Finite Group Databasebeta

← #6508305

Subgroups — #6508305

Order 512.

1878 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35006C211yesmaximal, characteristic, complemented
2C2×#2447C211yesmaximal, characteristic, complemented
3#29334C211yesmaximal, characteristic, complemented
4#29302C211yesmaximal, characteristic, complemented
5#29027C214yesmaximal, complemented
6#28893C214yesmaximal, complemented
7#26826C214yesmaximal, complemented
8#26258C214yesmaximal, complemented
9#25864C211yesmaximal, characteristic, complemented
10#25731C211yesmaximal, characteristic
11#24509C212yesmaximal, complemented
12#24506C212yesmaximal, complemented
13#22041C212yesmaximal, complemented
14C4×#447C212yesmaximal, complemented
15#20534C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C22×#124C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#496C2212yes—
20C2×#489C2212yes—
21C2×#448C2212yes—
22C2×#447C2212yes—
23C2×C4×D8C2214yes—
24C2×#432C2212yes—
25#2857C2211yescharacteristic
26#2856C2211yescharacteristic
27#2854C2212yes—
28#2639—21no—
29#2618—21no—
30#2560—22no—
31#2558—22no—
32#2512C2212yescomplemented
33#2506C2212yescomplemented
34#2452C2212yescomplemented
35#2447—22no—
36#2447—22no—
37#2447C2214yes—
38#2447C2214yes—
39#2445C2212yescomplemented
40C2×#397—21no—
41C2×#397—21no—
42#2325—21no—
43C4×#121C2212yes—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2211yescharacteristic
48C2×#385—21no—
49C2×#381—21no—
50C2×C4×C4⋊C4C2211yescharacteristic
51#2100C2212yescomplemented
52#2084C2212yescomplemented
53#2068C2214yescomplemented
54#2065C2214yes—
55#2057C2212yescomplemented
56#2055C2214yescomplemented
57#2054C2214yes—
58#2042C2214yes—
59#2041C2212yescomplemented
60#2012C2214yes—
61#1973C2214yescomplemented
62#1972C2214yes—
63#1968C2214yescomplemented
64#1959C2212yes—
65#1959C2212yes—
66#1944C2214yescomplemented
67#1936C2214yes—
68#1936C2214yescomplemented
69#1935C2214yescomplemented
70#1933C2214yescomplemented
71#1927C2214yescomplemented
72#1879C2214yes—
73#1877C2214yescomplemented
74#1874C2212yes—
75#1874C2212yes—
76#1862C2214yes—
77#1853C2212yes—
78#1852C2212yes—
79#1787C2212yes—
80#1786C2212yes—
81#1771C2214yescomplemented
82#1771C2214yescomplemented
83C4×#102C2214yes—
84#1749C2211yescharacteristic
85#1749C2211yescharacteristic
86C4×C22⋊C8C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C22×D8C2314yes—
92#539—48no—
93C2×C4⋊Q8C2311yescharacteristic
94C2×#127C2311yescharacteristic
95C2×#126—22no—
96C2×#125—22no—
97C2×#124—21no—
98C2×#124—21no—
99C2×#124—44no—
100C2×#124C2312yes—
101C2×#124C2312yes—
102C2×#123—22no—
103C2×#123—22no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—44no—
112C2×C4×D4C2314yes—
113C2×C4×D4C2312yes—
114C22×C4⋊C4—21no—
115C22×C4⋊C4D411yescharacteristic
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#496—44no—
128#489—44no—
129#448—28no—
130#447—28no—
131C4×D8—48no—
132#432—28no—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×#97C2314yes—
147C2×#97C2312yes—
148C2×#97C2312yes—
149C2×C4×C8C2314yes—
150#400—22no—
151#398—22no—
152#397—24no—
153#397—24no—
154#397—22no—
155#397—24no—
156#397—24no—
157#396—22no—
158#396—22no—
159#394—22no—
160#394—22no—
161#393—22no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391C2314yes—
166#391C2314yes—
167#390—21no—
168#390—21no—
169#390—22no—
170#390C2312yes—
171#390C2314yes—
172#390C2312yes—
173#390C2312yes—
174#390C2312yes—
175#389—21no—
176#389—21no—
177#389—21no—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#388—22no—
187#388—22no—
188#387—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#385—24no—
193#385—24no—
194#385—22no—
195#384C2311yescharacteristic
196#384C2311yescharacteristic
197#381—24no—
198#381—24no—
199#380—21no—
200#380C2312yes—
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C22⋊C4—22no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95—21no—
212C2×#95D411yescharacteristic
213C2×#95D411yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2312yes—
219#336C2312yes—
220#336C2314yes—
221C25—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—24no—
225C22×Q8—42no—
226C22×Q8—42no—
227C22×Q8C2412yes—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—24no—
234C22×D4—42no—
235C22×D4C2412yes—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—82no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252C2×SD16—28no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×D8—28no—
256C2×D8—28no—
257C2×D8—44no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260C4⋊Q8—42no—
261#127—42no—
262#126—48no—
263#125—48no—
