Finite Group Databasebeta

← #369126

Subgroups — #369126

Order 512.

2882 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2541C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1982C211yesmaximal, characteristic, complemented
4C2×#1957C211yesmaximal, characteristic, complemented
5C2×#1954C211yesmaximal, characteristic, complemented
6C2×#1874C214yesmaximal, complemented
7#10969C214yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9C4×#340C211yesmaximal, characteristic, complemented
10C22×#127—21no—
11C22×#127C2211yescharacteristic
12C22×#121C2214yes—
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#102C2212yes—
18C22×#102C2214yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2214yes—
21C22×C4×C8C2211yescharacteristic
22#2541—24no—
23#2541C418yescomplemented
24C2×#392—22no—
25D4×C4⋊C4—24no—
26D4×C4⋊C4—24no—
27C2×#390—21no—
28C2×#389—21no—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1982—24no—
35#1957—24no—
36#1954—24no—
37#1874—216no—
38C2×#340C2214yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C23×D4—24no—
42C23×D4—21no—
43C23×D4—21no—
44C23×D4D411yescharacteristic
45C23×D4C2312yes—
46C24×C4—22no—
47C24×C4C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#127—28no—
51C2×#127—22no—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—44no—
57C2×#127—44no—
58C2×#127C2×C414yescomplemented
59C2×#121—28no—
60C2×#121—28no—
61C2×#121—28no—
62C2×#121—44no—
63C2×#121—44no—
64C2×#121C2×C4116yescomplemented
65C2×C4×D4—28no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—44no—
71C2×C4×D4—44no—
72C2×C4×D4C2×C414yes—
73C2×C4×D4D414yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—28no—
80C22×C4⋊C4—44no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—22no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×#102—28no—
96C2×#102—28no—
97C2×#102—28no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—44no—
101C2×#102—44no—
102C2×C22⋊C8—28no—
103C2×C22⋊C8—28no—
104C2×C22⋊C8—44no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108#392—216no—
109#390—28no—
110#389—28no—
111#385—216no—
112#384—22no—
113#384D414yescomplemented
114#380—24no—
115#380—24no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C414yescomplemented
119C4×C4⋊C4D414yescomplemented
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123#340—28no—
124#340—24no—
125#340D418yescomplemented
126C25—22no—
127C25C2×D412yes—
128C22×D4—216no—
129C22×D4—24no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—24no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—24no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—24no—
144C22×D4—48no—
145C22×D4—48no—
146C22×D4—48no—
147C22×D4—48no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—82no—
157C22×D4—82no—
158C22×D4—84no—
159C22×D4—84no—
160C22×D4C22⋊C414yes—
161C22×D4D814yes—
162C22×D4SD1614yes—
163C22×D4C22×C414yes—
164C23×C4—22no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—28no—
177C23×C4—48no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—84no—
181C23×C4C22×C414yes—
182C23×C4C2×D412yes—
183C23×C4C2×D414yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—24no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—24no—
198C22×C8C2×D414yes—
199#127—28no—
200#127—28no—
201#127—432no—
202#127—44no—
203#127—44no—
204#121—232no—
205#121—416no—
206#121—416no—
207#121—416no—
208C4×D4—28no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—28no—
212C4×D4—432no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—416no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22⋊C414yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—216no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—48no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4C22⋊C418yes—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42C22×C412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C4⋊C8—44no—
266#102—416no—
267#102—48no—
268#102—48no—
269C22⋊C8—416no—
270C4×C8—44no—
271#95—28no—
272#95—24no—
273#95—24no—
274C24—22no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—42no—
284C24—42no—
285C24—44no—
286C24—44no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—44no—
292C24—44no—
293C24C2×C22⋊C414yes—
294C24C2×D814yes—
295C24C2×SD1614yes—
296C24C22×D411yescharacteristic
297C2×D4—28no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—28no—
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—28no—
305C2×D4—28no—
306C2×D4—28no—
307C2×D4—432no—
308C2×D4—432no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—816no—
332C2×D4—816no—
333C2×D4—816no—
334C2×D4—816no—
335C2×D4—816no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4#102116yescomplemented
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—416no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—416no—
