Finite Group Databasebeta

← #374272

Subgroups — #374272

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#459C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3C2×#1957C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic, complemented
5#14154C211yesmaximal, characteristic, complemented
6#11022C214yesmaximal, complemented
7#10931C214yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10834C211yesmaximal, characteristic, complemented
10C23×D8C2212yes—
11C22×#127C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C2×#459—24no—
15C2×#459—24no—
16C2×#459C418yescomplemented
17C22×C4⋊C8C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2214yes—
20C22×#102C2212yes—
21C22×C4×C8C2211yescharacteristic
22#2354—24no—
23#2354—24no—
24C2×#390—21no—
25C2×#390—21no—
26C2×#390C2212yes—
27C2×#390C2214yes—
28C2×#388—22no—
29C2×#384C2211yescharacteristic
30C2×#380—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1957—24no—
35C4×#102—24no—
36C2×#340C2212yes—
37C2×#340C2214yes—
38C2×#336C2214yes—
39#1438—22no—
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2312yes—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C22×D8—24no—
46C22×D8—24no—
47C22×D8—44no—
48C22×D8—44no—
49C22×D8C2×C418yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127C2×C414yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D414yes—
63C2×C4×D4D414yes—
64C2×#117—28no—
65C2×#117—44no—
66C2×#117—44no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—24no—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#459—232no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8C2×C414yes—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—28no—
93C2×#102—28no—
94C2×#102—44no—
95C2×#102C2×C418yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99#390—24no—
100#390—24no—
101#390—28no—
102#390—44no—
103#390—44no—
104#390—44no—
105#388—216no—
106#384—22no—
107#384D414yes—
108#380—28no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95D412yes—
118#340—24no—
119#340—44no—
120#336—44no—
121C25—21no—
122C25—22no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—82no—
150C22×D4—82no—
151C22×D4C22×C414yes—
152C22×D4C22×C412yes—
153C22×D4C2×D412yes—
154C22×D4C2×D412yes—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C2×D8—216no—
171C2×D8—416no—
172C2×D8—416no—
173C2×D8—48no—
174C2×D8—48no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—24no—
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8C22×C414yes—
186C22×C8#4114yes—
187#127—28no—
188#127—28no—
189#127—44no—
190#127—44no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4C22⋊C4116yescomplemented
195#117—432no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4D814yescomplemented
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—84no—
235C2×C22⋊C4—84no—
236C2×C22⋊C4—84no—
237C2×C42—28no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C4⋊C8—28no—
254#102—216no—
255#102—216no—
256#102—416no—
257C4×C8—44no—
258#95—28no—
259#95—24no—
260#95—42no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—24no—
265C24—42no—
266C24—42no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—44no—
276C24—44no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—44no—
282C24—44no—
283C24C22×D411yescharacteristic
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—28no—
289C2×D4—24no—
290C2×D4—28no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—88no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—88no—
327C2×D4—84no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4C2×C22⋊C414yes—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4#12012yes—
372C22×C4#12012yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4C2×D814yes—
376C22×C4C2×D812yes—
377C22×C4#13512yes—
378C22×C4AGL(1,ℤ/8)12yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390D8—432no—
391C2×C8—24no—
392C2×C8—28no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—44no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—84no—
417C4⋊C4C2×C22⋊C414yes—
418C22⋊C4—216no—
419C22⋊C4—416no—
420C22⋊C4—816no—
421C22⋊C4—88no—
422C22⋊C4—816no—
423C42—416no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—84no—
428C42C2×C22⋊C414yes—
429C42C4×D414yes—
430C42#12114yes—
431C42#12314yes—
432C23—24no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—48no—
448C23—44no—
449C23—44no—
450C23—48no—
451C23—44no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—82no—
460C23—84no—
461C23—84no—
462C23—84no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—84no—
478C23C2×C4×D411yescharacteristic
479C23C2×#12011yescharacteristic
480C23C2×#12111yescharacteristic
481C23#54612yes—
482C23#55012yes—
483C23C22×D811yescharacteristic
484C23C2×#13511yescharacteristic
485C23C2×AGL(1,ℤ/8)11yescharacteristic
486C23C2×AGL(1,ℤ/8)11yescharacteristic
487C23#57612yes—
488C23#57712yes—
489D4—24no—
490D4—24no—
491D4—44no—
492D4—44no—
493D4—48no—
494D4—44no—
495D4—42no—
496D4—42no—
497D4—44no—
498D4—816no—
499D4—816no—
500D4—84no—
501D4—82no—
502D4—82no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—432no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4#38614yes—
538C2×C4#38614yes—
539C2×C4C4×D814yes—
540C2×C4#44214yes—
541C2×C4#44714yes—
542C2×C4#44914yes—
543C2×C4#45914yes—
544C2×C4#46314yes—
545C2×C4C22×C22⋊C411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C4×#4112yes—
548C2×C4#51812yes—
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C2×C4#52512yes—
552C2×C4#53211yescharacteristic
553C2×C4#53211yescharacteristic
554C2×C4#53512yes—
555C2×C4#53511yescharacteristic
556C2×C4#53511yescharacteristic
557C2×C4#54612yes—
558C2×C4#55012yes—
559C2×C4#55312yes—
560C2×C4#55312yes—
561C2×C4#55512yes—
562C2×C4#55512yes—
563C8—44no—
564C8—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—82no—
575C22—82no—
576C22—88no—
577C22—84no—
578C22—88no—
579C22—84no—
580C22—84no—
581C22—162no—
582C22—162no—
583C22—164no—
584C22—164no—
585C22—162no—
586C22—162no—
587C22C2×#38611yescharacteristic
588C22#236312yes—
589C22#241412yes—
590C22#243111yescharacteristic
591C22#243111yescharacteristic
592C22C2×C4×D811yescharacteristic
593C22C2×#44211yescharacteristic
594C22#294612yes—
595C22C2×#44711yescharacteristic
596C22C2×#44911yescharacteristic
597C22#300811yescharacteristic
598C22#301011yescharacteristic
599C22C2×#45911yescharacteristic
600C22C2×#46311yescharacteristic
601C22#304512yes—
602C22#329412yes—
603C22#330512yes—
604C22#332512yes—
605C22#333212yes—
606C22#339112yes—
607C22#339412yes—
608C22#339712yes—
609C22#339912yes—
610C4—24no—
611C4—44no—
612C4—44no—
613C4—84no—
614C4—84no—
615C4—82no—
616C4#187918yes—
617C4#229711yescharacteristic
618C4#230911yescharacteristic
619C4#236312yes—
620C4#241412yes—
621C4#250011yescharacteristic
622C4#250011yescharacteristic
623C2—84no—
624C2—164no—
625C2#1413611yescharacteristic
626C2C2×#187911yescharacteristic
627C2#2093211yescharacteristic
628C2#2309312yes—
629C2#2309912yes—
630C2#2462612yes—
631C2#2463112yes—
632C2#2556711yescharacteristic
633C2#2557211yescharacteristic
634C2#2562811yescharacteristic
635C2#2563111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2354C2semidirect
3. C2×#1957C2semidirect
4. C2×C4×#102C2semidirect
5. #14154C2semidirect
6. #11022C2semidirect
7. #10931C2semidirect
8. #10849C2semidirect
9. #10834C2semidirect
16. C2×#459C4semidirect
194. C4×D4C22⋊C4semidirect
217. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C22×#4592
C2×#23542
C2×#19572
C2×C4×#1022
#141542
#110222
#109312
#108492
#108342