Finite Group Databasebeta

← #6457223

Subgroups — #6457223

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3097C212yesmaximal
2#34268C211yesmaximal, characteristic
3#34183C211yesmaximal, characteristic
4#27657C214yesmaximal, complemented
5#27656C214yesmaximal, complemented
6#27652C212yesmaximal, complemented
7#27652C212yesmaximal, complemented
8#27648C212yesmaximal, complemented
9#27647C212yesmaximal, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23656C211yesmaximal, characteristic, complemented
12#23629C211yesmaximal, characteristic, complemented
13C2×#2072C211yesmaximal, characteristic, complemented
14#21175C212yesmaximal, complemented
15#20703C212yesmaximal
16#20651C212yesmaximal, complemented
17#20630C211yesmaximal, characteristic
18C2×#536C2211yescharacteristic
19C2×#520C2212yes—
20C2×#519C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22#3097—216no—
23C2×#484C2214yes—
24C2×#485C2214yes—
25C2×#485C2214yes—
26C2×#483C2214yes—
27C2×#428C2212yes—
28C2×#427C2212yes—
29C2×#424C2212yes—
30C2×#419C2212yes—
31C22×#103C2212yes—
32C2×#408C2212yes—
33#2367—21no—
34#2367—21no—
35#2365—21no—
36#2365—21no—
37#2359—22no—
38#2359—22no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4—21no—
41D4×C4⋊C4—22no—
42D4×C4⋊C4—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45#2323C2212yes—
46#2323C2212yes—
47#2317C2214yes—
48#2317C2212yes—
49#2311C2211yescharacteristic
50C4×C22⋊Q8C2212yes—
51C4×C22⋊Q8C2212yes—
52#2296C2211yescharacteristic
53C2×#391C2211yescharacteristic
54#2285C2212yes—
55#2285C2211yescharacteristic
56C2×#384C2212yes—
57C2×#384C2211yescharacteristic
58#2274C2211yescharacteristic
59#2085C2212yes—
60#2085C2212yes—
61#2084C2211yescharacteristic
62#2084C2212yes—
63#2084C2212yes—
64#2084C2211yescharacteristic
65#2083C2211yescharacteristic
66#2083C2212yes—
67#2083C2211yescharacteristic
68#2083C2212yes—
69#2074C2214yes—
70#2072C2212yes—
71#2072C2212yes—
72#2072C2212yes—
73#2072C2212yes—
74#2072C2214yes—
75#1949C2214yes—
76#1949C2214yes—
77#1948C2214yes—
78#1947C2214yes—
79#1825C2214yes—
80#1825C2214yes—
81#1824C2214yes—
82#1823C2214yes—
83#1805C2214yes—
84#1805C2214yes—
85#1803C2214yes—
86#1803C2214yes—
87#1798C2214yes—
88#1797C2212yes—
89#1797C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2311yescharacteristic
97C23×Q8C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#536—24no—
101#536D418yes—
102C2×C4⋊Q8—21no—
103C2×#124—21no—
104C2×#123—22no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107#520—28no—
108#520—28no—
109#519—28no—
110C4×#41—28no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8D411yescharacteristic
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4—24no—
119C2×C4×D4—22no—
120C2×C4×D4D412yes—
121C2×C4×D4D412yes—
122C2×C4×D4D411yescharacteristic
123C2×C4×D4D411yescharacteristic
124C2×#117—22no—
125C2×#117C2311yescharacteristic
126C2×#117C2314yes—
127C2×#117C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C22×C4⋊C4—21no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#485—216no—
141#485—216no—
142#484—216no—
143#483—216no—
144#428—28no—
145#427—28no—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#106C2312yes—
150C2×#106C2314yes—
151C2×#106C2312yes—
152#424—28no—
153C2×C4⋊C8C2314yes—
154C2×C4⋊C8C2314yes—
155#419—28no—
156C2×#103—28no—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161#408—28no—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164#391—21no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#389—24no—
171#389—24no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389C2311yescharacteristic
177#389C2311yescharacteristic
178#389C2311yescharacteristic
179#389C2311yescharacteristic
180#388—22no—
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—21no—
190#385—21no—
191#385—22no—
192#385—22no—
193#385—24no—
194#385—24no—
195#385—24no—
196#384—22no—
197#384—22no—
198#384—21no—
199#384—21no—
200#384C2314yes—
201#384C2312yes—
202#384C2312yes—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382C2314yes—
206#382C2312yes—
207#380—21no—
208#380—21no—
209#380—22no—
210#380—22no—
211#380C2312yes—
212#380C2312yes—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C2×#95—21no—
229C2×#95—21no—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2312yes—
244#336C2312yes—
245C2×#41—24no—
246C2×#41—24no—
247C2×#41C2×D414yes—
248C2×#41#4114yes—
249C22×Q8—21no—
250C22×Q8—22no—
251C22×Q8—24no—
252C22×Q8—24no—
253C22×Q8C2×D414yes—