381C22×C4—44no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—88no—
390C22×C4—816no—
391C22×C4—816no—
392C22×C4C2×C22⋊C414yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4#12312yes—
398C22×C4#12312yes—
399C22×C4C2×D812yes—
400C22×C4C2×SD1612yes—
401C22×C4AGL(1,ℤ/8)12yes—
402C22×C4#13712yes—
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4112yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)12yes—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—44no—
424C2×C8—44no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—28no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—28no—
434C4⋊C4—416no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—48no—
439C4⋊C4—84no—
440C4⋊C4C4×D418yescomplemented
441C4⋊C4#12714yescomplemented
442C22⋊C4—216no—
443C22⋊C4—432no—
444C22⋊C4—416no—
445C42—48no—
446C42—48no—
447C42C4×D414yes—
448C42C4×D414yes—
449C42#12114yes—
450C42#12114yes—
451C23—24no—
452C23—24no—
453C23—28no—
454C23—24no—
455C23—28no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—24no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—48no—
474C23—44no—
475C23—44no—
476C23—48no—
477C23—44no—
478C23—84no—
479C23—84no—
480C23—82no—
481C23—82no—
482C23—84no—
483C23—82no—
484C23—82no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23—84no—
496C23C2×#10218yes—
497C23C22×C22⋊C411yescharacteristic
498C23C2×#12011yescharacteristic
499C23C2×#12311yescharacteristic
500C23D4212yes—
501C23#54612yes—
502C23C22×D811yescharacteristic
503C23C22×SD1611yescharacteristic
504C23C2×AGL(1,ℤ/8)11yescharacteristic
505C23C2×#13711yescharacteristic
506C23#57612yes—
507C23#57712yes—
508D4—232no—
509D4—44no—
510D4—44no—
511D4—48no—
512D4—816no—
513D4—864no—
514D4—816no—
515D4—84no—
516D4—82no—
517D4—82no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—832no—
552C2×C4—82no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—82no—
556C2×C4#38614yes—
557C2×C4#38614yes—
558C2×C4C2×#10214yes—
559C2×C4#41714yes—
560C2×C4#44714yes—
561C2×C4#44814yes—
562C2×C4#48014yes—
563C2×C4#48114yes—
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×C4×D412yes—
566C2×C4C2×C4×D411yescharacteristic
567C2×C4#51812yes—
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12111yescharacteristic
570C2×C4C2×#12711yescharacteristic
571C2×C4#53212yes—
572C2×C4#53211yescharacteristic
573C2×C4#53511yescharacteristic
574C2×C4#53511yescharacteristic
575C2×C4#54012yes—
576C2×C4D4212yes—
577C2×C4D4212yes—
578C2×C4#54612yes—
579C2×C4#55012yes—
580C2×C4#55012yes—
581C2×C4#55912yes—
582C8—44no—
583C8—82no—
584C22—216no—
585C22—28no—
586C22—48no—
587C22—48no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—88no—
596C22—88no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—164no—
601C22—164no—
602C22—162no—
603C22—162no—
604C22—162no—
605C22—162no—
606C22C2×#38611yescharacteristic
607C22D4×C22⋊C412yes—
608C22#243111yescharacteristic
609C22#243311yescharacteristic
610C22#244512yes—
611C22C22×#10211yescharacteristic
612C22C2×#41711yescharacteristic
613C22#289712yes—
614C22C2×#44711yescharacteristic
615C22C2×#44811yescharacteristic
616C22#300811yescharacteristic
617C22#301011yescharacteristic
618C22C2×#48011yescharacteristic
619C22C2×#48111yescharacteristic
620C22#309612yes—
621C22#327412yes—
622C22#327812yes—
623C22D4×D812yes—
624C22D4×SD1612yes—
625C22#329412yes—
626C22#329512yes—
627C22#333212yes—
628C22#333312yes—
629C4—24no—
630C4—48no—
631C4—44no—
632C4—44no—
633C4—84no—
634C4—82no—
635C4#187418yescomplemented
636C4C4×#12711yescharacteristic
637C4#232811yescharacteristic
638C4#236612yes—
639C4#242812yes—
640C4#254111yescharacteristic
641C4#254411yescharacteristic
642C2—48no—
643C2—164no—
644C2#1402511yescharacteristic
645C2C2×#187411yescharacteristic
646C2#2091411yescharacteristic
647C2D4×#10212yes—
648C2#2243212yes—
649C2#2556711yescharacteristic
650C2#2556911yescharacteristic
651C2#2575611yescharacteristic
652C2#2575711yescharacteristic
653C2#2623012yes—
654C2#2623212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2541C2
LCSG ▷ C22×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×D4×C4⋊C4C2semidirect
3. C2×#1982C2semidirect
4. C2×#1957C2semidirect
5. C2×#1954C2semidirect
6. C2×#1874C2semidirect
7. #10969C2semidirect
8. #10849C2semidirect
9. C4×#340C2semidirect
23. #2541C4semidirect
58. C2×#127C2×C4semidirect
64. C2×#121C2×C4semidirect
113. #384D4semidirect
118. C4×C4⋊C4C2×C4semidirect
119. C4×C4⋊C4D4semidirect
125. #340D4semidirect
340. C2×D4#102semidirect
440. C4⋊C4C4×D4semidirect
441. C4⋊C4#127semidirect
635. C4#1874semidirect

Maximal subgroups

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

subgroupindex
C2×#25412
C2×D4×C4⋊C42
C2×#19822
C2×#19572
C2×#19542
C2×#18742
#109692
#108492
C4×#3402