Finite Group Databasebeta

← #6629281

Subgroups — #6629281

Order 512.

4401 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#3108C212yesmaximal, complemented
3#36621C211yesmaximal, characteristic
4#33008C212yesmaximal, complemented
5#33003C212yesmaximal, complemented
6#32982C211yesmaximal, characteristic, complemented
7#32978C211yesmaximal, characteristic
8#32960C214yesmaximal, complemented
9#32909C214yesmaximal, complemented
10#32882C211yesmaximal, characteristic, complemented
11#32879C211yesmaximal, characteristic
12C4×#451C214yesmaximal, complemented
13C4×#448C214yesmaximal, complemented
14#20735C212yesmaximal, complemented
15#20714C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C2×#529C2211yescharacteristic
18C22×C4×Q8—22no—
19C22×C4×Q8—21no—
20C22×C4×Q8—22no—
21C22×C4×Q8—21no—
22C22×C4×Q8—216no—
23C22×C4×Q8C2214yes—
24C22×C4×Q8C2214yes—
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2212yes—
27C23×C4⋊C4—21no—
28C23×C4⋊C4C2211yescharacteristic
29C23×C42—21no—
30C23×C42C2211yescharacteristic
31#3108—216no—
32C2×#483C2212yes—
33C2×#480C2212yes—
34C2×#479C2214yescomplemented
35C2×#464C2214yescomplemented
36C2×#451C2212yes—
37C2×#451C2214yes—
38C2×#448C2212yes—
39C2×#448C2214yes—
40C2×C4×Q16C2214yescomplemented
41C2×C4×SD16C2214yescomplemented
42C2×#431C2212yes—
43#2855C2211yescharacteristic
44#2854C2211yescharacteristic
45#2853—21no—
46#2852—22no—
47#2851C2212yes—
48#2850C2212yes—
49#2849—21no—
50#2848C2211yescharacteristic
51#2847C2211yescharacteristic
52C4×C22⋊Q8C2212yes—
53C4×#123—21no—
54C4×#121C2212yes—
55C4×#120—21no—
56#2290—22no—
57#2283C2211yescharacteristic
58#1880C2218yescomplemented
59#1878C2218yescomplemented
60#1874C2214yes—
61#1872C2214yes—
62#1865C2218yescomplemented
63#1863C2218yescomplemented
64#1853C2214yes—
65#1852C2214yes—
66#1823C2212yes—
67#1823C2212yes—
68#1817C2212yes—
69#1805C2218yescomplemented
70#1797C2214yes—
71#1787C2214yes—
72#1787C2212yes—
73#1787C2214yes—
74#1787C2212yes—
75#1779C2211yescharacteristic
76#1779C2211yescharacteristic
77C4×#103C2218yescomplemented
78C4×C22⋊C8C2214yes—
79C23×Q8—28no—
80C23×Q8—216no—
81C23×Q8—22no—
82C23×Q8—21no—
83C23×Q8C2312yes—
84C23×Q8C2314yes—
85C24×C4—21no—
86C24×C4—21no—
87C24×C4D411yescharacteristic
88C24×C4D411yescharacteristic
89C24×C4C2311yescharacteristic
90C22×Q16C2314yes—
91C22×SD16C2314yes—
92#529—28no—
93C2×#126—21no—
94C2×#125C2311yescharacteristic
95C2×#124C2311yescharacteristic
96C2×#123—21no—
97C2×#123—21no—
98C2×C22⋊Q8C2311yescharacteristic
99C2×C22⋊Q8C2312yes—
100C2×#121C2311yescharacteristic
101C2×#121C2312yes—
102C2×#120—21no—
103C2×C4×Q8—216no—
104C2×C4×Q8—232no—
105C2×C4×Q8—232no—
106C2×C4×Q8—216no—
107C2×C4×Q8—28no—
108C2×C4×Q8—232no—
109C2×C4×Q8—232no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—24no—
116C2×C4×Q8—22no—
117C2×C4×Q8—24no—
118C2×C4×Q8—22no—
119C2×C4×Q8D4116yescomplemented
120C2×C4×Q8C2318yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2314yes—
123C2×C4×Q8C2314yes—
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C22×C4⋊C4—216no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4—24no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4—28no—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2312yes—
138C22×C42—216no—
139C22×C42—24no—
140C22×C42—22no—
141C22×C42—24no—
142C22×C42—22no—
143C22×C42—28no—
144C22×C42D412yes—
145C22×C42D411yescharacteristic
146C22×C42D412yes—
147C22×C42D411yescharacteristic
148C22×C42C2312yes—
149C22×C42C2311yescharacteristic
150C22×C42C2312yes—
151C22×C42C2311yescharacteristic
152#483—28no—
153#480—28no—
154#479—216no—
155#464—216no—
156#451—28no—
157#451—216no—
158#448—28no—
159#448—216no—
160C4×Q16—216no—
161C4×SD16—216no—
162#431—28no—
