Finite Group Databasebeta

← #6481798

Subgroups — #6481798

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C211yesmaximal, characteristic
2C2×#3101C211yesmaximal, characteristic, complemented
3#37633C211yesmaximal, characteristic, complemented
4#34186C211yesmaximal, characteristic, complemented
5#33121C212yesmaximal, complemented
6#33119C212yesmaximal, complemented
7#29727C212yesmaximal
8#29723C212yesmaximal, complemented
9#24686C214yesmaximal, complemented
10#24680C212yesmaximal, complemented
11#24677C212yesmaximal, complemented
12#24636C214yesmaximal, complemented
13#24630C212yesmaximal, complemented
14#24627C212yesmaximal, complemented
15#21952C211yesmaximal, characteristic, complemented
16#21864C211yesmaximal, characteristic, complemented
17#21862C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3105—28no—
22#3101—28no—
23C2×#484C2211yescharacteristic
24C2×#481C2211yescharacteristic
25C2×#483C2212yes—
26C2×#478C2212yes—
27C2×#476C2212yes—
28C2×#463C2212yes—
29C2×#461C2212yes—
30C2×#450C2211yescharacteristic
31C2×#450C2212yes—
32C2×#447C2211yescharacteristic
33C2×C4×SD16C2212yes—
34C2×C4×SD16C2214yes—
35C2×C4×D8C2212yes—
36C2×#431C2211yescharacteristic
37C2×#431C2211yescharacteristic
38#2814—22no—
39#2637—22no—
40#2629—22no—
41#2621C2212yes—
42#2620C2212yes—
43#2567C2211yescharacteristic, complemented
44#2565C2212yescomplemented
45#2563C2211yescharacteristic, complemented
46#2547C2211yescharacteristic, complemented
47#2546C2211yescharacteristic, complemented
48#2545C2212yescomplemented
49#2485—22no—
50#2357—22no—
51#2357C2214yes—
52#2354—22no—
53#2354C2212yes—
54#2354C2212yes—
55#2312—22no—
56#2309C2212yes—
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59#2280—22no—
60#2274C2212yes—
61#2073C2214yes—
62#2071C2214yes—
63#2055C2214yes—
64#2055C2214yes—
65#2055C2214yes—
66#2054C2214yes—
67#2013C2214yes—
68#2012C2214yes—
69#1991C2212yescomplemented
70#1990C2212yescomplemented
71#1983C2214yescomplemented
72#1981C2214yescomplemented
73#1958C2212yescomplemented
74#1956C2212yescomplemented
75#1949C2214yescomplemented
76#1948C2214yescomplemented
77#1939C2212yescomplemented
78#1939C2212yescomplemented
79#1939C2214yescomplemented
80#1928C2214yescomplemented
81#1927C2214yescomplemented
82#1927C2214yescomplemented
83#1926C2214yescomplemented
84#1875C2212yes—
85#1873C2212yes—
86#1873C2212yes—
87#1873C2212yes—
88#1805C2214yescomplemented
89#1803C2212yescomplemented
90#1802C2212yescomplemented
91#1798C2212yes—
92#1798C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×SD16C2314yes—
96C22×SD16C2312yes—
97C22×D8C2312yes—
98#533—24no—
99#533—24no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103#529—24no—
104#529—24no—
105C2×#126—21no—
106C2×#125—21no—
107C2×#125C2311yescharacteristic
108C2×#124C2311yescharacteristic
109C2×#124C2312yes—
110C2×#123—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8C2312yes—
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121C2311yescharacteristic
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—24no—
122C2×C4×D4D418yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117—22no—
128C2×#117—22no—
129C2×#117—21no—
130C2×#117C2312yes—
131C2×#117C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C42C2311yescharacteristic
135#484—24no—
136#483—28no—
137#481—24no—
138#478—28no—
139#476—28no—
140#463—28no—
141#461—28no—
142#450—28no—
143#450—24no—
144#447—24no—
145C4×SD16—216no—
146C4×SD16—28no—
147C4×D8—28no—
148#431—24no—
149#431—24no—
150C2×#107C2312yes—
151C2×#106C2312yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2314yes—
156C2×C4⋊C8C2314yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×C22⋊C8C2312yes—
168C2×C22⋊C8C2311yescharacteristic
169C2×C22⋊C8C2311yescharacteristic
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172#398—24no—
173#398—24no—
174#398—22no—
175#397—24no—
176#397—22no—
177#396—24no—
178#396—24no—
179#395—22no—
180#394—24no—
181#393—22no—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#390—24no—
186#390C2312yes—
187#390C2314yes—
188#390C2314yes—
189#389—22no—
190#389—22no—
191#388—24no—
192#388—24no—
193#388—24no—
194#388—24no—
195#388—24no—
196#387—24no—
197#384D412yes—
198#384C2312yes—
199#384C2312yes—
200#382D412yes—
201#382C2312yes—
202#382C2312yes—
203#381—24no—
204#380—21no—
205#380—21no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380D412yes—
211#380D412yes—
212#379—24no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C22⋊C4—24no—
222C4×C22⋊C4—24no—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2314yes—
228#336C2314yes—
229C25—21no—
230C22×Q8—21no—
231C22×Q8C2411yescharacteristic
232C22×Q8C2412yes—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4—42no—
237C22×D4—42no—
238C22×D4C2×D414yes—
239C22×D4C2411yescharacteristic
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—28no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—84no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C2×SD16—28no—
