Finite Group Databasebeta

← #7510013

Subgroups — #7510013

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35629C211yesmaximal, characteristic, complemented
2#35266C211yesmaximal, characteristic, complemented
3#32541C214yesmaximal, complemented
4#32538C214yesmaximal, complemented
5#32531C214yesmaximal, complemented
6#32529C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic, complemented
8C2×#2092C211yesmaximal, characteristic, complemented
9#23614C211yesmaximal, characteristic, complemented
10#23610C211yesmaximal, characteristic, complemented
11#23546C211yesmaximal, characteristic
12#23544C211yesmaximal, characteristic
13#21235C211yesmaximal, characteristic
14#21174C211yesmaximal, characteristic, complemented
15#21172C211yesmaximal, characteristic, complemented
16#21158C211yesmaximal, characteristic, complemented
17#21148C211yesmaximal, characteristic, complemented
18#21065C211yesmaximal, characteristic, complemented
19#21058C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C2×#428C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#424C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#405C2211yescharacteristic
36#2860—22no—
37#2644—22no—
38#2514C2214yescomplemented
39#2513C2214yes—
40#2513C2214yescomplemented
41#2512C2214yes—
42#2510—24no—
43#2495—22no—
44#2494—22no—
45#2480—24no—
46#2317C2211yescharacteristic
47#2316C2211yescharacteristic
48#2316C2211yescharacteristic
49#2311C2211yescharacteristic
50#2311C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2274C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2097C2212yescomplemented
60#2095C2212yescomplemented
61#2093C2212yescomplemented
62#2092C2214yes—
63#2092C2214yes—
64#2092C2212yescomplemented
65#2067C2214yes—
66#2066C2214yescomplemented
67#2065C2214yes—
68#2065C2214yescomplemented
69#2055C2214yes—
70#2055C2214yes—
71#2054C2214yes—
72#2054C2214yes—
73#1973C2212yes—
74#1972C2212yes—
75#1970C2212yes—
76#1968C2212yes—
77#1949C2214yescomplemented
78#1949C2214yescomplemented
79#1949C2214yescomplemented
80#1948C2214yescomplemented
81#1943C2214yescomplemented
82#1942C2214yescomplemented
83#1940C2214yescomplemented
84#1939C2214yescomplemented
85#1928C2212yescomplemented
86#1927C2212yescomplemented
87#1926C2214yes—
88#1926C2212yescomplemented
89#1925C2214yes—
90#1925C2212yescomplemented
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C2×AGL(1,ℤ/8)—24no—
100C2×AGL(1,ℤ/8)—42no—
101C2×AGL(1,ℤ/8)—42no—
102C22×SD16C2312yes—
103C22×D8C2312yes—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C2×#126—24no—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—22no—
118C22×C42C2311yescharacteristic
119#428—42no—
120#428—42no—
121#427—42no—
122C2×#107—24no—
123C2×#107C2312yes—
124C2×#107C2312yes—
125C2×#107C2312yes—
126C2×#107C2312yes—
127C2×#107C2312yes—
128C2×#106—42no—
129C2×#106C2312yes—
130C2×#106C2314yes—
131C2×#106C2312yes—
132C2×#106C2312yes—
133#424—24no—
134#424—42no—
135#423—24no—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2314yes—
138C2×C4⋊C8C2314yes—
139#418—42no—
140#417—42no—
141#416—24no—
142#416—42no—
143C2×#103C2311yescharacteristic
144C2×#103C2311yescharacteristic
145C2×#103C2312yes—
146C2×#103C2312yes—
147C2×#102—24no—
148C2×#102—42no—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2311yescharacteristic
152C2×#102C2311yescharacteristic
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157#405—42no—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#401—24no—
161#400—24no—
162#397—24no—
163#397—24no—
164#397—24no—
165#397—24no—
166#397—24no—
167#396—24no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#390—22no—
172#390—22no—
173#390—21no—
174#390—21no—
175#390C2314yes—
176#390C2312yes—
177#390C2312yes—
178#390C2312yes—
179#390C2312yes—
180#390C2314yes—
181#390C2312yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389D412yes—
186#389D412yes—
187#389C2314yes—
188#389C2314yes—
189#388—24no—
190#388—24no—
191#388—24no—
192#388—24no—
193#388—24no—
194#387—24no—
195#386—22no—
196#386—22no—
197#386—22no—
198#385—22no—
199#385D414yes—
200#384C2312yes—
201#383—24no—
202#382—21no—
203#382—21no—
204#382C2312yes—
205#382C2314yes—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#379—22no—
211#379—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2314yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—21no—
221#376—21no—
222C2×#95—22no—
223C43—21no—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C25—21no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—22no—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4C2411yescharacteristic
250C22×D4C2412yes—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2411yescharacteristic
257AGL(1,ℤ/8)—416no—
258C2×SD16—24no—
259C2×SD16—42no—
260C2×SD16—42no—
261C2×D8—24no—
262C2×D8—42no—
263C2×D8—42no—
264C2×M16—22no—
