Finite Group Databasebeta

← #6393471

Subgroups — #6393471

Order 512.

3298 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3538C211yesmaximal, characteristic
2C2×#2929C211yesmaximal, characteristic
3#34360C211yesmaximal, characteristic
4#22491C214yesmaximal, complemented
5#22486C214yesmaximal, complemented
6#22425C214yesmaximal, complemented
7D4×#103C214yesmaximal, complemented
8#21898C212yesmaximal, complemented
9#21771C212yesmaximal, complemented
10#21765C212yesmaximal, complemented
11#21413C212yesmaximal, complemented
12#20974C212yesmaximal, complemented
13C2×#1892C212yesmaximal, complemented
14C22×#131C2211yescharacteristic
15C24×C8C2211yescharacteristic
16#3538—28no—
17#3538C4116yescomplemented
18C2×#563—22no—
19C2×#552C2214yes—
20C2×#551—22no—
21C2×D4×Q8C2214yes—
22C2×#544—22no—
23C2×#541C2212yes—
24C2×#539—22no—
25C2×#536C2212yes—
26C2×#525C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28#2929—28no—
29C2×#435C2214yes—
30C2×C8×D4C2214yes—
31C2×#424C2211yescharacteristic
32C2×#419C2212yescomplemented
33C22×#103C2212yes—
34C22×#103C2212yes—
35C22×#103C2212yescomplemented
36C2×#408C2211yescharacteristic
37C2×#407C2211yescharacteristic
38C2×#405C2211yescharacteristic
39#2376—22no—
40#2358—22no—
41#2352C2214yes—
42D4×C4⋊C4C2214yes—
43#2329—22no—
44#2325—22no—
45#2314C2212yes—
46#2311C2212yes—
47C2×#380C2212yes—
48#1992C2212yescomplemented
49#1991C2212yescomplemented
50#1989C2214yescomplemented
51#1981C2214yescomplemented
52#1960C2212yescomplemented
53#1956C2212yescomplemented
54#1894C2214yes—
55#1894C2214yes—
56#1894C2214yescomplemented
57#1892C2214yes—
58#1892C2214yes—
59#1892C2214yes—
60#1892C2214yes—
61#1892C2214yescomplemented
62#1873C2214yescomplemented
63#1872C2214yes—
64#1872C2214yes—
65#1872C2214yescomplemented
66#1825C2214yescomplemented
67#1823C2214yes—
68#1805C2214yescomplemented
69#1803C2214yescomplemented
70#1787C2214yes—
71#1787C2214yescomplemented
72#1763C2214yescomplemented
73C4×#103C2214yescomplemented
74#1736C2212yescomplemented
75#1735C2212yescomplemented
76C2×#340C2212yes—
77C2×#340C2212yes—
78C22×#41C2311yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4C2311yescharacteristic
82C2×#131—24no—
83C2×#131—42no—
84C2×#131—42no—
85C22×M16C2312yes—
86C22×M16C2311yescharacteristic
87C22×M16C2311yescharacteristic
88C23×C8—24no—
89C23×C8—42no—
90C23×C8—42no—
91C23×C8C2311yescharacteristic
92C23×C8C2312yes—
93C23×C8C2312yes—
94C23×C8C2312yes—
95C23×C8C2311yescharacteristic
96#563—232no—
97#552—216no—
98#552C2×C4132yescomplemented
99#551—232no—
100D4×Q8—216no—
101D4×Q8C2×C4132yescomplemented
102#544—232no—
103#541—28no—
104#541C2×C4116yescomplemented
105#539—232no—
106#536—28no—
107#536C2×C4116yescomplemented
108C2×C4⋊Q8—22no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8C2312yes—
111C2×#126—24no—
112C2×#126—24no—
113C2×#125—24no—
114C2×#125—24no—
115C2×#125C2312yes—
116C2×#124—22no—
117#525C2×C418yes—
118#525D418yes—
119C2×#123—22no—
120C2×#123—22no—
121C2×#123—24no—
122C2×#123D412yes—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—28no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8—24no—
128C2×C22⋊Q8C2×C4116yescomplemented
129C2×C22⋊Q8D412yes—
130C2×C22⋊Q8C2314yes—
131C2×C22⋊Q8C2314yes—
132C2×C22⋊Q8C2314yes—
133C2×#121D412yes—
134C2×#120D412yes—
135C2×C4×Q8C2314yes—
136C2×C4×D4—24no—
137C2×C4×D4—24no—
138C2×C4×D4C2314yes—
139C2×#117—24no—
140C2×#117C2312yes—
141C2×#117C2312yes—
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—22no—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C22⋊C4—22no—
149C22×C22⋊C4—22no—
150#435—216no—
151C8×D4—216no—
152#424—24no—
153C2×C4⋊C8C2314yes—
154#419—44no—
155C2×#103—44no—
