Finite Group Databasebeta

← #412473

Subgroups — #412473

Order 512.

2324 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×#2041C214yesmaximal, complemented
3C2×#1982C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic, complemented
5C2×#1802C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic, complemented
7#13910C211yesmaximal, characteristic, complemented
8#11066C214yesmaximal, complemented
9#10834C211yesmaximal, characteristic, complemented
10C22×#127C2211yescharacteristic
11C22×#127C2212yes—
12C22×#121—21no—
13C22×#121—21no—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2214yes—
17C22×#102C2212yes—
18C22×#102C2214yes—
19C22×#102C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2541—24no—
22#2541—24no—
23C2×#392—22no—
24C2×#390C2212yes—
25C2×#389C2214yes—
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2041—216no—
31#1982—24no—
32#1957—24no—
33#1802—24no—
34C4×#102—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2214yes—
38#1444—22no—
39C4⋊#95—21no—
40C23×D4—21no—
41C23×D4—22no—
42C23×D4—22no—
43C23×D4—21no—
44C23×D4C2312yes—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#127—24no—
50C2×#127—24no—
51C2×#127—28no—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—44no—
58C2×#127D414yes—
59C2×#121—28no—
60C2×#121—28no—
61C2×#121—44no—
62C2×#121—44no—
63C2×#121—44no—
64C2×#121—44no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4D414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
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×#102—28no—
93C2×#102—28no—
94C2×#102—28no—
95C2×#102—28no—
96C2×#102—28no—
97C2×#102—44no—
98C2×#102D418yescomplemented
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102#392—216no—
103#390—24no—
104#390—24no—
105#389—28no—
106#389—44no—
107#384—22no—
108#384—22no—
109#384—44no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yescomplemented
114C2×#95—21no—
115C2×#95—24no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#336—28no—
120C25—21no—
121C25—22no—
122C22×D4—28no—
123C22×D4—28no—
124C22×D4—24no—
125C22×D4—24no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—24no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—48no—
144C22×D4—48no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—44no—
152C22×D4—42no—
153C22×D4—44no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—82no—
158C22×D4—82no—
159C22×D4—84no—
160C22×D4—84no—
161C22×D4C2×D414yes—
162C22×D4C2×D412yes—
163C22×D4C2×D414yes—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2411yescharacteristic
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—24no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8C2×D414yes—
191#127—28no—
192#127—28no—
193#127—416no—
194#127—48no—
195#127—48no—
196#127—44no—
197#127—44no—
198#121—432no—
199#121—432no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—44no—
203C4×D4—44no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—84no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4D814yes—
229C2×C4⋊C4SD1614yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4#4114yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C4⋊C8—44no—
257#102—216no—
258#102—416no—
259#102—48no—
260#102—416no—
261C4×C8—44no—
262#95—24no—
263#95—24no—
264#95—48no—
265C24—22no—
266C24—24no—
267C24—24no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—44no—
280C24—44no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—44no—
286C24—44no—
287C24C22×D411yescharacteristic
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—28no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—28no—
302C2×D4—416no—
303C2×D4—416no—
304C2×D4—416no—
305C2×D4—416no—
306C2×D4—41no—
307C2×D4—44no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—42no—
324C2×D4—44no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—816no—
334C2×D4—816no—
335C2×D4—88no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—816no—
340C2×D4—816no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4#12714yes—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—84no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4#12012yes—
383C22×C4#12012yes—
384C22×C4#12112yes—
385C22×C4#12112yes—
