—21no—
34#2487—21no—
35#2454C2212yes—
36#2450—22no—
37C2×#398—22no—
38#2369C2211yescharacteristic
39D4×C4⋊C4—21no—
40D4×C4⋊C4C2211yescharacteristic
41#2345—21no—
42C4×#121C2212yes—
43C4×#121C2211yescharacteristic
44C2×#389—21no—
45C2×#389C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2100C2214yes—
52#2100C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2083C2212yes—
57#2083C2214yes—
58#2071C2214yes—
59#2071C2212yes—
60#2059C2214yes—
61#1992C2214yes—
62#1990C2212yes—
63#1987C2212yes—
64#1948C2212yes—
65#1947C2214yes—
66#1947C2212yes—
67#1943C2214yes—
68#1939C2212yes—
69#1936C2214yes—
70#1925C2212yes—
71#1921C2214yes—
72#1920C2214yes—
73#1918C2212yes—
74#1917C2214yes—
75#1874C2211yescharacteristic
76#1864C2214yes—
77#1853C2212yes—
78#1852C2211yescharacteristic
79#1836C2214yes—
80#1834C2212yes—
81#1823C2212yes—
82#1823C2212yes—
83#1823C2212yes—
84#1823C2211yescharacteristic
85#1823C2212yes—
86#1802C2212yes—
87#1802C2214yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90C4×#107C2212yes—
91#1770C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—22no—
97C2×#125—22no—
98C2×#125—24no—
99C2×#125—48no—
100C2×#125C2312yes—
101C2×C22⋊Q8—21no—
102C2×#121C2311yescharacteristic
103C2×C4×D4—21no—
104C2×C4×D4—22no—
105C2×C4×D4C2311yescharacteristic
106C2×C4×D4C2312yes—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—22no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C42—21no—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#480—28no—
122#480—24no—
123C2×#107C2312yes—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#102C2314yes—
130C2×#102C2312yes—
131C2×#102C2312yes—
132C2×C22⋊C8C2311yescharacteristic
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2311yescharacteristic
135C2×#97C2314yes—
136C2×C4×C8C2312yes—
137#400—22no—
138#400—22no—
139#399—22no—
140#398—22no—
141#398—24no—
142#398—24no—
143#398—24no—
144#398—24no—
145#396—22no—
146#393—22no—
147#391—21no—
148#391—22no—
149#390C2312yes—
150#390C2311yescharacteristic
151#389—22no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—24no—
156#389—22no—
157#389—21no—
158#389—22no—
159#389—22no—
160#389C2314yes—
161#389C2311yescharacteristic
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#388—21no—
168#388—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#386—21no—
173#386—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384—21no—
181#384—22no—
182#384—22no—
183#384C2312yes—
184#384C2314yes—
185#384C2314yes—
186#384C2312yes—
187#384C2311yescharacteristic
188#382C2312yes—
189#380—21no—
190#380—21no—
191#380—21no—
192#380—21no—
193#380—22no—
194#380C2311yescharacteristic
195#380C2312yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—21no—
211C2×#95—21no—
212C2×#95—22no—
213C2×#95D412yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×D4—21no—
233C22×D4C2411yescharacteristic
234C23×C4—22no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—22no—
239C23×C4C2×D412yes—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D412yes—
243C23×C4C2411yescharacteristic
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2414yes—
247C4⋊Q8—24no—
248#125—44no—
249#125—416no—
250#125—48no—
251#125—48no—
252C22⋊Q8—82no—
253#121—22no—
254#121#4114yes—
255C4×D4—24no—
256C4×D4—44no—
257C4×D4—44no—
258C4×D4—84no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—84no—
294C2×C4⋊C4—82no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C42—24no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2412yes—
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2411yescharacteristic
333#107—24no—
334#107—28no—
335#107—42no—
336#106—44no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339#102—28no—
340#102—24no—
341#102—42no—
342C22⋊C8—22no—
343C22⋊C8—24no—
344C22⋊C8—22no—
345#97—44no—
346C4×C8—24no—
347#95—22no—
348#95—22no—
349#95—24no—
350#95—24no—
351#95—21no—
352#95—22no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C24—21no—
367C24C22×D411yescharacteristic
368C2×Q8—42no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4C2×#4112yes—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4AGL(1,ℤ/8)14yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4112yes—
425C22×C42−1+412yes—
426C22×C4C2511yescharacteristic
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—82no—
466C4⋊C4—84no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—81no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4#13714yes—
476C4⋊C4C2×#4112yes—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—44no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—24no—
490C42—24no—
491C42—42no—
492C42—42no—
493C42—44no—
494C42—44no—
495C42—42no—
496C42—82no—
497C422−1+414yes—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23#54812yes—
510C23C2×AGL(1,ℤ/8)12yes—
511C23#57612yes—
512C23C23×D411yescharacteristic
513Q8—162no—
514D4—24no—
515D4—48no—
516D4—42no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—48no—
542C2×C4—44no—
543C2×C4—48no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—88no—
570C2×C4—84no—
571C2×C4—164no—
572C2×C4#53612yes—
573C2×C4#53612yes—
574C2×C4#54812yes—
575C2×C4C2×AGL(1,ℤ/8)12yes—
576C2×C4C2×AGL(1,ℤ/8)12yes—
577C2×C4C2×#13714yes—
578C2×C4C2×#13712yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C2×2−1+412yes—
581C2×C4C2×2−1+411yescharacteristic
582C2×C4#58512yes—
583C8—42no—
584C8—44no—
585C8—42no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22#336614yes—
600C22C2×#53611yescharacteristic
601C22C2×#54811yescharacteristic
602C22#348511yescharacteristic
603C22C22×AGL(1,ℤ/8)11yescharacteristic
604C22C22×#13711yescharacteristic
605C22C2×#57611yescharacteristic
606C22#358611yescharacteristic
607C4—24no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—81no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—84no—
621C4—81no—
622C4—82no—
623C4#323814yes—
624C4#351512yes—
625C4#356312yes—
626C2—24no—
627C2—82no—
628C2#3912211yescharacteristic
629C2C2×#323811yescharacteristic
630C2#6087111yescharacteristic
631C2C2×#336611yescharacteristic
632C2#6132211yescharacteristic
633C2#6171711yescharacteristic
634C2#6172611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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×#2552C2semidirect
3. #33148C2semidirect
4. #32326C2semidirect
5. #31470C2semidirect
6. #29972C2semidirect
7. #28798C2semidirect
8. #27966C2semidirect
9. #27960C2semidirect
10. #27735C2semidirect
13. #26512C2semidirect
14. #26110C2semidirect
15. #22629C2semidirect

Maximal subgroups

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

subgroupindex
#365512
C2×#25522
#331482
#323262
#314702
#299722
#287982
#279662
#279602
#277352
#276442
#265122
#261102
#226292
#220812
C4×#4802