Finite Group Databasebeta

← #6466545

Subgroups — #6466545

Order 512.

2004 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3137C212yesmaximal
2#35275C211yesmaximal, characteristic, complemented
3#34276C211yesmaximal, characteristic
4#28730C214yesmaximal, complemented
5#28729C214yesmaximal, complemented
6#28726C212yesmaximal, complemented
7#28723C212yesmaximal, complemented
8#28720C212yesmaximal, complemented
9#28717C212yesmaximal, complemented
10#23630C211yesmaximal, characteristic
11#23629C211yesmaximal, characteristic, complemented
12#23627C211yesmaximal, characteristic
13C2×#2072C211yesmaximal, characteristic, complemented
14#21326C212yesmaximal, complemented
15#20652C212yesmaximal
16#20651C212yesmaximal, complemented
17#20430C211yesmaximal, characteristic, complemented
18C2×#544C2212yes—
19C2×#536C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#519C2211yescharacteristic
22#3137—216no—
23C2×#491C2214yes—
24C2×#489C2214yes—
25C2×#488C2214yes—
26C2×#487C2214yes—
27C2×#424C2212yes—
28C2×#417C2212yes—
29C2×#419C2212yes—
30C2×#416C2212yes—
31C22×#103C2212yes—
32C2×#405C2212yes—
33#2868C2212yes—
34#2859C2212yes—
35#2660C2214yes—
36#2552—21no—
37#2552—21no—
38#2551—21no—
39#2551—21no—
40#2440—22no—
41#2440—22no—
42#2362—21no—
43#2362—21no—
44#2360—21no—
45#2360—21no—
46#2359—22no—
47#2359—22no—
48#2327C2212yes—
49#2326C2212yes—
50#2317C2211yescharacteristic
51#2317C2212yes—
52C4×C4⋊Q8C2212yes—
53C4×#124C2212yes—
54#2297C2211yescharacteristic
55C2×#391C2211yescharacteristic
56#2285C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2074C2212yes—
60#2074C2214yes—
61#2073C2212yes—
62#2072C2211yescharacteristic
63#2072C2212yes—
64#2072C2212yes—
65#2072C2212yes—
66#2072C2212yes—
67#2072C2211yescharacteristic
68#2072C2212yes—
69#2072C2212yes—
70#2072C2214yes—
71#2071C2211yescharacteristic
72#2071C2212yes—
73#2071C2211yescharacteristic
74#2071C2212yes—
75#1993C2214yes—
76#1992C2214yes—
77#1991C2214yes—
78#1990C2214yes—
79#1805C2214yes—
80#1805C2214yes—
81#1805C2214yes—
82#1804C2214yes—
83#1803C2214yes—
84#1803C2214yes—
85#1803C2214yes—
86#1802C2214yes—
87#1768C2214yes—
88#1767C2212yes—
89C4×C4⋊C8C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#544—28no—
101#544—28no—
102#536—24no—
103#536D418yes—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—22no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2314yes—
109C2×#125—22no—
110C2×#125C2314yes—
111C2×#124C2312yes—
112C2×#123—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—24no—
116#520—28no—
117#519—28no—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×D4—22no—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117—22no—
127C2×#117C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#491—216no—
138#489—216no—
139#488—216no—
140#487—216no—
141#424—28no—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144#419—28no—
145#417—28no—
146#416—28no—
147C2×#103—28no—
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158#405—28no—
159C2×#97C2314yes—
160C2×C4×C8C2314yes—
161#398—24no—
162#398—24no—
163#395—24no—
164#395—24no—
165#391—21no—
166#391—21no—
167#391C2311yescharacteristic
168#391C2312yes—
169#391C2312yes—
170#391C2311yescharacteristic
171#390C2311yescharacteristic
172#390C2311yescharacteristic
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#386—21no—
192#386—21no—
193#385—22no—
194#385—24no—
195#385—22no—
196#385—22no—
197#384—22no—
198#384—22no—
199#384—21no—
200#384—21no—
201#384C2312yes—
202#384C2314yes—
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2311yescharacteristic
206#384C2312yes—
207#382C2314yes—
208#382C2312yes—
209#382C2312yes—
210#380—21no—
211#380—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4D411yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226#376C2312yes—
227C2×#95—21no—
228C2×#95—21no—
229C43C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2312yes—
240#336C2311yescharacteristic
241#336C2311yescharacteristic
242#336C2312yes—
243#336C2312yes—
244#336C2312yes—
245C2×#41—22no—
246C2×#41—24no—
247C2×#41—24no—
248C2×#41C2×D414yes—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—22no—
252C22×Q8—24no—
253C22×Q8—24no—
254C22×Q8C2×D414yes—
255C22×Q8C2412yes—
256C22×Q8C2411yescharacteristic
257C22×D4—21no—
258C22×D4C2411yescharacteristic
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—21no—
