Finite Group Databasebeta

← #376987

Subgroups — #376987

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1934C212yesmaximal
4#14700C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11267C211yesmaximal, characteristic, complemented
7#11049C211yesmaximal, characteristic
8#10931C212yesmaximal
9#10884C212yesmaximal, complemented
10#10880C212yesmaximal, complemented
11C4×#336C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×C4×C8C2212yes—
17C4×C4⋊Q8—28no—
18C2×#391—21no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2212yes—
25C2×#388—21no—
26C2×#388—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C43C2211yescharacteristic
31#2023C418yescomplemented
32#1934—28no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1437—21no—
43#1437—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4×Q8—28no—
55C2×C4×Q8—44no—
56C2×C4×Q8—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C42—22no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—42no—
86C2×#102C2×C418yes—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—42no—
90#391—28no—
91#390—24no—
92#390—22no—
93#390—24no—
94#390C2×C414yes—
95#389—28no—
96#389—24no—
97#389—24no—
98#389D418yes—
99#388—44no—
100#388—44no—
101#384—24no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C43—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340C2×C418yes—
119#336—22no—
120#336—24no—
121#336C2×C414yes—
122C25—21no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—24no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22×C412yes—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C22×C414yes—
165C22×C8#4114yes—
166C4⋊Q8—216no—
167C4⋊Q8—216no—
168C4×Q8—432no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4#4114yes—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—28no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—41no—
244C2×C42—41no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42D814yes—
248C2×C42SD1614yes—
249C2×C42C22×C412yes—
250C2×C42C2×D414yes—
251C2×C42#4112yes—
252C2×C42#4114yes—
253#102—216no—
254#102—48no—
255C4×C8—48no—
256#95—28no—
257#95—28no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—42no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24C22×D411yescharacteristic
279C2×Q8—216no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—48no—
283C2×Q8—48no—
284C2×Q8—48no—
285C2×Q8—48no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12412yes—
368C22×C4#12412yes—
369C22×C4C2×D812yes—
370C22×C4C2×D812yes—
371C22×C4C2×D812yes—
372C22×C4C2×SD1612yes—
373C22×C4C2×SD1612yes—
374C22×C4C2×SD1612yes—
375C22×C4#13512yes—
376C22×C4#13512yes—
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4#13712yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D412yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4111yescharacteristic
385C22×C4H(5,𝔽2)11yescharacteristic
386C22×C42−1+411yescharacteristic
387C22×C42−1+411yescharacteristic
388C22×C42−1+411yescharacteristic
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—216no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—416no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—84no—
421C4⋊C4C4×D418yes—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—28no—
435C42—28no—
436C42—44no—
437C42—44no—
438C42—416no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23C2×C4×D411yescharacteristic
475C23C2×#12011yescharacteristic
476C23#52511yescharacteristic
477C23C2×#12411yescharacteristic
478C23#53211yescharacteristic
479C23#53611yescharacteristic
480C23#53711yescharacteristic
481C23C22×D811yescharacteristic
482C23C22×SD1611yescharacteristic
483C23C2×#13511yescharacteristic
484C23C2×#13511yescharacteristic
485C23C2×AGL(1,ℤ/8)11yescharacteristic
486C23C2×#13711yescharacteristic
487C23#57511yescharacteristic
488C23#57511yescharacteristic
489Q8—432no—
490D4—44no—
491D4—44no—
492D4—82no—
493D4—82no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—28no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4C4×D814yes—
567C2×C4C4×SD1614yes—
568C2×C4#44714yes—
569C2×C4#44814yes—
570C2×C4#48614yes—
571C2×C4C2×C4×D412yes—
572C2×C4C4×#4112yes—
573C2×C4#51911yescharacteristic
574C2×C4#52011yescharacteristic
575C2×C4#52512yes—
576C2×C4#52912yes—
577C2×C4#54111yescharacteristic
578C2×C4#54211yescharacteristic
579C2×C4#54612yes—
580C2×C4#54712yes—
581C2×C4#54812yes—
582C2×C4#54812yes—
583C2×C4#55212yes—
584C2×C4#55612yes—
585C2×C4#55612yes—
586C2×C4#55912yes—
587C8—44no—
588C8—44no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—161no—
597C22—161no—
598C22—161no—
599C22—161no—
600C22—161no—
601C22—161no—
602C22—161no—
603C22—161no—
604C22C4×#12011yescharacteristic
605C22C4×#12411yescharacteristic
606C22#232311yescharacteristic
607C22#243011yescharacteristic
608C22#243611yescharacteristic
609C22#259211yescharacteristic
610C22#260411yescharacteristic
611C22C2×C4×D811yescharacteristic
612C22C2×C4×SD1611yescharacteristic
613C22C4×AGL(1,ℤ/8)11yescharacteristic
614C22C4×#13711yescharacteristic
615C22C2×#44711yescharacteristic
616C22C2×#44811yescharacteristic
617C22#300211yescharacteristic
618C22#300211yescharacteristic
619C22#310011yescharacteristic
620C22#310311yescharacteristic
621C22#310511yescharacteristic
622C22#310811yescharacteristic
623C22C2×#48611yescharacteristic
624C22#313111yescharacteristic
625C22#314011yescharacteristic
626C22#314111yescharacteristic
627C22#316711yescharacteristic
628C22#316911yescharacteristic
629C22#318011yescharacteristic
630C22#318111yescharacteristic
631C22#322611yescharacteristic
632C22#322711yescharacteristic
633C22#323311yescharacteristic
634C22#323311yescharacteristic
635C22#324511yescharacteristic
636C22#324611yescharacteristic
637C22#324711yescharacteristic
638C22#324811yescharacteristic
639C4—24no—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—42no—
645C4—48no—
646C4—82no—
647C4—84no—
648C4—161no—
649C4—161no—
650C4#236212yes—
651C4#236712yes—
652C4#245812yes—
653C4#246912yes—
654C2—162no—
655C2#1418811yescharacteristic
656C2C4×#44711yescharacteristic
657C2C4×#44811yescharacteristic
658C2C4×#48611yescharacteristic
659C2#2165511yescharacteristic
660C2#2211011yescharacteristic
661C2#2211111yescharacteristic
662C2#2549611yescharacteristic
663C2#2549711yescharacteristic
664C2#2590311yescharacteristic
665C2#2590611yescharacteristic
666C2#2934611yescharacteristic
667C2#2935011yescharacteristic
668C2#2951311yescharacteristic
669C2#2951411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×C4⋊Q8C2semidirect
5. #11384C2semidirect
6. #11267C2semidirect
9. #10884C2semidirect
10. #10880C2semidirect
11. C4×#336C2semidirect
31. #2023C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20232
C2×#19342
#147002
#113842
#112672
#110492
#109312
#108842
#108802
C4×#3362