Finite Group Databasebeta

← #6472810

Subgroups — #6472810

Order 512.

2518 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2616C211yesmaximal, characteristic
2#34175C211yesmaximal, characteristic
3#32989C214yesmaximal, complemented
4#29901C214yesmaximal, complemented
5#27602C212yesmaximal, complemented
6#27600C212yesmaximal, complemented
7#24564C214yesmaximal, complemented
8#24563C214yesmaximal, complemented
9#24010C212yesmaximal, complemented
10#24006C212yesmaximal, complemented
11#21944C211yesmaximal, characteristic, complemented
12#21940C211yesmaximal, characteristic, complemented
13#21765C211yesmaximal, characteristic, complemented
14#21762C211yesmaximal, characteristic, complemented
15#20581C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C2×#520C2212yes—
20C2×#484C2214yescomplemented
21C2×#485C2214yescomplemented
22C2×#408C2214yes—
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25#2616—21no—
26#2616—22no—
27#2616—21no—
28#2616C2212yes—
29#2616C2212yes—
30#2616C2214yes—
31C2×#395—21no—
32C2×#394—21no—
33C2×#394—22no—
34#2341—21no—
35Q8×C22⋊C4—21no—
36#2339—21no—
37#2339—22no—
38D4×C22⋊C4—21no—
39D4×C22⋊C4—22no—
40#2311C2212yes—
41#2311C2212yes—
42#2311C2212yes—
43#2280C2211yescharacteristic
44#2280C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#380C2212yes—
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2075C2214yescomplemented
52#2075C2214yescomplemented
53#2036C2214yescomplemented
54#2036C2214yescomplemented
55#2036C2214yescomplemented
56#2035C2214yescomplemented
57#1894C2212yescomplemented
58#1894C2212yescomplemented
59#1892C2214yescomplemented
60#1873C2214yescomplemented
61#1873C2214yescomplemented
62#1872C2214yescomplemented
63#1872C2214yescomplemented
64#1854C2214yescomplemented
65#1854C2214yescomplemented
66#1853C2212yes—
67#1852C2212yes—
68#1852C2212yes—
69#1852C2212yes—
70#1852C2214yescomplemented
71#1852C2214yescomplemented
72#1825C2214yescomplemented
73#1823C2211yescharacteristic
74#1823C2211yescharacteristic
75#1823C2212yes—
76#1823C2214yescomplemented
77#1809C2214yescomplemented
78#1808C2212yescomplemented
79#1808C2212yescomplemented
80#1787C2211yescharacteristic
81#1787C2211yescharacteristic
82#1787C2212yes—
83#1780C2212yes—
84#1780C2212yes—
85#1779C2212yes—
86#1779C2212yes—
87C22×#41—24no—
88C22×#41—24no—
89C22×#41—48no—
90C22×#41D411yescharacteristic
91C22×#41D412yes—
92C22×#41D411yescharacteristic
93C22×#41C2311yescharacteristic
94C22×#41C2312yes—
95C22×#41C2311yescharacteristic
96C23×Q8D411yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4D412yes—
99C23×D4D411yescharacteristic
100C24×C4C2312yes—
101C24×C4C2311yescharacteristic
102C22×M16C2312yes—
103C23×C8C2312yes—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—48no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2314yes—
115#520—48no—
116C2×C4×Q8—22no—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—22no—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4—22no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—21no—
136#485—48no—
137#484—48no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146#408—48no—
147#407—24no—
148C2×C22⋊C8—24no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2311yescharacteristic
152C2×C22⋊C8C2311yescharacteristic
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158#395—22no—
159#395—24no—
160#395—22no—
161#394—22no—
162#394—24no—
163#394—24no—
164#394—24no—
165#394—24no—
166#394—24no—
167#394—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#390—22no—
172#390—22no—
173#390—22no—
174#386—22no—
175#386—22no—
176#386—22no—
177#386—22no—
178#386—22no—
179#386—22no—
180#381—22no—
181#381—21no—
182#380—22no—
183#380—21no—
184#380—22no—
185#380—21no—
186#380—22no—
187#380—22no—
188#380—21no—
189#380—21no—
190#380C2311yescharacteristic