254C22×Q8C2×D411yescharacteristic
255C22×Q8C2412yes—
256C22×D4C2×D412yes—
257C22×D4C2×D411yescharacteristic
258C23×C4—21no—
259C23×C4—24no—
260C23×C4—22no—
261C23×C4—21no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D412yes—
266C23×C4C2412yes—
267C23×C4C2411yescharacteristic
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C4⋊Q8—28no—
278#124—28no—
279#123—216no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8C2×D418yes—
283C22⋊Q8C2×D418yes—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—48no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—416no—
296C4×D4—48no—
297#117—22no—
298#117—28no—
299#117—24no—
300#117—28no—
301#117—24no—
302#117—28no—
303#117—24no—
304#117—48no—
305#117C2×D414yes—
306#117#4118yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—84no—
334C2×C4⋊C4—82no—
335C2×C4⋊C4—84no—
336C2×C4⋊C4—82no—
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2414yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—21no—
361C2×C42—24no—
362C2×C42—28no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2412yes—
377#107—24no—
378#107—24no—
379#107—28no—
380#106—24no—
381#106—24no—
382#106—28no—
383C4⋊C8—28no—
384C4⋊C8—28no—
385#103—28no—
386#103—28no—
387#103—28no—
388#103—28no—
389C22⋊C8—28no—
390C22⋊C8—28no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95—22no—
404#95—22no—
405#95—21no—
406#95—21no—
407C24C22×D412yes—
408C24C22×D411yescharacteristic
409#41—216no—
410#41—216no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—216no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—48no—
421C2×Q8#12114yes—
422C2×Q8C22×D414yes—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4#12118yes—
428C2×D4#12114yes—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—41no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—82no—
459C22×C4—84no—
460C22×C4—82no—
461C22×C4—84no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C42−1+414yes—
478C22×C4C2511yescharacteristic
479M16—48no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—48no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4#12314yes—
525C4⋊C4#12314yes—
526C4⋊C4C22×D412yes—
527C4⋊C4C22×D412yes—
528C22⋊C4—24no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—48no—
534C22⋊C4—44no—
535C22⋊C4—42no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4#12318yes—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—48no—
547C42—44no—
548C42—42no—
549C42—44no—
550C42—44no—
551C42—82no—
552C42—82no—
553C42C2×#4118yes—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23C2×#12114yes—
565C23C2×#12112yes—
566C23C2×#12312yes—
567C23#53612yes—
568C23C23×D411yescharacteristic
569Q8—28no—
570Q8—28no—
571Q8—44no—
572Q8—48no—
573D4—216no—
574D4—28no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—164no—
626C2×C4—168no—
627C2×C4—162no—
628C2×C4C2×#12114yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12314yes—
631C2×C4C2×#12314yes—
632C2×C4C2×#12312yes—
633C2×C4#53612yes—
634C2×C4#54718yes—
635C2×C4#55218yes—
636C2×C4#57712yes—
637C2×C4#57812yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4112yes—
640C2×C4C22×#4111yescharacteristic
641C2×C4C2×2−1+412yes—
642C8—42no—
643C8—42no—
644C8—44no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22#248614yes—
659C22C22×#12111yescharacteristic
660C22C22×#12311yescharacteristic
661C22C2×#53611yescharacteristic
662C22C2×#57711yescharacteristic
663C22C2×#57811yescharacteristic
664C22#358511yescharacteristic
665C22#358611yescharacteristic
666C4—22no—
667C4—24no—
668C4—22no—
669C4—24no—
670C4—24no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—41no—
675C4—44no—
676C4—84no—
677C4—82no—
678C4—82no—
679C4—81no—
680C4—81no—
681C4—82no—
682C4#248614yes—
683C4C2×#54712yes—
684C4C2×#55212yes—
685C2—24no—
686C2—44no—
687C2C2×#248611yescharacteristic
688C2#6043111yescharacteristic
689C2#6043211yescharacteristic
690C2#6052211yescharacteristic
691C2#6052611yescharacteristic
692C2#6087811yescharacteristic
693C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #27657C2semidirect
5. #27656C2semidirect
6. #27652C2semidirect
7. #27652C2semidirect
8. #27648C2semidirect
9. #27647C2semidirect
10. #23658C2semidirect
11. #23656C2semidirect
12. #23629C2semidirect
13. C2×#2072C2semidirect
14. #21175C2semidirect
16. #20651C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30972
#342682
#341832
#276572
#276562
#276522
#276482
#276472
#236582
#236562
#236292
C2×#20722
#211752
#207032
#206512
#206302