Finite Group Databasebeta

← #6508501

Subgroups — #6508501

Order 512.

1712 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic
2C2×#2453C211yesmaximal, characteristic, complemented
3#29111C211yesmaximal, characteristic, complemented
4#29086C214yesmaximal, complemented
5#28955C214yesmaximal, complemented
6#28314C211yesmaximal, characteristic
7#26841C214yesmaximal, complemented
8#26499C214yesmaximal, complemented
9#25913C211yesmaximal, characteristic
10#25819C211yesmaximal, characteristic, complemented
11#24600C212yesmaximal, complemented
12#24503C212yesmaximal, complemented
13#22081C212yesmaximal, complemented
14#21454C212yesmaximal, complemented
15#20536C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#501C2212yes—
20C2×#490C2212yes—
21C2×#480C2212yes—
22C2×#448C2212yes—
23C2×#440C2214yes—
24C2×#431C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yes—
33#2506C2212yes—
34#2453—22no—
35#2453—22no—
36#2453C2214yes—
37#2453C2214yes—
38#2452C2212yes—
39#2445C2212yes—
40C2×#400—21no—
41C2×#398—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#389—21no—
47C2×#389C2211yescharacteristic
48C2×#389C2211yescharacteristic
49C2×#387—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2095C2212yes—
52#2086C2214yes—
53#2072C2212yes—
54#2067C2214yes—
55#2066C2214yes—
56#2059C2214yes—
57#2055C2214yes—
58#2025C2214yes—
59#2024C2212yes—
60#2014C2212yes—
61#1992C2212yes—
62#1992C2212yes—
63#1973C2214yes—
64#1968C2214yes—
65#1943C2214yes—
66#1942C2214yes—
67#1936C2214yes—
68#1935C2214yes—
69#1934C2214yes—
70#1933C2214yes—
71#1921C2214yes—
72#1880C2214yes—
73#1874C2212yes—
74#1874C2212yes—
75#1864C2214yes—
76#1863C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1823C2212yes—
80#1787C2212yes—
81#1770C2214yes—
82#1770C2214yes—
83#1763C2214yes—
84#1749C2211yescharacteristic
85#1749C2212yes—
86#1749C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C22×SD16C2314yes—
89#539—48no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#125—22no—
94C2×#125—21no—
95C2×#125—21no—
96C2×#125—44no—
97C2×#125C2312yes—
98C2×#125C2312yes—
99C2×#123—22no—
100C2×#123—22no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—22no—
103C2×#121—21no—
104C2×#121—21no—
105C2×#121C2312yes—
106C2×C4×Q8—22no—
107C2×C4×Q8—44no—
108C2×C4×Q8C2314yes—
109C2×C4×D4—22no—
110C2×C4×D4C2312yes—
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#501—44no—
126#490—44no—
127#480—28no—
128#448—28no—
129#440—48no—
130#431—28no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×#97C2314yes—
146C2×C4×C8C2314yes—
147#400—22no—
148#400—24no—
149#400—24no—
150#398—22no—
151#398—24no—
152#398—24no—
153#397—22no—
154#396—22no—
155#396—22no—
156#394—22no—
157#394—22no—
158#393—22no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—22no—
163#391C2314yes—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—24no—
173#389—24no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389C2312yes—
178#389C2312yes—
179#389C2314yes—
180#389C2312yes—
181#389C2312yes—
182#389C2314yes—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—24no—
189#387—24no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#380—21no—
197#380C2312yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D412yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C22⋊C4—22no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C22×Q8—24no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8C2412yes—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4C2412yes—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—22no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C2×SD16—28no—
244C2×SD16—28no—
245C2×SD16—44no—
246C22×C8C2414yes—
247C22×C8C2414yes—
248C4⋊Q8—42no—
249#127—42no—
250#126—48no—
251#125—24no—
252#125—24no—
253#125—42no—
254#125—42no—
255#125—44no—
256#125—44no—
257#125—48no—
258#123—48no—
259#123—48no—
260C22⋊Q8—48no—
261C22⋊Q8—44no—
262#121—24no—
263#121—24no—
264#121—44no—
265#121—44no—
266C4×Q8—28no—
267C4×Q8—44no—
268C4×Q8—48no—
269C4×D4—44no—
270C4×D4—48no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
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—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×C22⋊C4—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—44no—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2412yes—
335#107—28no—
336#107—28no—
337#106—44no—
338C4⋊C8—28no—
339#103—28no—
340#103—28no—
341#103—44no—
342#102—28no—
343#102—28no—
344#102—44no—
345C22⋊C8—24no—
346C22⋊C8#4118yes—
347#97—44no—
348#97—42no—
349#97—42no—
350C4×C8—28no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—22no—
358#95—24no—
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—
370C24C22×D411yescharacteristic
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—416no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—81no—
380C2×Q8—81no—
381C2×Q82−1+414yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—24no—
385C2×D4—24no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—81no—
392C2×D4—81no—
393C2×D4—81no—
394C2×D4—81no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
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—41no—
414C22×C4—41no—
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—44no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4AGL(1,ℤ/8)14yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C2×#4114yes—
447C22×C4C2511yescharacteristic
448SD16—416no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—24no—
454C2×C8C2×#4118yes—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—24no—
464C4⋊C4—22no—
465C4⋊C4—24no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4H(5,𝔽2)14yes—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—82no—
497C22⋊C4—82no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—42no—
505C42—41no—
506C42—41no—
507C42—48no—
508C42—44no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23#53212yes—
521C23C2×AGL(1,ℤ/8)12yes—
522C23#57512yes—
523C23C23×D411yescharacteristic
524Q8—28no—
525Q8—48no—
526Q8—44no—
527Q8—162no—
528D4—44no—
529D4—44no—
530D4—88no—
531D4—162no—
532D4—162no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—48no—
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—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—164no—
591C2×C4#53212yes—
592C2×C4C2×AGL(1,ℤ/8)14yes—
593C2×C4C22×#4112yes—
594C2×C4C2×H(5,𝔽2)12yes—
595C2×C4C2×2−1+412yes—
596C8—28no—
597C8—44no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22#318014yes—
612C22C2×#53211yescharacteristic
613C22#348411yescharacteristic
614C22#348611yescharacteristic
615C22C22×AGL(1,ℤ/8)11yescharacteristic
616C22C2×#57511yescharacteristic
617C22#358611yescharacteristic
618C22#358611yescharacteristic
619C4—24no—
620C4—22no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4#352014yes—
636C2—24no—
637C2—162no—
638C2#3539711yescharacteristic
639C2C2×#318011yescharacteristic
640C2#6070511yescharacteristic
641C2#6158311yescharacteristic
642C2#6158511yescharacteristic
643C2#6182811yescharacteristic
644C2#6183211yescharacteristic
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×#2453C2semidirect
3. #29111C2semidirect
4. #29086C2semidirect
5. #28955C2semidirect
7. #26841C2semidirect
8. #26499C2semidirect
10. #25819C2semidirect
11. #24600C2semidirect
12. #24503C2semidirect
13. #22081C2semidirect
14. #21454C2semidirect
15. #20536C2semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#24532
#291112
#290862
#289552
#283142
#268412
#264992
#259132
#258192
#246002
#245032
#220812
#214542
#205362