Finite Group Databasebeta

← #376114

Subgroups — #376114

Order 512.

1772 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q16C211yesmaximal, characteristic, complemented
2C2×#2001C211yesmaximal, characteristic
3C2×#1892C212yesmaximal
4C2×C8×C22⋊C4C211yesmaximal, characteristic
5#13969C212yesmaximal
6#11256C211yesmaximal, characteristic, complemented
7#11096C211yesmaximal, characteristic
8#11024C211yesmaximal, characteristic, complemented
9#10926C212yesmaximal, complemented
10#10820C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C4×Q8C2212yes—
16C2×C4×Q16—28no—
17C2×C4×Q16—24no—
18C22×#107C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#2001C418yescomplemented
29#1892—28no—
30C8×C22⋊C4—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1441—22no—
39#1439—22no—
40#1437—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×Q8C2312yes—
45C24×C4C2311yescharacteristic
46C22×Q16—24no—
47C22×Q16—24no—
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16C2×C414yes—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C2×C22⋊Q8C2×C418yes—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8—42no—
71C2×C4×Q8—42no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42—22no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C4×Q16—232no—
85C2×#107—22no—
86C2×#107—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103C2×C418yes—
95C2×#103D418yes—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8C2×C414yes—
99C2×C4×C8—24no—
100C2×C4×C8—22no—
101C2×C4×C8—42no—
102C2×C4×C8D414yes—
103#391—24no—
104#391—42no—
105#380—24no—
106#380—42no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95D411yescharacteristic
117C2×#95D411yescharacteristic
118#340—24no—
119#340—22no—
120#340—24no—
121#340—42no—
122#340C2×C414yes—
123#336—22no—
124#336—22no—
125C25C2×D411yescharacteristic
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C22×C414yes—
138C22×Q8C2×D414yes—
139C22×Q8#4114yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C22×C412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C2×Q16—28no—
161C2×Q16—216no—
162C2×Q16—28no—
163C2×Q16—216no—
164C2×Q16—28no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—28no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—48no—
180C22×C8—48no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8C22×C412yes—
184C22×C8C22×C412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D414yes—
187C22×C8C2×D412yes—
188C22×C8#4112yes—
189C22×C8#4112yes—
190C22⋊Q8—216no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194C4×Q8—216no—
195C4×Q8—48no—
196C4×Q8—48no—
197C4×Q8—48no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4Q1614yescomplemented
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42C2×D412yes—
244C2×C42#4112yes—
245#107—28no—
246#103—216no—
247#103—48no—
248#103—48no—
249C22⋊C8—28no—
250C4×C8—28no—
251C4×C8—44no—
252#95—22no—
253#95—22no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—82no—
260#95—82no—
261C24—21no—
262C24—21no—
263C24—22no—
264C24—22no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24C4×D412yes—
270C24C2×Q1612yes—
271C24#13512yes—
272C24C22×D411yescharacteristic
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8C4×D418yes—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4C4×D412yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4C2×Q1612yes—
369C22×C4C2×Q1612yes—
370C22×C4#13512yes—
371C22×C4#13512yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)12yes—
382Q16—232no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—22no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—416no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8C4×D414yes—
403C2×C8#12014yes—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—416no—
409C4⋊C4—416no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C22⋊C4—28no—
419C22⋊C4—28no—
420C22⋊C4—416no—
421C22⋊C4—88no—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—416no—
426C42—41no—
427C42—41no—
428C42—42no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23C4×Q1614yescomplemented
452C23C2×C4×D411yescharacteristic
453C23C2×#12111yescharacteristic
454C23#53211yescharacteristic
455C23D4212yes—
456C23#54612yes—
457C23C22×Q1611yescharacteristic
458C23C2×#13511yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×#13511yescharacteristic
461C23#57712yes—
462C23#57812yes—
463Q8—28no—
464Q8—28no—
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4#46714yes—
530C2×C4C2×C4×D411yescharacteristic
531C2×C4C2×C4×D412yes—
532C2×C4C4×#4111yescharacteristic
533C2×C4#51812yes—
534C2×C4C2×#12011yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4#52512yes—
538C2×C4#52911yescharacteristic
539C2×C4#53511yescharacteristic
540C2×C4#53511yescharacteristic
541C2×C4#54012yes—
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4#54612yes—
545C2×C4#55112yes—
546C2×C4#55312yes—
547C2×C4#55312yes—
548C8—24no—
549C8—24no—
550C8—42no—
551C8—42no—
552C22—24no—
553C22—28no—
554C22—42no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22C4×#12111yescharacteristic
564C22#235812yes—
565C22#240811yescharacteristic
566C22#246412yes—
567C22#253511yescharacteristic
568C22C2×C4×Q1611yescharacteristic
569C22C4×#13511yescharacteristic
570C22#294712yes—
571C22C2×#46711yescharacteristic
572C22#305011yescharacteristic
573C22#305911yescharacteristic
574C22#306011yescharacteristic
575C22#316811yescharacteristic
576C22#317011yescharacteristic
577C22#317812yes—
578C22#328212yes—
579C22#328412yes—
580C22#328512yes—
581C22D4×Q1612yes—
582C22#329912yes—
583C22#330012yes—
584C22#330012yes—
585C22#330412yes—
586C4—24no—
587C4—42no—
588C4—48no—
589C4—42no—
590C4—42no—
591C4—42no—
592C4—84no—
593C4—84no—
594C4—162no—
595C4C4×#12011yescharacteristic
596C4#232811yescharacteristic
597C4#236412yes—
598C4#240411yescharacteristic
599C4#245712yes—
600C4#259811yescharacteristic
601C2—44no—
602C2#1417511yescharacteristic
603C2C4×#46711yescharacteristic
604C2#2153711yescharacteristic
605C2#2287112yes—
606C2#2287312yes—
607C2#2426511yescharacteristic
608C2#2426811yescharacteristic
609C2#2696412yes—
610C2#2696512yes—
611C2#2814211yescharacteristic
612C2#2814511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C4×Q16C2semidirect
6. #11256C2semidirect
8. #11024C2semidirect
9. #10926C2semidirect
10. #10820C2semidirect
11. #10789C2semidirect
28. #2001C4semidirect
233. C2×C22⋊C4Q16semidirect
451. C23C4×Q16semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q162
C2×#20012
C2×#18922
C2×C8×C22⋊C42
#139692
#112562
#110962
#110242
#109262
#108202
#107892