263C23×C4—22no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—28no—
277C4⋊Q8—24no—
278C4⋊Q8—28no—
279C4⋊Q8—48no—
280C4⋊Q8—84no—
281C4⋊Q8—84no—
282C4⋊Q8C2×D418yes—
283#125—28no—
284#125—28no—
285#125—48no—
286#124—24no—
287#124C2×D418yes—
288#123—216no—
289C22⋊Q8—28no—
290C22⋊Q8—28no—
291C22⋊Q8—416no—
292C4×Q8—28no—
293C4×Q8—28no—
294C4×Q8—28no—
295C4×Q8—44no—
296C4×Q8—44no—
297C4×D4—28no—
298C4×D4—48no—
299#117—24no—
300#117—24no—
301#117—24no—
302#117—44no—
303#117—48no—
304#117C2×D414yes—
305#117#4114yes—
306#117#4118yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—82no—
334C2×C4⋊C4—82no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—22no—
362C2×C42C2×D412yes—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2412yes—
372C4⋊C8—28no—
373C4⋊C8—28no—
374#103—28no—
375#103—28no—
376#103—24no—
377#103—24no—
378#103—28no—
379#103—28no—
380#103—28no—
381#102—24no—
382#102—24no—
383#102—28no—
384#97—28no—
385C4×C8—28no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—21no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—24no—
399#95—24no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—21no—
405C24C22×D411yescharacteristic
406#41—216no—
407#41—48no—
408C2×Q8—24no—
409C2×Q8—22no—
410C2×Q8—21no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—48no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—48no—
420C2×Q8—84no—
421C2×Q8—84no—
422C2×Q8C22×D414yes—
423C2×Q8C22×D412yes—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4C22×D412yes—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—42no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D412yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4114yes—
472C22×C42−1+414yes—
473C22×C4C2511yescharacteristic
474M16—48no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—48no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—42no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—42no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—82no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4#12114yes—
517C4⋊C4#12114yes—
518C4⋊C4#12414yes—
519C4⋊C4#12414yes—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—81no—
529C22⋊C4#12118yes—
530C22⋊C4#12418yes—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—42no—
538C42—42no—
539C42—44no—
540C42—42no—
541C42—44no—
542C42—44no—
543C42—44no—
544C42C22×D414yes—
545C42C2×#4114yes—
546C42C2×#4118yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×#12112yes—
559C23C2×#12412yes—
560C23#53612yes—
561C23C23×D411yescharacteristic
562Q8—28no—
563Q8—24no—
564Q8—44no—
565Q8—42no—
566Q8—48no—
567Q8—168no—
568D4—24no—
569D4—48no—
570D4—42no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—164no—
618C2×C4—164no—
619C2×C4—162no—
620C2×C4C2×#12114yes—
621C2×C4C2×#12114yes—
622C2×C4C2×#12112yes—
623C2×C4C2×#12414yes—
624C2×C4C2×#12414yes—
625C2×C4C2×#12412yes—
626C2×C4#53612yes—
627C2×C4#54718yes—
628C2×C4#55618yes—
629C2×C4#57712yes—
630C2×C4#57812yes—
631C2×C4C23×D411yescharacteristic
632C2×C4C22×#4112yes—
633C2×C4C22×#4111yescharacteristic
634C2×C4C2×2−1+412yes—
635C8—42no—
636C8—42no—
637C8—44no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22#254714yes—
652C22C22×#12111yescharacteristic
653C22C22×#12411yescharacteristic
654C22C2×#53611yescharacteristic
655C22C2×#57711yescharacteristic
656C22C2×#57811yescharacteristic
657C22#358511yescharacteristic
658C22#358611yescharacteristic
659C4—24no—
660C4—24no—
661C4—22no—
662C4—22no—
663C4—24no—
664C4—24no—
665C4—42no—
666C4—44no—
667C4—41no—
668C4—41no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4#254714yes—
676C4C2×#54712yes—
677C4C2×#55612yes—
678C2—24no—
679C2—82no—
680C2C2×#254711yescharacteristic
681C2#6043111yescharacteristic
682C2#6043211yescharacteristic
683C2#6062311yescharacteristic
684C2#6062511yescharacteristic
685C2#6087211yescharacteristic
686C2#6087411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35275C2semidirect
4. #28730C2semidirect
5. #28729C2semidirect
6. #28726C2semidirect
7. #28723C2semidirect
8. #28720C2semidirect
9. #28717C2semidirect
11. #23629C2semidirect
13. C2×#2072C2semidirect
14. #21326C2semidirect
16. #20651C2semidirect
17. #20430C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31372
#352752
#342762
#287302
#287292
#287262
#287232
#287202
#287172
#236302
#236292
#236272
C2×#20722
#213262
#206522
#206512
#204302