Finite Group Databasebeta

← #6376069

Subgroups — #6376069

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C211yesmaximal, characteristic
2#34220C211yesmaximal, characteristic, complemented
3C2×#2319C211yesmaximal, characteristic, complemented
4#22157C214yesmaximal, complemented
5#22126C214yesmaximal, complemented
6#22082C212yesmaximal, complemented
7#22079C212yesmaximal, complemented
8#22049C212yesmaximal, complemented
9#22040C212yesmaximal
10#21391C211yesmaximal, characteristic, complemented
11#21358C211yesmaximal, characteristic, complemented
12#21285C212yesmaximal, complemented
13#21282C212yesmaximal, complemented
14#21136C212yesmaximal, complemented
15#21129C212yesmaximal, complemented
16#20738C211yesmaximal, characteristic, complemented
17#20469C211yesmaximal, characteristic, complemented
18C2×#529C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3105C4116yescomplemented
21C2×#484C2211yescharacteristic
22C2×#481C2211yescharacteristic
23C2×#478C2212yes—
24C2×#463C2212yes—
25C2×#450C2211yescharacteristic
26C2×#447C2211yescharacteristic
27C2×C4×SD16C2212yes—
28C2×C4×D8C2212yes—
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2373—22no—
32#2359—22no—
33#2353—21no—
34#2349—21no—
35#2346—21no—
36#2343—21no—
37#2330C2212yescomplemented
38#2326C2212yescomplemented
39#2320C2211yescharacteristic, complemented
40#2319—21no—
41#2319—21no—
42#2319—22no—
43#2319C2212yes—
44#2319C2214yes—
45#2319C2212yes—
46#2319C2211yescharacteristic, complemented
47C4×C22⋊Q8C2212yes—
48C4×#121C2212yes—
49C2×#390C2212yes—
50C2×#389C2212yes—
51C2×#388—22no—
52#2285C2211yescharacteristic
53C2×#381—21no—
54C2×#380—21no—
55#2276C2211yescharacteristic, complemented
56#2275C2211yescharacteristic, complemented
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59#1989C2212yescomplemented
60#1983C2212yescomplemented
61#1973C2214yes—
62#1970C2214yes—
63#1947C2214yescomplemented
64#1944C2214yescomplemented
65#1942C2214yescomplemented
66#1936C2214yes—
67#1936C2214yescomplemented
68#1934C2214yes—
69#1933C2214yescomplemented
70#1925C2214yes—
71#1925C2214yes—
72#1921C2214yescomplemented
73#1919C2214yescomplemented
74#1880C2214yes—
75#1879C2214yes—
76#1874C2212yes—
77#1872C2212yes—
78#1863C2214yescomplemented
79#1862C2214yescomplemented
80#1852C2212yes—
81#1852C2212yes—
82#1824C2212yes—
83#1802C2214yes—
84#1786C2212yes—
85C4×#107C2212yescomplemented
86C4×#107C2212yescomplemented
87#1768C2214yescomplemented
88#1763C2212yescomplemented
89#1762C2212yes—
90#1762C2212yes—
91#1762C2212yescomplemented
92#1749C2212yes—
93C4×C22⋊C8C2212yes—
94#1746C2212yescomplemented
95#1726C2212yescomplemented
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×D8C2312yes—
100#529—24no—
101#529C2×C418yes—
102C2×#126—21no—
103C2×#125C2311yescharacteristic
104C2×#124C2311yescharacteristic
105C2×#123—21no—
106C2×C22⋊Q8C2311yescharacteristic
107C2×#121C2311yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4—22no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4C2×C418yescomplemented
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×#117—21no—
121C2×#117—21no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—22no—
127C22×C22⋊C4—22no—
128C22×C42—21no—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#484C2×C418yescomplemented
132#481C2×C418yes—
133#478C2×C4116yescomplemented
134#463C2×C4116yescomplemented
135#450C2×C418yescomplemented
136#447C2×C418yes—
137C4×SD16C2×C4116yescomplemented
138C4×D8C2×C4116yescomplemented
139#432—24no—
140#431C2×C418yescomplemented
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2311yescharacteristic
158C2×#97C2312yes—
159C2×#97C2314yes—