163C2×#107C2314yes—
164C2×#106C2314yes—
165C2×C4⋊C8C2314yes—
166C2×#103C2314yes—
167C2×#103C2314yes—
168C2×#103C2318yes—
169C2×#103C2318yes—
170C2×#103C2314yes—
171C2×#102C2314yes—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2314yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2312yes—
176C2×C4×C8C2314yes—
177#391C2312yes—
178#390C2312yes—
179#389C2312yes—
180#389C2312yes—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#386—22no—
190#386—21no—
191#386—22no—
192#386—22no—
193#386—22no—
194#386—21no—
195#385—22no—
196#385—21no—
197#385—22no—
198#385—22no—
199#385—22no—
200#385—21no—
201#383—21no—
202#382C2312yes—
203#381—21no—
204#381—21no—
205#380C2311yescharacteristic
206#380C2311yescharacteristic
207#380C2312yes—
208#380C2312yes—
209#379—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4C2312yes—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215#376—21no—
216#340C2318yes—
217#340C2318yes—
218#336C2318yes—
219C25C2×D411yescharacteristic
220C22×Q8—216no—
221C22×Q8—28no—
222C22×Q8—2128no—
223C22×Q8—216no—
224C22×Q8—216no—
225C22×Q8—216no—
226C22×Q8—216no—
227C22×Q8—216no—
228C22×Q8—216no—
229C22×Q8—24no—
230C22×Q8—22no—
231C22×Q8—24no—
232C22×Q8—28no—
233C22×Q8C2×D418yes—
234C22×Q8#41116yes—
235C22×Q8C2411yescharacteristic
236C22×Q8C2414yes—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4C2411yescharacteristic
240C23×C4—28no—
241C23×C4—216no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—24no—
245C23×C4—28no—
246C23×C4—24no—
247C23×C4—28no—
248C23×C4—28no—
249C23×C4—24no—
250C23×C4—24no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—21no—
254C23×C4—42no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D412yes—
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C23×C4C2412yes—
269C2×Q16—28no—
270C2×Q16—216no—
271C2×SD16—28no—
272C2×SD16—216no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275#126—44no—
276#125—24no—
277#124—24no—
278#123—44no—
279#123—44no—
280C22⋊Q8—24no—
281C22⋊Q8—28no—
282#121—24no—
283#121—28no—
284#120—44no—
285C4×Q8—232no—
286C4×Q8—232no—
287C4×Q8—232no—
288C4×Q8—232no—
289C4×Q8—232no—
290C4×Q8—216no—
291C4×Q8—216no—
292C4×Q8—216no—
293C4×Q8—216no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8C2×D4116yes—
298C4×Q8C2×D4116yes—
299C4×D4—24no—
300C4×D4—44no—
301C4×D4—48no—
302#117—44no—
303C2×C4⋊C4—216no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—232no—
306C2×C4⋊C4—232no—
307C2×C4⋊C4—216no—
308C2×C4⋊C4—216no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—216no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4C2×D418yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C42—216no—
343C2×C42—232no—
344C2×C42—232no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—216no—
348C2×C42—24no—
349C2×C42—28no—
350C2×C42—24no—
351C2×C42—24no—
352C2×C42—28no—
353C2×C42—28no—
354C2×C42—28no—
355C2×C42—28no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D418yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369#107—28no—
370#106—28no—
371C4⋊C8—28no—
372#103—28no—
373#103—28no—
374#103—216no—
375#103—216no—
376#103—28no—
377#102—28no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—28no—
381C22⋊C8—28no—
382C4×C8—28no—
383#95—21no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394C24—22no—
395C24—24no—
396C24—24no—
397C24—22no—
398C24—21no—
399C24#13514yes—
400C24C22×D411yescharacteristic
401C24C22×D412yes—
402C2×Q8—216no—
403C2×Q8—264no—
404C2×Q8—2128no—
405C2×Q8—232no—
406C2×Q8—264no—
407C2×Q8—216no—
408C2×Q8—216no—
409C2×Q8—232no—
410C2×Q8—264no—
411C2×Q8—28no—
412C2×Q8—216no—
413C2×Q8—24no—
414C2×Q8—28no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8C22×D418yes—
