Finite Group Databasebeta

← #412364

Subgroups — #412364

Order 512.

2070 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#2034C214yesmaximal, complemented
3C2×#1954C211yesmaximal, characteristic, complemented
4C2×#1844C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#13906C211yesmaximal, characteristic, complemented
8#11095C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C22×#121—21no—
11C22×#121C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#106C2214yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×C22⋊C8C2214yes—
19C22×C4×C8C2211yescharacteristic
20#2537—24no—
21#2537—24no—
22C2×#394—22no—
23C2×#390—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27C2×#380C2212yes—
28C2×#380C2214yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2034—216no—
31#1954—24no—
32#1844—24no—
33#1802—24no—
34C4×#102—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2214yes—
38#1439—22no—
39C4⋊#95—21no—
40C23×D4—22no—
41C23×D4D411yescharacteristic
42C23×D4C2311yescharacteristic
43C24×C4—21no—
44C24×C4C2312yes—
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#121—28no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—28no—
51C2×#121—44no—
52C2×#121—44no—
53C2×#121—44no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C2×#106—28no—
83C2×#106—28no—
84C2×#106—44no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—28no—
92C2×#102D418yescomplemented
93C2×C22⋊C8—28no—
94C2×C22⋊C8—28no—
95C2×C22⋊C8—28no—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8D414yescomplemented
99#394—216no—
100#390—44no—
101#384—22no—
102#384—22no—
103#384—22no—
104#384D414yes—
105#380—28no—
106#380—24no—
107#380—24no—
108#380—28no—
109#380—44no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C2×#95—22no—
113C2×#95—24no—
114C2×#95—21no—
115#340—24no—
116#340—24no—
117#340—28no—
118C25—21no—
119C25C2×D412yes—
120C22×D4—28no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—48no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—84no—
139C22×D4SD1614yes—
140C22×D4C2×D412yes—
141C22×D4C2×D414yes—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—24no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—24no—
175C22×C8—22no—
176C22×C8C2×D414yes—
177C22×C8C2×D414yes—
178C22×C8#4114yes—
179#121—416no—
180#121—48no—
181#121—432no—
182#121—48no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—44no—
188C4×D4—44no—
189C4×D4—44no—
190C4×D4—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—48no—
237C2×C22⋊C4—84no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#106—416no—
248C4⋊C8—44no—
249#102—216no—
250#102—48no—
251C22⋊C8—416no—
252C4×C8—28no—
253#95—28no—
254#95—48no—
255#95—42no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—24no—
267C24—24no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24C2×SD1614yes—
278C24C22×D411yescharacteristic
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—28no—
290C2×D4—28no—
291C2×D4—28no—
292C2×D4—416no—
293C2×D4—416no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—88no—
306C2×D4—816no—
307C2×D4—816no—
308C2×D4#12714yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4C2×SD1612yes—
372C22×C4#13512yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D412yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4112yes—
382C22×C4C2×#4111yescharacteristic
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+412yes—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—44no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—44no—
395C2×C8—42no—
396C2×C8#12118yescomplemented
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—28no—
401C4⋊C4—24no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C22⋊C4—416no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—816no—
421C42—84no—
422C42#12114yes—
423C42#12114yes—
424C42#12314yes—
425C42#12414yes—
426C23—28no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—28no—
431C23—24no—
432C23—44no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—42no—
461C23—44no—
462C23—82no—
463C23—82no—
464C23—81no—
465C23—81no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—81no—
470C23—81no—
471C23—82no—
472C23C2×#12011yescharacteristic
473C23C2×#12111yescharacteristic
474C23C2×#12311yescharacteristic
475C23D4212yes—
476C23#54712yes—
477C23C22×SD1611yescharacteristic
478C23C2×#13511yescharacteristic
479C23C2×AGL(1,ℤ/8)11yescharacteristic
480C23C2×AGL(1,ℤ/8)11yescharacteristic
481C23#57712yes—
482C23#57712yes—
483D4—24no—
484D4—24no—
485D4—42no—
486D4—44no—
487D4—44no—
488D4—42no—
489D4—416no—
490D4—832no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—816no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—84no—
530C2×C4—84no—
531C2×C4#39414yes—
532C2×C4#39414yes—
533C2×C4#44914yes—
534C2×C4#45014yes—
535C2×C4#46514yes—
536C2×C4#46914yes—
537C2×C4#48114yes—
538C2×C4#48214yes—
539C2×C4C2×#12111yescharacteristic
540C2×C4C2×#12112yes—
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12311yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4C2×#12711yescharacteristic
545C2×C4#53212yes—
546C2×C4#53211yescharacteristic
547C2×C4#53511yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#53712yes—
550C2×C4#54212yes—
551C2×C4D4212yes—
552C2×C4#54712yes—
553C2×C4#54712yes—
554C2×C4#54812yes—
555C2×C4#55312yes—
556C2×C4#55512yes—
557C8—28no—
558C8—82no—
559C22—44no—
560C22—48no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—48no—
566C22—44no—
567C22—82no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—84no—
576C22—82no—
577C22—84no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22C2×#39411yescharacteristic
582C22#242212yes—
583C22#243211yescharacteristic
584C22#243311yescharacteristic
585C22#253712yes—
586C22C2×#45011yescharacteristic
587C22C2×#44911yescharacteristic
588C22#300911yescharacteristic
589C22#301011yescharacteristic
590C22C2×#46511yescharacteristic
591C22C2×#46911yescharacteristic
592C22#306012yes—
593C22C2×#48211yescharacteristic
594C22C2×#48111yescharacteristic
595C22#309912yes—
596C22#327212yes—
597C22#327212yes—
598C22D4×SD1612yes—
599C22#328412yes—
600C22#333512yes—
601C22#334012yes—
602C22#335412yes—
603C22#335412yes—
604C4—24no—
605C4—82no—
606C4—84no—
607C4—84no—
608C4—82no—
609C4—162no—
610C4#203418yescomplemented
611C4#241612yes—
612C4#249112yes—
613C4#250011yescharacteristic
614C4#250111yescharacteristic
615C4#262011yescharacteristic
616C4#285611yescharacteristic
617C2—84no—
618C2—84no—
619C2#1513011yescharacteristic
620C2C2×#203411yescharacteristic
621C2#2348911yescharacteristic
622C2#2501212yes—
623C2#2501312yes—
624C2#2569911yescharacteristic
625C2#2570011yescharacteristic
626C2#2575311yescharacteristic
627C2#2575411yescharacteristic
628C2#2822312yes—
629C2#2823112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2537C2
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2034C2semidirect
3. C2×#1954C2semidirect
4. C2×#1844C2semidirect
5. C2×#1802C2semidirect
6. C2×C4×#102C2semidirect
7. #13906C2semidirect
8. #11095C2semidirect
9. #10849C2semidirect
92. C2×#102D4semidirect
98. C2×C4×C8D4semidirect
396. C2×C8#121semidirect
610. C4#2034semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#20342
C2×#19542
C2×#18442
C2×#18022
C2×C4×#1022
#139062
#110952
#108492