265C2×M16—22no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16#4114yes—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8#4114yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2412yes—
281#126—88no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—84no—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—24no—
345#107—24no—
346#107—42no—
347#107—42no—
348#107—42no—
349#106—44no—
350#106—42no—
351#106—42no—
352#106—42no—
353C4⋊C8—28no—
354C4⋊C8—28no—
355C4⋊C8—44no—
356#103—22no—
357#103—22no—
358#103—42no—
359#103—42no—
360#102—24no—
361#102—24no—
362#102—22no—
363#102—22no—
364#102—44no—
365#102—42no—
366#102—42no—
367#102—44no—
368#97—42no—
369C4×C8—42no—
370#95—24no—
371#95—24no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95C2×D414yes—
383#95C2×D414yes—
384C24—21no—
385C24—22no—
386C24—42no—
387C24—42no—
388C24—42no—
389C24—82no—
390C24—82no—
391C24C22×D411yescharacteristic
392#41—44no—
393#41—44no—
394#41—44no—
395#41—44no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×D4—22no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—44no—
423C2×D4—44no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—82no—
442C22×C4—82no—
443C22×C4—82no—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C42−1+412yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456SD16—48no—
457D8—48no—
458M16—28no—
459M16—44no—
460C2×C8—28no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—22no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—44no—
474C2×C8C2×#4112yes—
475C2×C8C2×#4112yes—
476C2×C8C2×#4114yes—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—84no—
499C4⋊C4—84no—
500C4⋊C4—84no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—48no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—84no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—82no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—84no—
529C42—82no—
530C23—22no—
531C23—22no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—41no—
537C23—42no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—88no—
543C23—84no—
544C23—164no—
545C23—162no—
546C23—162no—
547C23#53712yes—
548C23#54712yes—
549C23#55012yes—
550C23C23×D411yescharacteristic
551Q8—41no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555D4—41no—
556D4—42no—
557D4—42no—
558D4—41no—
559D4—42no—
560D4—41no—
561D4—42no—
562D4—41no—
563D4—84no—
564D4—84no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—162no—
599C2×C4—162no—
600C2×C4—162no—
601C2×C4#53712yes—
602C2×C4#53814yes—
603C2×C4#53914yes—
604C2×C4#54712yes—
605C2×C4#55012yes—
606C2×C4#55614yes—
607C2×C4#56314yes—
608C2×C4C22×#4111yescharacteristic
609C2×C4C22×#4111yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×H(5,𝔽2)11yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C2×C4C2×2−1+411yescharacteristic
614C8—24no—
615C8—22no—
616C8—22no—
617C8—42no—
618C8—42no—
619C22—24no—
620C22—24no—
621C22—22no—
622C22—22no—
623C22—82no—
624C22—84no—
625C22—82no—
626C22—84no—
627C22—81no—
628C22—82no—
629C22—81no—
630C22—82no—
631C22—81no—
632C22—82no—
633C22—168no—
634C22#272714yes—
635C22C2×#53711yescharacteristic
636C22C2×#54711yescharacteristic
637C22C2×#55011yescharacteristic
638C22#358412yes—
639C22#358512yes—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—42no—
644C4—82no—
645C4—82no—
646C4—81no—
647C4—81no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—82no—
652C4#272714yes—
653C4C2×#53811yescharacteristic
654C4C2×#53911yescharacteristic
655C4C2×#55611yescharacteristic
656C4C2×#56311yescharacteristic
657C2—24no—
658C2—161no—
659C2—162no—
660C2C2×#272711yescharacteristic
661C2#6093312yes—
662C2#6127412yes—
663C2#6141311yescharacteristic
664C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×AGL(1,ℤ/8)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35629C2semidirect
2. #35266C2semidirect
3. #32541C2semidirect
4. #32538C2semidirect
5. #32531C2semidirect
6. #32529C2semidirect
7. #23737C2semidirect
8. C2×#2092C2semidirect
9. #23614C2semidirect
10. #23610C2semidirect
14. #21174C2semidirect
15. #21172C2semidirect
16. #21158C2semidirect
17. #21148C2semidirect
18. #21065C2semidirect
19. #21058C2semidirect
38. #2514C22semidirect
40. #2513C22semidirect
59. #2097C22semidirect
60. #2095C22semidirect
61. #2093C22semidirect
64. #2092C22semidirect
66. #2066C22semidirect
68. #2065C22semidirect
77. #1949C22semidirect
78. #1949C22semidirect
79. #1949C22semidirect
80. #1948C22semidirect
81. #1943C22semidirect
82. #1942C22semidirect
83. #1940C22semidirect
84. #1939C22semidirect
85. #1928C22semidirect
86. #1927C22semidirect
88. #1926C22semidirect
90. #1925C22semidirect

Maximal subgroups

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

subgroupindex
#356292
#352662
#325412
#325382
#325312
#325292
#237372
C2×#20922
#236142
#236102
#235462
#235442
#212352
#211742
#211722
#211582
#211482
#210652
#210582