156C2×#103—44no—
157C2×#103—44no—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166#408—24no—
167#407—24no—
168C2×C22⋊C8C2314yes—
169C2×C22⋊C8C2314yes—
170#405—24no—
171C2×#97C2312yes—
172C2×C4×C8C2312yes—
173#391—22no—
174#390—22no—
175#389—24no—
176#388—24no—
177#388—24no—
178#387—24no—
179#387—24no—
180#386—22no—
181#386—22no—
182#386—24no—
183#385—24no—
184#385—24no—
185#384C2312yes—
186#382C2312yes—
187#380—22no—
188#380—22no—
189#380C2314yes—
190#380C2314yes—
191#380C2314yes—
192C4×C4⋊C4C2314yes—
193C4×C22⋊C4—24no—
194C2×#95—22no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203C25C2×D411yescharacteristic
204C2×#41—41no—
205C2×#41—41no—
206C2×#41—41no—
207C2×#41—41no—
208C2×#41C22×C414yes—
209C2×#41C2×D414yes—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8—42no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—42no—
222C22×Q8C22×C418yes—
223C22×Q8C22×C418yes—
224C22×Q8C2×D411yescharacteristic
225C22×Q8C2414yes—
226C22×D4C2×D414yes—
227C22×D4C2×D412yes—
228C22×D4C2×D411yescharacteristic
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—41no—
236C23×C4—41no—
237C23×C4—41no—
238C23×C4—41no—
239C23×C4C22×C414yes—
240C23×C4C2×D412yes—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246#131—416no—
247C2×M16—22no—
248C2×M16—24no—
249C2×M16—22no—
250C2×M16—24no—
251C2×M16—41no—
252C2×M16—41no—
253C2×M16—42no—
254C2×M16—41no—
255C2×M16—41no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—24no—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8—42no—
267C22×C8—42no—
268C22×C8—41no—
269C22×C8—41no—
270C22×C8—42no—
271C22×C8—416no—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2411yescharacteristic
276C22×C8C2411yescharacteristic
277C4⋊Q8—216no—
278C4⋊Q8—232no—
279C4⋊Q8—24no—
280C4⋊Q8C22×C418yes—
281#126—232no—
282#126—232no—
283#125—232no—
284#125—232no—
285#125—24no—
286#125C22×C418yes—
287#124—216no—
288#123—216no—
289#123—216no—
290#123—232no—
291#123C22⋊C4116yes—
292C22⋊Q8—216no—
293C22⋊Q8—216no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—232no—
298C22⋊Q8—232no—
299C22⋊Q8C22⋊C4116yes—
300C22⋊Q8C22×C4116yes—
301C22⋊Q8C22×C4116yes—
302C22⋊Q8C22×C4116yes—
303#121C22⋊C4116yes—
304#120C22⋊C4116yes—
305C4×Q8—28no—
306C4×Q8C22×C4116yes—
307C4×D4—232no—
308C4×D4—88no—
309C4×D4C22×C4116yes—
310C4×D4C2×D4116yes—
311#117—24no—
312#117—232no—
313#117—42no—
314#117—42no—
315#117C22×C418yes—
316#117C22×C414yes—
317#117C2×D414yes—
318C2×C4⋊C4—216no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—42no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4C22×C418yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2414yes—
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2414yes—
343C2×C22⋊C4—216no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—84no—
353C2×C22⋊C4C2×D414yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D414yes—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42C2412yes—
361C4⋊C8—28no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#103—44no—
369#103—44no—
370C22⋊C8—28no—
371C22⋊C8—28no—
372#97—24no—
373C4×C8—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379C24—24no—
380C24—24no—
381C24—41no—
382C24—41no—
383C24C2×C22⋊C414yes—
384C24C22×D411yescharacteristic
385C24C22×D412yes—
386C24C22×D412yes—
387#41—44no—
388#41—44no—
389#41—44no—
390#41—44no—
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—82no—
408C2×Q8—84no—
409C2×Q8—84no—
410C2×Q8—84no—
411C2×Q8—84no—
412C2×Q8C2×C22⋊C414yes—
413C2×Q8C23×C418yes—
414C2×D4—28no—
415C2×D4—28no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4C2×C22⋊C4116yes—
421C2×D4C2×C22⋊C418yes—
422C2×D4C2×C22⋊C414yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4C2×C22⋊C418yes—