251C2×SD16—24no—
252C2×SD16—44no—
253C2×SD16—42no—
254C2×SD16—42no—
255C2×SD16—44no—
256C2×D8—24no—
257C2×D8—42no—
258C2×D8—42no—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2414yes—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—44no—
267#126—44no—
268#125—22no—
269#125—22no—
270#125—44no—
271#124—22no—
272#124—22no—
273#124—44no—
274#123—44no—
275C22⋊Q8—22no—
276C22⋊Q8—24no—
277C22⋊Q8—22no—
278C22⋊Q8—24no—
279C22⋊Q8—44no—
280#121—22no—
281#121—22no—
282C4×Q8—22no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—44no—
287C4×D4—28no—
288C4×D4—28no—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—44no—
294C4×D4C2×D418yes—
295C4×D4C2×D418yes—
296#117—44no—
297#117—44no—
298#117—44no—
299#117—44no—
300#117—84no—
301#117—84no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4C2×D414yes—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D414yes—
352C2×C42#4114yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#106—28no—
359#106—24no—
360C4⋊C8—24no—
361C4⋊C8—24no—
362C4⋊C8—44no—
363C4⋊C8—44no—
364#103—28no—
365#103—24no—
366#103—44no—
367#103—44no—
368#103—44no—
369#102—24no—
370#102—24no—
371#102—28no—
372#102—24no—
373#102—44no—
374C22⋊C8—22no—
375C22⋊C8—22no—
376C22⋊C8—24no—
377C4×C8—24no—
378C4×C8—24no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95C2×D414yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—22no—
391C24—24no—
392C24—21no—
393C24—42no—
394C24—42no—
395C24—82no—
396C24—82no—
397C24C22×D411yescharacteristic
398C2×Q8—21no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—82no—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—48no—
422C2×D4—42no—
423C2×D4—82no—
424C2×D4C22×D414yes—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—816no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4#12114yes—
448C22×C4#12114yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C2×#4112yes—
458C22×C4C2511yescharacteristic
459SD16—48no—
460SD16—416no—
461D8—48no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—24no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4C22×D414yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C42—24no—
516C42—24no—
517C42—44no—
518C42—42no—
519C42—44no—
520C42—84no—
521C42—84no—
522C42#12114yes—
523C42#12114yes—
524C42C22×D412yes—
525C42C22×D412yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—24no—
533C23—48no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—84no—
546C23—84no—
547C23—88no—
548C23—88no—
549C23C2×#12112yes—
550C23C23×D411yescharacteristic
551Q8—42no—
552Q8—44no—
553Q8—42no—
554Q8—44no—
555Q8—84no—
556D4—24no—
557D4—24no—
558D4—48no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—84no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—416no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—88no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C2×#12112yes—
604C2×C4C2×#12112yes—
605C2×C4C2×#12112yes—
606C2×C4#57714yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4#58512yes—
610C2×C4#58512yes—
611C8—42no—
612C8—42no—
613C8—44no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—816no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22C22×#12111yescharacteristic
633C22#348511yescharacteristic
634C22#348711yescharacteristic
635C22C2×#57712yes—
636C22#358412yes—
637C4—22no—
638C4—22no—
639C4—22no—
640C4—22no—
641C4—44no—
642C4—44no—
643C4—44no—
644C4—81no—
645C4—81no—
646C4—82no—
647C4—81no—
648C4—82no—
649C4—84no—
650C4#304214yes—
651C4#344912yes—
652C4#349312yes—
653C2—42no—
654C2—84no—
655C2—161no—
656C2#3515011yescharacteristic
657C2C2×#304211yescharacteristic
658C2#6042211yescharacteristic
659C2#6178512yes—
660C2#6192712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3101C2semidirect
3. #37633C2semidirect
4. #34186C2semidirect
5. #33121C2semidirect
6. #33119C2semidirect
8. #29723C2semidirect
9. #24686C2semidirect
10. #24680C2semidirect
11. #24677C2semidirect
12. #24636C2semidirect
13. #24630C2semidirect
14. #24627C2semidirect
15. #21952C2semidirect
16. #21864C2semidirect
17. #21862C2semidirect
43. #2567C22semidirect
44. #2565C22semidirect
45. #2563C22semidirect
46. #2547C22semidirect
47. #2546C22semidirect
48. #2545C22semidirect
69. #1991C22semidirect
70. #1990C22semidirect
71. #1983C22semidirect
72. #1981C22semidirect
73. #1958C22semidirect
74. #1956C22semidirect
75. #1949C22semidirect
76. #1948C22semidirect
77. #1939C22semidirect
78. #1939C22semidirect
79. #1939C22semidirect
80. #1928C22semidirect
81. #1927C22semidirect
82. #1927C22semidirect
83. #1926C22semidirect
88. #1805C22semidirect
89. #1803C22semidirect
90. #1802C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31052
C2×#31012
#376332
#341862
#331212
#331192
#297272
#297232
#246862
#246802
#246772
#246362
#246302
#246272
#219522
#218642
#218622