264#124—24no—
265#124—24no—
266#124—42no—
267#124—42no—
268#124—44no—
269#124—44no—
270#123—48no—
271#123—48no—
272C22⋊Q8—48no—
273C22⋊Q8—44no—
274#121—24no—
275#121—24no—
276#121—44no—
277#121—44no—
278C4×D4—28no—
279C4×D4—44no—
280C4×D4—44no—
281C4×D4—48no—
282C4×D4—48no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
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×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—42no—
336C2×C42—42no—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2412yes—
345#107—44no—
346#106—44no—
347C4⋊C8—28no—
348#103—28no—
349#103—44no—
350#103—44no—
351#102—28no—
352#102—28no—
353#102—44no—
354#102—44no—
355#102—44no—
356C22⋊C8—24no—
357C22⋊C8#4118yes—
358#97—28no—
359#97—42no—
360#97—42no—
361C4×C8—44no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95C2×D412yes—
375#95C2×D412yes—
376#95C2×D412yes—
377#95C2×D412yes—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24—42no—
386C24—42no—
387C24C22×D411yescharacteristic
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—81no—
397C2×Q8—81no—
398C2×Q8—88no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—416no—
408C2×D4—42no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—81no—
416C2×D4—81no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4H(5,𝔽2)14yes—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—84no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—816no—
461C22×C4AGL(1,ℤ/8)14yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C2×#4114yes—
472C22×C4C2511yescharacteristic
473SD16—88no—
474D8—416no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8C2×#4118yes—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—81no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—48no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—82no—
519C22⋊C4—82no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—82no—
523C42—24no—
524C42—22no—
525C42—22no—
526C42—24no—
527C42—22no—
528C42—22no—
529C42—42no—
530C42—41no—
531C42—41no—
532C42—44no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—24no—
538C23—48no—
539C23—48no—
540C23—48no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—84no—
555C23—84no—
556C23—84no—
557C23—84no—
558C23#53212yes—
559C23C2×AGL(1,ℤ/8)12yes—
560C23#57512yes—
561C23C23×D411yescharacteristic
562Q8—44no—
563Q8—44no—
564Q8—84no—
565Q8—162no—
566D4—28no—
567D4—48no—
568D4—44no—
569D4—44no—
570D4—44no—
571D4—88no—
572D4—162no—
573D4—162no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—816no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—164no—
629C2×C4#53212yes—
630C2×C4C2×AGL(1,ℤ/8)14yes—
631C2×C4C22×#4112yes—
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4#58512yes—
634C8—28no—
635C8—44no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—816no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22#317114yes—
658C22C2×#53211yescharacteristic
659C22#348411yescharacteristic
660C22#348711yescharacteristic
661C22C22×AGL(1,ℤ/8)11yescharacteristic
662C22C2×#57511yescharacteristic
663C22#358511yescharacteristic
664C22#358611yescharacteristic
665C4—22no—
666C4—24no—
667C4—42no—
668C4—42no—
669C4—41no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#351314yes—
681C2—24no—
682C2—84no—
683C2—162no—
684C2#3539611yescharacteristic
685C2C2×#317111yescharacteristic
686C2#6070311yescharacteristic
687C2#6156911yescharacteristic
688C2#6157511yescharacteristic
689C2#6188811yescharacteristic
690C2#6189511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#25731C2
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. #35006C2semidirect
2. C2×#2447C2semidirect
3. #29334C2semidirect
4. #29302C2semidirect
5. #29027C2semidirect
6. #28893C2semidirect
7. #26826C2semidirect
8. #26258C2semidirect
9. #25864C2semidirect
11. #24509C2semidirect
12. #24506C2semidirect
13. #22041C2semidirect
14. C4×#447C2semidirect
15. #20534C2semidirect
32. #2512C22semidirect
33. #2506C22semidirect
34. #2452C22semidirect
39. #2445C22semidirect
51. #2100C22semidirect
52. #2084C22semidirect
53. #2068C22semidirect
55. #2057C22semidirect
56. #2055C22semidirect
59. #2041C22semidirect
61. #1973C22semidirect
63. #1968C22semidirect
66. #1944C22semidirect
68. #1936C22semidirect
69. #1935C22semidirect
70. #1933C22semidirect
71. #1927C22semidirect
73. #1877C22semidirect
81. #1771C22semidirect
82. #1771C22semidirect

Maximal subgroups

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

subgroupindex
#350062
C2×#24472
#293342
#293022
#290272
#288932
#268262
#262582
#258642
#257312
#245092
#245062
#220412
C4×#4472
#205342