465C22×C4C2×C22⋊C414yes—
466C22×C4C23×C414yes—
467C22×C4C23×C414yes—
468C22×C4C23×C412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C2511yescharacteristic
477M16—48no—
478M16—44no—
479M16—44no—
480C2×C8—28no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—48no—
495C2×C8—48no—
496C2×C8—48no—
497C2×C8—44no—
498C2×C8H(5,𝔽2)12yes—
499C2×C8H(5,𝔽2)12yes—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—28no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4C2×C22⋊C418yes—
517C4⋊C4C2×C22⋊C4116yes—
518C4⋊C4C23×C418yes—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4—28no—
522C22⋊C4—216no—
523C22⋊C4—48no—
524C22⋊C4—84no—
525C22⋊C4—82no—
526C22⋊C4—84no—
527C22⋊C4C2×C22⋊C4116yes—
528C22⋊C4C2×C22⋊C418yes—
529C22⋊C4C2×C22⋊C4116yes—
530C42—28no—
531C42—216no—
532C42—82no—
533C42—84no—
534C42C23×C414yes—
535C42C22×D414yes—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—216no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—22no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—48no—
553C23C22×C22⋊C412yes—
554C23C22×C22⋊C414yes—
555C23C22×C22⋊C414yes—
556C23#57714yes—
557C23#57814yes—
558C23C23×D411yescharacteristic
559Q8—28no—
560Q8—28no—
561Q8—44no—
562Q8—44no—
563Q8—44no—
564Q8—88no—
565Q8—88no—
566D4—232no—
567D4—44no—
568D4—48no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—48no—
592C2×C4—44no—
593C2×C4—416no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—88no—
605C2×C4—88no—
606C2×C4—164no—
607C2×C4—164no—
608C2×C4—164no—
609C2×C4C22×C22⋊C418yes—
610C2×C4C22×C22⋊C414yes—
611C2×C4C22×C22⋊C414yes—
612C2×C4C22×C22⋊C412yes—
613C2×C4C24×C411yescharacteristic
614C2×C4C23×D411yescharacteristic
615C2×C4C2×H(5,𝔽2)11yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4#58511yescharacteristic
618C2×C4#58511yescharacteristic
619C8—22no—
620C8—22no—
621C8—42no—
622C8—42no—
623C8—42no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—28no—
631C22—24no—
632C22—22no—
633C22—24no—
634C22—416no—
635C22—42no—
636C22—41no—
637C22—42no—
638C22—41no—
639C22—44no—
640C22—42no—
641C22#289818yes—
642C22C23×C22⋊C411yescharacteristic
643C22#349611yescharacteristic
644C22#349611yescharacteristic
645C22C2×#57712yes—
646C22C2×#57812yes—
647C4—28no—
648C4—22no—
649C4—24no—
650C4—24no—
651C4—48no—
652C4—48no—
653C4—84no—
654C4—84no—
655C4C4×H(5,𝔽2)11yescharacteristic
656C4C4×H(5,𝔽2)11yescharacteristic
657C4#348411yescharacteristic
658C4#348511yescharacteristic
659C2—28no—
660C2—44no—
661C2—42no—
662C2#3425611yescharacteristic
663C2C2×#289812yes—
664C2#6226811yescharacteristic
665C2#6227111yescharacteristic
666C2#6228112yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #22491C2semidirect
5. #22486C2semidirect
6. #22425C2semidirect
7. D4×#103C2semidirect
8. #21898C2semidirect
9. #21771C2semidirect
10. #21765C2semidirect
11. #21413C2semidirect
12. #20974C2semidirect
13. C2×#1892C2semidirect
17. #3538C4semidirect
32. C2×#419C22semidirect
35. C22×#103C22semidirect
48. #1992C22semidirect
49. #1991C22semidirect
50. #1989C22semidirect
51. #1981C22semidirect
52. #1960C22semidirect
53. #1956C22semidirect
56. #1894C22semidirect
61. #1892C22semidirect
62. #1873C22semidirect
65. #1872C22semidirect
66. #1825C22semidirect
68. #1805C22semidirect
69. #1803C22semidirect
71. #1787C22semidirect
72. #1763C22semidirect
73. C4×#103C22semidirect
74. #1736C22semidirect
75. #1735C22semidirect
98. #552C2×C4semidirect
101. D4×Q8C2×C4semidirect
104. #541C2×C4semidirect
107. #536C2×C4semidirect
128. C2×C22⋊Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#35382
C2×#29292
#343602
#224912
#224862
#224252
D4×#1032
#218982
#217712
#217652
#214132
#209742
C2×#18922