Finite Group Databasebeta

← #6498516

Subgroups — #6498516

Order 512.

2090 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic, complemented
2C2×#2965C211yesmaximal, characteristic, complemented
3#35280C211yesmaximal, characteristic, complemented
4#35273C211yesmaximal, characteristic
5#33049C212yesmaximal, complemented
6#33048C212yesmaximal, complemented
7#29877C212yesmaximal, complemented
8#29807C212yesmaximal, complemented
9#28565C218yesmaximal, complemented
10#28195C214yesmaximal, complemented
11#28150C214yesmaximal, complemented
12#21860C211yesmaximal, characteristic, complemented
13#21859C211yesmaximal, characteristic, complemented
14C2×#1834C211yesmaximal, characteristic, complemented
15C2×#537—21no—
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#3167—28no—
20C2×#498C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#465C2214yes—
25#2965—28no—
26C2×#446C2212yes—
27C2×C4×SD16C2214yes—
28C2×#436C2212yes—
29C2×#435C2214yes—
30C2×#431C2211yescharacteristic
31C2×#431C2212yes—
32C2×C4×M16C2212yes—
33C22×C4×C8C2211yescharacteristic
34#2857C2212yes—
35#2857C2212yes—
36#2857C2212yes—
37#2856C2212yes—
38#2629—22no—
39#2620—22no—
40#2582—21no—
41#2582C2212yes—
42#2538—21no—
43#2538—21no—
44#2538C2212yes—
45#2538C2212yes—
46#2537—21no—
47#2537C2212yes—
48#2417—22no—
49#2417C2214yes—
50#2416—22no—
51#2416C2214yes—
52#2316—21no—
53#2311—21no—
54#2311—21no—
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2088C2218yes—
58#2077C2218yes—
59#2055C2218yes—
60#2000C2218yes—
61#1990C2212yes—
62#1990C2212yes—
63#1957C2212yes—
64#1956C2212yes—
65#1936C2218yes—
66#1935C2218yes—
67#1895C2218yes—
68#1894C2218yes—
69#1874C2214yes—
70#1872C2214yes—
71#1834C2212yes—
72#1834C2214yes—
73#1834C2212yes—
74#1803C2214yes—
75#1802C2214yes—
76C2×#340C2214yes—
77C22×#41—21no—
78C22×#41D411yescharacteristic
79C22×SD16C2314yes—
80C22×M16C2314yes—
81C23×C8C2312yes—
82#537—48no—
83#533—24no—
84#533—24no—
85#532—24no—
86#532—24no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2311yescharacteristic
90C2×#127C2311yescharacteristic
91C2×#125—21no—
92C2×#124—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2314yes—
99C2×#121—24no—
100C2×#121—22no—
101C2×#121—22no—
102C2×#121—21no—
103C2×#121—22no—
104C2×#121C2312yes—
105C2×#121C2314yes—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×#117—21no—
119C2×#117D411yescharacteristic
120C2×#117D412yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#498—24no—
130#492—24no—
131#487—24no—
132#486—24no—
133#465—216no—
134#446—28no—
135C4×SD16—216no—
136#436—28no—
137#435—216no—
138#431—24no—
139#431—28no—
140C2×#106C2314yes—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2314yes—
144C2×#103C2318yes—
145C2×#102C2314yes—
146C2×#102C2318yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C4×M16—28no—
150C2×#97C2314yes—
151C2×C4×C8D418yescomplemented
152C2×C4×C8C2312yes—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156#398—24no—
157#398—24no—
158#396—24no—
159#396—24no—
160#396—24no—
161#396—24no—
162#394—24no—
163#394—24no—
164#391—22no—
165#391—22no—
166#391—22no—
167#391C2314yes—
168#390—22no—
169#390—22no—
170#390—22no—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#386—24no—
181#386—24no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2312yes—
187#384C2312yes—
188#384C2312yes—
189#384C2312yes—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—24no—
197#380C2314yes—
198#380C2314yes—
199#340C2318yes—
200#340C2318yes—
201#340C2318yes—
202#340C2318yes—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—24no—
206C2×#41—44no—
207C2×#41—44no—
208C2×#41—44no—
209C22×Q8—21no—
210C22×Q8—21no—
211C22×Q8C2×D411yescharacteristic
212C22×Q8C2412yes—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—21no—
216C22×D4—21no—
217C22×D4C2×D412yes—
218C22×D4C2×D411yescharacteristic
219C22×D4C2412yes—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4C2×D412yes—
224C23×C4C2×D412yes—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C2×SD16—28no—
230C2×SD16—44no—
231C2×SD16—44no—
232C2×M16—28no—
233C2×M16—44no—
234C2×M16—44no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8C2×D418yes—
238C22×C8C2414yes—
239C22×C8C2412yes—