386C22×C4#12412yes—
387C22×C4#12412yes—
388C22×C4C2×D814yes—
389C22×C4C2×D812yes—
390C22×C4C2×SD1614yes—
391C22×C4C2×SD1612yes—
392C22×C4AGL(1,ℤ/8)12yes—
393C22×C4AGL(1,ℤ/8)12yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4112yes—
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—44no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—44no—
415C2×C8—42no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—816no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C4⋊C4#12118yescomplemented
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—48no—
435C42—42no—
436C42—42no—
437C42—44no—
438C42—48no—
439C42#12114yes—
440C42#12114yes—
441C42#12314yes—
442C42#12314yes—
443C23—24no—
444C23—42no—
445C23—44no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—81no—
468C23—81no—
469C23—82no—
470C23—81no—
471C23—81no—
472C23—82no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—82no—
478C23—82no—
479C23—84no—
480C23—84no—
481C23—84no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—84no—
489C23C2×#12011yescharacteristic
490C23C2×#12111yescharacteristic
491C23C2×#12411yescharacteristic
492C23D4212yes—
493C23#54812yes—
494C23C22×D811yescharacteristic
495C23C22×SD1611yescharacteristic
496C23C2×AGL(1,ℤ/8)11yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23#57612yes—
499C23#57712yes—
500D4—24no—
501D4—24no—
502D4—42no—
503D4—44no—
504D4—48no—
505D4—42no—
506D4—44no—
507D4—48no—
508D4—816no—
509D4—832no—
510D4—832no—
511D4—82no—
512D4—82no—
513D4—82no—
514D4—82no—
515D4—816no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—816no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4#39414yes—
551C2×C4#39414yes—
552C2×C4#44714yes—
553C2×C4#45014yes—
554C2×C4#45914yes—
555C2×C4#46014yes—
556C2×C4#48614yes—
557C2×C4#49014yes—
558C2×C4C2×#12111yescharacteristic
559C2×C4C2×#12112yes—
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4C2×#12711yescharacteristic
564C2×C4#53212yes—
565C2×C4#53211yescharacteristic
566C2×C4#53511yescharacteristic
567C2×C4#53611yescharacteristic
568C2×C4#53712yes—
569C2×C4#54212yes—
570C2×C4D4212yes—
571C2×C4#54712yes—
572C2×C4#54712yes—
573C2×C4#54712yes—
574C2×C4#55312yes—
575C2×C4#55512yes—
576C8—44no—
577C8—82no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—82no—
582C22—82no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—82no—
588C22—82no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—164no—
597C22—164no—
598C22—164no—
599C22—164no—
600C22C2×#39411yescharacteristic
601C22#241612yes—
602C22#243211yescharacteristic
603C22#243311yescharacteristic
604C22#249112yes—
605C22C2×#44711yescharacteristic
606C22C2×#45011yescharacteristic
607C22#300811yescharacteristic
608C22#300911yescharacteristic
609C22C2×#45911yescharacteristic
610C22C2×#46011yescharacteristic
611C22#304512yes—
612C22C2×#48611yescharacteristic
613C22C2×#49011yescharacteristic
614C22#313612yes—
615C22#327212yes—
616C22#327412yes—
617C22D4×D812yes—
618C22D4×SD1612yes—
619C22#335812yes—
620C22#336212yes—
621C22#336612yes—
622C22#336812yes—
623C4—24no—
624C4—48no—
625C4—44no—
626C4—44no—
627C4—82no—
628C4—162no—
629C4#204118yescomplemented
630C4#242212yes—
631C4#250011yescharacteristic
632C4#250111yescharacteristic
633C4#253712yes—
634C4#262011yescharacteristic
635C4#285611yescharacteristic
636C2—84no—
637C2—164no—
638C2#1513011yescharacteristic
639C2C2×#204111yescharacteristic
640C2#2350911yescharacteristic
641C2#2472612yes—
642C2#2472812yes—
643C2#2564511yescharacteristic
644C2#2564911yescharacteristic
645C2#2575311yescharacteristic
646C2#2575711yescharacteristic
647C2#2776712yes—
648C2#2777512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2541C2
LCSG ▷ C23×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×#2041C2semidirect
3. C2×#1982C2semidirect
4. C2×#1957C2semidirect
5. C2×#1802C2semidirect
6. C2×C4×#102C2semidirect
7. #13910C2semidirect
8. #11066C2semidirect
9. #10834C2semidirect
98. C2×#102D4semidirect
113. C4×C4⋊C4D4semidirect
431. C4⋊C4#121semidirect
629. C4#2041semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×#20412
C2×#19822
C2×#19572
C2×#18022
C2×C4×#1022
#139102
#110662
#108342