191#380C2311yescharacteristic
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2314yes—
199#380C2311yescharacteristic
200#380C2312yes—
201#380C2311yescharacteristic
202#380C2314yes—
203#380C2312yes—
204#379—22no—
205#379—21no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—21no—
210C2×#95—22no—
211C2×#95—21no—
212C2×#95—22no—
213C2×#95—21no—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222C25C2×D412yes—
223C25C2×D411yescharacteristic
224C2×#41—28no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—24no—
230C2×#41—24no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—28no—
234C2×#41—28no—
235C2×#41—44no—
236C2×#41—44no—
237C2×#41—44no—
238C2×#41—44no—
239C2×#41—44no—
240C2×#41—44no—
241C2×#41—432no—
242C2×#41—432no—
243C22×Q8—24no—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—44no—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2×D412yes—
251C22×Q8C2×D411yescharacteristic
252C22×Q8C2412yes—
253C22×D4—28no—
254C22×D4—24no—
255C22×D4—44no—
256C22×D4—44no—
257C22×D4—44no—
258C22×D4C2×D411yescharacteristic
259C22×D4C2×D412yes—
260C22×D4C2×D412yes—
261C22×D4C2×D412yes—
262C22×D4C2×D412yes—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C23×C4—24no—
266C23×C4—24no—
267C23×C4—28no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—21no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—44no—
275C23×C4—44no—
276C23×C4C2×D412yes—
277C23×C4C2×D412yes—
278C23×C4C2×D412yes—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4#4114yes—
283C23×C4#4114yes—
284C23×C4C2412yes—
285C23×C4C2411yescharacteristic
286C23×C4C2411yescharacteristic
287C23×C4C2411yescharacteristic
288C23×C4C2412yes—
289C2×M16—24no—
290C2×M16—44no—
291C22×C8—24no—
292C22×C8—44no—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2414yes—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—28no—
301C22⋊Q8—28no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C22⋊Q8—48no—
305C22⋊Q8—48no—
306C22⋊Q8—44no—
307C22⋊Q8—44no—
308C4×Q8—48no—
309C4×D4—48no—
310C4×D4—48no—
311C4×D4—48no—
312#117—28no—
313#117—44no—
314#117—44no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—28no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—48no—
331C2×C4⋊C4—44no—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
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—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4—82no—
362C2×C22⋊C4—84no—
363C2×C22⋊C4—82no—
364C2×C22⋊C4—84no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368#107—44no—
369#107—44no—
370#107—44no—
371#106—44no—
372#103—44no—
373#103—44no—
374#103—44no—
375#103—44no—
376C22⋊C8—22no—
377C22⋊C8—22no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—24no—
381C22⋊C8—28no—
382C22⋊C8—28no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—24no—
392#95—22no—
393#95—22no—
394#95—24no—
395#95—21no—
396#95—22no—
397#95—22no—
398#95—22no—
399C24—22no—
400C24—22no—
401C24—24no—
402C24—24no—
403C24—24no—
404C24—24no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24#12314yes—
411C24C22×D412yes—
412C24C22×D411yescharacteristic
413C24C22×D411yescharacteristic
414C24C22×D412yes—
415C24C22×D411yescharacteristic
416#41—216no—
417#41—216no—
418#41—416no—
419#41—416no—
420#41—48no—
421#41—48no—
422#41—432no—
423#41—416no—
424#41—416no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—416no—
438C2×Q8—42no—
439C2×Q8—84no—
440C2×Q8—84no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—24no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—24no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—416no—
461C2×D4—432no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—416no—
494C22×C4—416no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—82no—
509C22×C4—82no—
510C22×C4—82no—
511C22×C4—82no—
512C22×C4#12314yes—
513C22×C4#12314yes—