240C22×C8C2412yes—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247#127—22no—
248#127—22no—
249#125—44no—
250#124—44no—
251C22⋊Q8—24no—
252C22⋊Q8—24no—
253C22⋊Q8—48no—
254C22⋊Q8—48no—
255C22⋊Q8—44no—
256C22⋊Q8—48no—
257C22⋊Q8—84no—
258#121—24no—
259#121—24no—
260#121—48no—
261#121—416no—
262#121—48no—
263#121—44no—
264#121—48no—
265#121—84no—
266C4×Q8—24no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—44no—
272C4×D4—28no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—24no—
278C4×D4—22no—
279C4×D4—24no—
280C4×D4—44no—
281#117—24no—
282#117—28no—
283#117—44no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2414yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—22no—
332C2×C42C2×D412yes—
333C2×C42C2×D412yes—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D414yes—
337C2×C42#4114yes—
338C2×C42C2412yes—
339C2×C42C2411yescharacteristic
340C2×C42C2411yescharacteristic
341#106—28no—
342C4⋊C8—28no—
343C4⋊C8—24no—
344#103—28no—
345#103—48no—
346#102—28no—
347#102—48no—
348C22⋊C8—24no—
349C22⋊C8—28no—
350#97—28no—
351C4×C8—24no—
352C4×C8C2×D414yes—
353C4×C8C2×D414yes—
354C4×C8C2×D418yes—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361C24—22no—
362C24C22×D412yes—
363C24C22×D411yescharacteristic
364#41—416no—
365#41—816no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—48no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—84no—
400C2×D4—84no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4#12114yes—
430C22×C4#12114yes—
431C22×C4AGL(1,ℤ/8)14yes—
432C22×C4#13714yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2511yescharacteristic
444SD16—416no—
445M16—416no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C2×C8—48no—
454C2×C8C22×D414yes—
455C2×C8C22×D414yes—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—48no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C22⋊C4—24no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—84no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C42—24no—
498C42—24no—
499C42—22no—
500C42—22no—
501C42—44no—
502C42#12114yes—
503C42#12114yes—
504C42C22×D412yes—
505C42C22×D412yes—
506C42C2×#4112yes—
507C42C2×#4112yes—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C2×#12112yes—
527C23C2×AGL(1,ℤ/8)12yes—
528C23C2×#13712yes—
529C23C23×D411yescharacteristic
530Q8—44no—
531Q8—44no—
532Q8—44no—
533Q8—84no—
534Q8—84no—
535D4—48no—
536D4—44no—
537D4—44no—
538D4—44no—
539D4—88no—
540D4—84no—
541D4—84no—
542D4—168no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—88no—
579C2×C4—164no—
580C2×C4—164no—
581C2×C4#468116yes—
582C2×C4C2×#12112yes—
583C2×C4C2×#12112yes—
584C2×C4C2×#12112yes—
585C2×C4C2×#12112yes—
586C2×C4C2×#12112yes—
587C2×C4C2×AGL(1,ℤ/8)12yes—
588C2×C4C2×AGL(1,ℤ/8)12yes—
589C2×C4C2×#13712yes—
590C2×C4C2×#13712yes—
591C2×C4#57712yes—
592C2×C4#57712yes—
593C2×C4C23×D411yescharacteristic
594C2×C4C22×#4111yescharacteristic
595C2×C4#58512yes—
596C2×C4#58512yes—
597C8—24no—
598C8—24no—
599C8—44no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22C2×#46814yes—
621C22C22×#12111yescharacteristic
622C22#350011yescharacteristic
623C22#350011yescharacteristic
624C22C22×AGL(1,ℤ/8)11yescharacteristic
625C22C22×#13711yescharacteristic
626C22C2×#57711yescharacteristic
627C22C2×#57711yescharacteristic
628C4—22no—
629C4—22no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—84no—
637C4—84no—
638C4C2×#46814yes—
639C4C2×#46814yes—
640C4#306014yes—
641C4#344912yes—
642C4#351212yes—
643C2—24no—
644C2—82no—
645C2—162no—
646C2#3532811yescharacteristic
647C2C22×#46811yescharacteristic
648C2C2×#306011yescharacteristic
649C2#6253911yescharacteristic
650C2#6254011yescharacteristic
651C2#6254111yescharacteristic
652C2#6254211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3167C2semidirect
2. C2×#2965C2semidirect
3. #35280C2semidirect
5. #33049C2semidirect
6. #33048C2semidirect
7. #29877C2semidirect
8. #29807C2semidirect
9. #28565C2semidirect
10. #28195C2semidirect
11. #28150C2semidirect
12. #21860C2semidirect
13. #21859C2semidirect
14. C2×#1834C2semidirect
151. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
C2×#29652
#352802
#352732
#330492
#330482
#298772
#298072
#285652
#281952
#281502
#218602
#218592
C2×#18342