160C2×C4×C8C2314yes—
161C2×C4×C8C2312yes—
162#391C2312yes—
163#390—22no—
164#390—22no—
165#390C2314yes—
166#390C2314yes—
167#390C2312yes—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389C2314yes—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—24no—
178#388—24no—
179#388—24no—
180#388—24no—
181#388—22no—
182#388—22no—
183#388—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#386—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384C2312yes—
193#383—21no—
194#382—21no—
195#382—21no—
196#381—22no—
197#381—22no—
198#381—24no—
199#381—21no—
200#380—21no—
201#380—22no—
202#380—22no—
203#380—24no—
204#380—21no—
205#380C2312yes—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222#376—21no—
223#376D411yescharacteristic
224#376D411yescharacteristic
225#376D412yes—
226#376C2311yescharacteristic
227#376C2312yes—
228#376C2311yescharacteristic
229C2×#95—21no—
230C2×#95—22no—
231C2×#95—21no—
232C2×#95D411yescharacteristic
233C2×#95D411yescharacteristic
234C2×#95D412yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C25—21no—
244C22×Q8C2411yescharacteristic
245C22×D4—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—44no—
249C22×D4C22×C414yes—
250C22×D4C2411yescharacteristic
251C22×D4C2412yes—
252C23×C4—21no—
253C23×C4—28no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—22no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—84no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D411yescharacteristic
264C23×C4C2411yescharacteristic
265C2×SD16—44no—
266C2×SD16C22×C418yes—
267C2×D8—44no—
268C2×D8C22×C418yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272#126—28no—
273#125—22no—
274#125C22×C414yes—
275#124—22no—
276#124C22×C414yes—
277#123—28no—
278C22⋊Q8—22no—
279C22⋊Q8C22×C414yes—
280#121—22no—
281#121C22×C414yes—
282C4×Q8—22no—
283C4×Q8—44no—
284C4×Q8C22×C414yes—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—22no—
291C4×D4—44no—
292C4×D4—84no—
293C4×D4C22×C418yes—
294C4×D4C22×C418yes—
295C4×D4C22×C414yes—
296#117—28no—
297#117—82no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—84no—
314C2×C4⋊C4C22×C414yes—
315C2×C4⋊C4C2414yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C22⋊C4—21no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—84no—
338C2×C22⋊C4—84no—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—21no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42—82no—
349C2×C42C22×C414yes—
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42#4114yes—
355C2×C42C2412yes—
356C2×C42C2412yes—
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#107—44no—
360#107—44no—
361#107C22×C418yes—
362#106C22×C418yes—
363C4⋊C8—24no—
364C4⋊C8C22×C418yes—
365#103—44no—
366#103C22×C418yes—
367#102—44no—
368#102—44no—
369#102—44no—
370#102C22×C418yes—
371#102C22×C418yes—
372#102C22×C418yes—
373C22⋊C8—24no—
374C22⋊C8C22×C414yes—
375C22⋊C8C22×C414yes—
376#97—24no—
377#97—44no—
378C4×C8—44no—
379C4×C8C22×C418yes—
380#95—21no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—24no—
385#95—24no—
386#95—21no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D411yescharacteristic
394#95C2×D411yescharacteristic
395#95C2×D412yes—
396#95C2×D411yescharacteristic
397#95C2×D412yes—
398#95C2×D411yescharacteristic
399#95C2×D412yes—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403C24—24no—
404C24—21no—
405C24—22no—
406C24—44no—
407C24—44no—
408C24—42no—
409C24—42no—
410C24C22×D411yescharacteristic
411C2×Q8—21no—
412C2×Q8—24no—
413C2×Q8C23×C412yes—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—24no—
421C2×D4—48no—
422C2×D4—44no—
423C2×D4—48no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4C23×C414yes—