514C22×C4#12314yes—
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D412yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C4C2×#4112yes—
523C22×C4C2×#4112yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4112yes—
526C22×C4H(5,𝔽2)18yes—
527C22×C4C2511yescharacteristic
528M16—84no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—24no—
532C2×C8—24no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8—84no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—48no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—48no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—48no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—81no—
568C22⋊C4—81no—
569C22⋊C4—82no—
570C22⋊C4—82no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—84no—
574C22⋊C4—84no—
575C42—44no—
576C42—44no—
577C42—44no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—21no—
589C23—22no—
590C23—21no—
591C23—22no—
592C23—21no—
593C23—21no—
594C23—28no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—44no—
603C23—44no—
604C23—48no—
605C23—48no—
606C23—48no—
607C23—48no—
608C23—84no—
609C23—84no—
610C23—84no—
611C23—84no—
612C23—84no—
613C23C2×#12312yes—
614C23C2×#12312yes—
615C23C2×#12312yes—
616C23#53414yes—
617C23#57612yes—
618C23#57812yes—
619C23C23×D411yescharacteristic
620Q8—28no—
621Q8—48no—
622Q8—44no—
623Q8—44no—
624Q8—42no—
625Q8—42no—
626Q8—84no—
627D4—28no—
628D4—216no—
629D4—48no—
630D4—48no—
631D4—44no—
632D4—48no—
633D4—48no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—28no—
646C2×C4—28no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—48no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—88no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—164no—
682C2×C4—164no—
683C2×C4—164no—
684C2×C4C2×#12312yes—
685C2×C4C2×#12312yes—
686C2×C4C2×#12312yes—
687C2×C4#53414yes—
688C2×C4C22×#4111yescharacteristic
689C2×C4C22×#4111yescharacteristic
690C2×C4C2×H(5,𝔽2)14yes—
691C8—42no—
692C8—42no—
693C8—44no—
694C22—24no—
695C22—24no—
696C22—22no—
697C22—22no—
698C22—24no—
699C22—24no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—24no—
706C22—22no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—88no—
723C22—816no—
724C22#309814yes—
725C22C22×#12311yescharacteristic
726C22C2×#53412yes—
727C22C2×#57611yescharacteristic
728C22C2×#57811yescharacteristic
729C22#358612yes—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—24no—
734C4—42no—
735C4—82no—
736C4—84no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4#349012yes—
743C4#349012yes—
744C2—24no—
745C2—44no—
746C2—42no—
747C2—42no—
748C2—82no—
749C2#3499211yescharacteristic
750C2C2×#309811yescharacteristic
751C2#6051911yescharacteristic
752C2#6075912yes—
753C2#6076212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32989C2semidirect
4. #29901C2semidirect
5. #27602C2semidirect
6. #27600C2semidirect
7. #24564C2semidirect
8. #24563C2semidirect
9. #24010C2semidirect
10. #24006C2semidirect
11. #21944C2semidirect
12. #21940C2semidirect
13. #21765C2semidirect
14. #21762C2semidirect
20. C2×#484C22semidirect
21. C2×#485C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2075C22semidirect
52. #2075C22semidirect
53. #2036C22semidirect
54. #2036C22semidirect
55. #2036C22semidirect
56. #2035C22semidirect
57. #1894C22semidirect
58. #1894C22semidirect
59. #1892C22semidirect
60. #1873C22semidirect
61. #1873C22semidirect
62. #1872C22semidirect
63. #1872C22semidirect
64. #1854C22semidirect
65. #1854C22semidirect
70. #1852C22semidirect
71. #1852C22semidirect
72. #1825C22semidirect
76. #1823C22semidirect
77. #1809C22semidirect
78. #1808C22semidirect
79. #1808C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26162
#341752
#329892
#299012
#276022
#276002
#245642
#245632
#240102
#240062
#219442
#219402
#217652
#217622
#205812