Finite Group Databasebeta

← #360504

Subgroups — #360504

Order 512.

3318 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊C8C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3C2×#1852C212yesmaximal, complemented
4C2×#1808C212yesmaximal, complemented
5#14680C211yesmaximal, characteristic
6#11096C214yesmaximal, complemented
7#10730C212yesmaximal, complemented
8#10721C212yesmaximal, complemented
9C25×C4C2211yescharacteristic
10C24×C8C2212yes—
11C23×C4⋊C4C2212yes—
12C23×C22⋊C4—21no—
13C22×#107C2214yescomplemented
14C22×C22⋊C8—28no—
15C22×C22⋊C8C418yescomplemented
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#380—28no—
20C2×#380—21no—
21C2×#380—21no—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#379—21no—
29C22×#95—21no—
30#1852—28no—
31#1808—28no—
32C2×#340C2214yescomplemented
33C2×#340C2214yescomplemented
34C2×#340C2214yescomplemented
35C22⋊#95—21no—
36C22⋊#95—21no—
37C22⋊#95—21no—
38C26D411yescharacteristic
39C24×C4—24no—
40C24×C4—24no—
41C24×C4—24no—
42C24×C4—24no—
43C24×C4—22no—
44C24×C4—44no—
45C24×C4C2×C414yes—
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4Q814yes—
50C24×C4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C23×C8—24no—
54C23×C8—24no—
55C23×C8—28no—
56C23×C8—44no—
57C23×C8C2×C418yescomplemented
58C23×C8C2312yes—
59C23×C8C2312yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—28no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—44no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2314yes—
68C22×C22⋊C4—28no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—44no—
76C22×C22⋊C4—44no—
77C2×#107—28no—
78C2×#107—28no—
79C2×#107—44no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—28no—
82C2×C22⋊C8—232no—
83C2×C22⋊C8C2×C414yes—
84C2×C22⋊C8C2×C414yes—
85C2×C22⋊C8C2×C418yescomplemented
86C2×C22⋊C8D418yes—
87#380—28no—
88#380—28no—
89#380—24no—
90#380—28no—
91#380—24no—
92#380—24no—
93#380—24no—
94#380—24no—
95#380—44no—
96#380D418yes—
97#379—44no—
98C2×#95—28no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105#340—28no—
106#340—28no—
107#340—44no—
108C25—22no—
109C25—24no—
110C25—22no—
111C25—24no—
112C25—22no—
113C25—22no—
114C25—42no—
115C25—42no—
116C25C4⋊C414yes—
117C25D814yes—
118C25Q1614yes—
119C25C2×D411yescharacteristic
120C25C2×D411yescharacteristic
121C25C2×D411yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—22no—
131C23×C4—28no—
132C23×C4—28no—
133C23×C4—216no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—216no—
139C23×C4—216no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—48no—
152C23×C4—48no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—416no—
157C23×C4—416no—
158C23×C4—416no—
159C23×C4—416no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4C4⋊C414yes—
167C23×C4C4⋊C414yes—
168C23×C4C4⋊C414yes—
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D414yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D414yes—
177C23×C4C2×Q812yes—
178C23×C4C2×Q812yes—
179C23×C4C2×Q812yes—
180C23×C4#4114yes—
181C23×C4#4114yes—
182C23×C4C2411yescharacteristic
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—24no—
186C22×C8—216no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8—24no—
191C22×C8—216no—
192C22×C8—48no—
193C22×C8—416no—
194C22×C8—416no—
195C22×C8C22×C414yes—
196C22×C8C22×C414yes—
197C22×C8C2×D418yes—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—416no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C2×D418yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—432no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236#107—416no—
237C22⋊C8—28no—
238C22⋊C8—216no—
239C22⋊C8C22⋊C4116yescomplemented
240#95—28no—
241#95—28no—
242#95—44no—
243#95—42no—
244#95—44no—
245#95—42no—
246C24—21no—
247C24—22no—
248C24—21no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24—21no—
257C24—22no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—28no—
265C24—28no—
266C24—28no—
267C24—28no—
268C24—28no—
269C24—28no—
270C24—216no—
271C24—216no—
272C24—28no—
273C24—28no—
274C24—44no—
275C24—44no—
276C24—48no—
277C24—48no—
278C24—48no—
279C24—48no—
280C24—48no—
281C24—48no—
282C24—48no—
283C24—48no—
284C24—48no—
285C24—48no—
286C24#107116yescomplemented
287C24C2×C4⋊C412yes—
288C24C2×C4⋊C412yes—
289C24C2×C4⋊C412yes—
290C24#12012yes—
291C24#12114yes—
292C24#12312yes—
293C24C2×D812yes—
294C24C2×D812yes—
295C24C2×D812yes—
296C24C2×Q1612yes—
297C24C2×Q1612yes—
298C24C2×Q1612yes—
299C24#13514yes—
300C24AGL(1,ℤ/8)12yes—
301C24#13712yes—