427C2×D4C23×C412yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—816no—
460C22×C4C4×D414yes—
461C22×C4C4×D414yes—
462C22×C4C23×C412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4112yes—
473C22×C4C2511yescharacteristic
474SD16—48no—
475D8—48no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—24no—
481C2×C8C23×C414yes—
482C2×C8C23×C414yes—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—82no—
498C4⋊C4—82no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4C23×C412yes—
504C4⋊C4C23×C412yes—
505C4⋊C4C23×C414yes—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C22⋊C4—24no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—82no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—82no—
521C42—24no—
522C42—24no—
523C42—24no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—82no—
528C42—82no—
529C42—82no—
530C42—81no—
531C42—81no—
532C42—82no—
533C42C4×D414yes—
534C42C4×D414yes—
535C42C23×C412yes—
536C42C23×C412yes—
537C42C2×#4112yes—
538C42C2×#4112yes—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—48no—
544C23—48no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—48no—
550C23—42no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—88no—
555C23—88no—
556C23C2×C4×D412yes—
557C23#57612yes—
558C23#57812yes—
559C23C23×D411yescharacteristic
560Q8—24no—
561Q8—42no—
562D4—24no—
563D4—24no—
564D4—24no—
565D4—48no—
566D4—48no—
567D4—42no—
568D4—44no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—88no—
617C2×C4—164no—
618C2×C4C2×C4×D412yes—
619C2×C4C2×C4×D412yes—
620C2×C4C2×C4×D412yes—
621C2×C4C2×C4×D412yes—
622C2×C4C2×C4×D412yes—
623C2×C4C24×C411yescharacteristic
624C2×C4C22×#4111yescharacteristic
625C2×C4#58514yes—
626C8—24no—
627C8—24no—
628C8—44no—
629C22—22no—
630C22—22no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—44no—
638C22—42no—
639C22—816no—
640C22—82no—
641C22—84no—
642C22—82no—
643C22—84no—
644C22#295114yes—
645C22C22×C4×D411yescharacteristic
646C22#344912yes—
647C22C2×#57611yescharacteristic
648C22C2×#57811yescharacteristic
649C22#358512yes—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4#342812yes—
667C4#345512yes—
668C2—24no—
669C2—82no—
670C2—84no—
671C2#3406511yescharacteristic
672C2C2×#295111yescharacteristic
673C2#6016811yescharacteristic
674C2#6097812yes—
675C2#6098112yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34220C2semidirect
3. C2×#2319C2semidirect
4. #22157C2semidirect
5. #22126C2semidirect
6. #22082C2semidirect
7. #22079C2semidirect
8. #22049C2semidirect
10. #21391C2semidirect
11. #21358C2semidirect
12. #21285C2semidirect
13. #21282C2semidirect
14. #21136C2semidirect
15. #21129C2semidirect
16. #20738C2semidirect
17. #20469C2semidirect
20. #3105C4semidirect
37. #2330C22semidirect
38. #2326C22semidirect
39. #2320C22semidirect
46. #2319C22semidirect
55. #2276C22semidirect
56. #2275C22semidirect
59. #1989C22semidirect
60. #1983C22semidirect
63. #1947C22semidirect
64. #1944C22semidirect
65. #1942C22semidirect
67. #1936C22semidirect
69. #1933C22semidirect
72. #1921C22semidirect
73. #1919C22semidirect
78. #1863C22semidirect
79. #1862C22semidirect
85. C4×#107C22semidirect
86. C4×#107C22semidirect
87. #1768C22semidirect
88. #1763C22semidirect
91. #1762C22semidirect
94. #1746C22semidirect
95. #1726C22semidirect
115. C2×C4×D4C2×C4semidirect
131. #484C2×C4semidirect
133. #478C2×C4semidirect
134. #463C2×C4semidirect
135. #450C2×C4semidirect
137. C4×SD16C2×C4semidirect
138. C4×D8C2×C4semidirect
140. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31052
#342202
C2×#23192
#221572
#221262
#220822
#220792
#220492
#220402
#213912
#213582
#212852
#212822
#211362
#211292
#207382
#204692