418C2×Q8C2×#41116yes—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—42no—
425C2×D4—44no—
426C2×D4—42no—
427C22×C4—216no—
428C22×C4—232no—
429C22×C4—264no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—232no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—216no—
455C22×C4—22no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—416no—
472C22×C4—432no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—44no—
479C22×C4#12014yes—
480C22×C4#12014yes—
481C22×C4#13718yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2×#4118yes—
492C22×C4C2511yescharacteristic
493Q16—416no—
494SD16—416no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—28no—
498C2×C8—28no—
499C4⋊C4—232no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4C22×D418yes—
517C22⋊C4—44no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—42no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C42—232no—
531C42—216no—
532C42—216no—
533C42—24no—
534C42—24no—
535C42—24no—
536C42—24no—
537C42—24no—
538C42—24no—
539C42—42no—
540C42—44no—
541C42#12014yes—
542C42#12014yes—
543C42C22×D414yes—
544C42C22×D414yes—
545C23—28no—
546C23—28no—
547C23—216no—
548C23—216no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—28no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—42no—
560C23—41no—
561C23C2×#12012yes—
562C23#52514yes—
563C23C2×#13514yes—
564C23C2×#13512yes—
565C23C2×#13714yes—
566C23#57512yes—
567C23C23×D411yescharacteristic
568Q8—232no—
569Q8—264no—
570Q8—28no—
571Q8—216no—
572Q8—44no—
573Q8#525116yes—
574D4—44no—
575D4—88no—
576D4—84no—
577C2×C4—216no—
578C2×C4—28no—
579C2×C4—232no—
580C2×C4—264no—
581C2×C4—216no—
582C2×C4—28no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—216no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—216no—
598C2×C4—28no—
599C2×C4—216no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—432no—
607C2×C4—44no—
608C2×C4—416no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—84no—
625C2×C4C2×#12012yes—
626C2×C4C2×#12012yes—
627C2×C4C2×#12012yes—
628C2×C4C2×#12012yes—
629C2×C4C2×#12012yes—
630C2×C4#52514yes—
631C2×C4C2×#13714yes—
632C2×C4C2×#13714yes—
633C2×C4C23×D412yes—
634C2×C4C22×#4114yes—
635C8—44no—
636C8—44no—
637C22—232no—
638C22—22no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—22no—
647C22—24no—
648C22—81no—
649C22—82no—
650C22—82no—
651C22—81no—
652C22—81no—
653C22#300214yes—
654C22#310818yescomplemented
655C22C22×#12011yescharacteristic
656C22C2×#52512yes—
657C22C22×#13511yescharacteristic
658C22C22×#13712yes—
659C22C2×#57511yescharacteristic
660C4—216no—
661C4—22no—
662C4—24no—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—24no—
667C4—42no—
668C4—48no—
669C4—81no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—81no—
674C4#300414yes—
675C4C2×#52514yes—
676C2—28no—
677C2—24no—
678C2—161no—
679C2#3661411yescharacteristic
680C2C2×#300211yescharacteristic
681C2C2×#300411yescharacteristic
682C2C2×#310812yes—
683C2#6056012yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×Q8C2semidirect
2. C2×#3108C2semidirect
4. #33008C2semidirect
5. #33003C2semidirect
6. #32982C2semidirect
8. #32960C2semidirect
9. #32909C2semidirect
10. #32882C2semidirect
12. C4×#451C2semidirect
13. C4×#448C2semidirect
14. #20735C2semidirect
15. #20714C2semidirect
34. C2×#479C22semidirect
35. C2×#464C22semidirect
40. C2×C4×Q16C22semidirect
41. C2×C4×SD16C22semidirect
58. #1880C22semidirect
59. #1878C22semidirect
62. #1865C22semidirect
63. #1863C22semidirect
69. #1805C22semidirect
77. C4×#103C22semidirect
119. C2×C4×Q8D4semidirect
654. C22#3108semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Q82
C2×#31082
#366212
#330082
#330032
#329822
#329782
#329602
#329092
#328822
#328792
C4×#4512
C4×#4482
#207352
#207142