302C24C22×D411yescharacteristic
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—432no—
349C22×C4—432no—
350C22×C4—464no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—416no—
360C22×C4—416no—
361C22×C4—416no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4C2×C22⋊C414yes—
387C22×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C2×C4⋊C412yes—
390C22×C4#11714yes—
391C22×C4C4×D418yes—
392C22×C4#12012yes—
393C22×C4#12114yes—
394C22×C4C22⋊Q818yes—
395C22×C4C22⋊Q818yes—
396C22×C4#12312yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×Q811yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4H(5,𝔽2)14yes—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—28no—
407C2×C8—28no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—432no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8C2×C22⋊C418yescomplemented
415C4⋊C4—28no—
416C4⋊C4—28no—
417C4⋊C4—416no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C22⋊C4—416no—
425C22⋊C4—416no—
426C22⋊C4—88no—
427C22⋊C4—84no—
428C22⋊C4—88no—
429C22⋊C4—84no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—232no—
467C23—22no—
468C23—232no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—416no—
481C23—416no—
482C23—432no—
483C23—432no—
484C23—416no—
485C23—416no—
486C23—416no—
487C23—432no—
488C23—432no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23#38014yes—
498C23#38514yes—
499C23C2×#10714yes—
500C23C2×#10714yes—
501C23C2×#10714yes—
502C23#42714yes—
503C23#42814yes—
504C23#44714yes—
505C23#45114yes—
506C23#46618yes—
507C23#46718yes—
508C23#48014yes—
509C23#48414yes—
510C23C22×C4⋊C411yescharacteristic
511C23C2×#12011yescharacteristic
512C23C2×#12112yes—
513C23C2×#12311yescharacteristic
514C23#53412yes—
515C23C22×D811yescharacteristic
516C23C22×Q1611yescharacteristic
517C23C2×#13512yes—
518C23C2×AGL(1,ℤ/8)11yescharacteristic
519C23C2×#13711yescharacteristic
520C23#57512yes—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—216no—
534C2×C4—216no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—832no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4#38014yes—
569C2×C4#38514yes—
570C2×C4C22×C22⋊C411yescharacteristic
571C2×C4C2×#11711yescharacteristic
572C2×C4C2×C4×D412yes—
573C2×C4#51812yes—
574C2×C4C2×C22⋊Q812yes—
575C2×C4C2×C22⋊Q812yes—
576C2×C4#54312yes—
577C2×C4D4214yes—
578C2×C4#54614yes—
579C2×C4#54614yes—
580C2×C4#55114yes—
581C8—28no—
582C8—44no—
583C22—24no—
584C22—24no—
585C22—28no—
586C22—28no—
587C22—28no—
588C22—28no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—28no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—464no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22#180818yescomplemented
616C22#185218yescomplemented
617C22C2×#38011yescharacteristic
618C22C2×#38511yescharacteristic
619C22#231511yescharacteristic
620C22#241012yes—
621C22#241212yes—
622C22C22×#10711yescharacteristic
623C22C2×#42711yescharacteristic
624C22C2×#42811yescharacteristic
625C22#291111yescharacteristic
626C22C2×#44711yescharacteristic
627C22C2×#45111yescharacteristic
628C22#300212yes—
629C22C2×#46612yes—
630C22C2×#46712yes—
631C22#305512yes—
632C22#305612yes—
633C22C2×#48011yescharacteristic
634C22C2×#48411yescharacteristic
635C22#309112yes—
636C22#318612yes—
637C22#318812yes—
638C22#318912yes—
639C22#319012yes—
640C4—24no—
641C4—24no—
642C4—42no—
643C4—88no—
644C4—162no—
645C4—162no—
646C4D4×C22⋊C412yes—
647C4#234212yes—
648C4#243714yes—
649C2—28no—
650C2—28no—
651C2—44no—
652C2#1387711yescharacteristic
653C2C2×#180811yescharacteristic
654C2#2066111yescharacteristic
655C2C2×#185211yescharacteristic
656C2#2084311yescharacteristic
657C2#2194011yescharacteristic
658C2#2194111yescharacteristic
659C2#2437812yes—
660C2#2438012yes—
661C2#2455012yes—
662C2#2455212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC26D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C25×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1852C2semidirect
4. C2×#1808C2semidirect
6. #11096C2semidirect
7. #10730C2semidirect
8. #10721C2semidirect
13. C22×#107C22semidirect
15. C22×C22⋊C8C4semidirect
32. C2×#340C22semidirect
33. C2×#340C22semidirect
34. C2×#340C22semidirect
57. C23×C8C2×C4semidirect
85. C2×C22⋊C8C2×C4semidirect
239. C22⋊C8C22⋊C4semidirect
286. C24#107semidirect
414. C2×C8C2×C22⋊C4semidirect
615. C22#1808semidirect
616. C22#1852semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊C82
C22×#3802
C2×#18522
C2×#18082
#146802
#110962
#107302
#107212