Finite Group Databasebeta

← #6375862

Subgroups — #6375862

Order 512.

2334 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3036C211yesmaximal, characteristic
2#34208C211yesmaximal, characteristic, complemented
3#34058C211yesmaximal, characteristic, complemented
4#22162C214yesmaximal, complemented
5#22158C214yesmaximal, complemented
6#22155C214yesmaximal, complemented
7#22149C214yesmaximal, complemented
8#21257C211yesmaximal, characteristic, complemented
9#21235C212yesmaximal, complemented
10#21172C211yesmaximal, characteristic, complemented
11#21092C212yesmaximal, complemented
12#20932C212yesmaximal, complemented
13#20878C212yesmaximal, complemented
14C2×#1804C211yesmaximal, characteristic, complemented
15#20431C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C2×#520C2212yes—
19C22×C4×D4C2211yescharacteristic
20#3036C4116yescomplemented
21C2×#463C2214yes—
22C2×#461C2212yes—
23C2×#459C2212yes—
24C2×#428C2212yes—
25C2×#423C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C2×#417C2212yes—
28C2×#416C2212yes—
29C2×#416C2212yes—
30C22×#102C2212yes—
31C2×#405C2212yes—
32#2364—22no—
33#2364—22no—
34#2352—22no—
35#2352—22no—
36#2328C2212yescomplemented
37#2327C2212yescomplemented
38#2319—22no—
39#2319C2214yes—
40#2316C2211yescharacteristic
41#2316C2211yescharacteristic
42#2312—22no—
43#2311C2212yes—
44#2309C2211yescharacteristic
45#2309C2211yescharacteristic
46C4×#124C2214yescomplemented
47#2293—22no—
48#2293C2214yes—
49C2×#382C2211yescharacteristic
50#2280—22no—
51C4×#117C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#1983C2212yescomplemented
54#1983C2212yescomplemented
55#1982C2212yescomplemented
56#1981C2212yescomplemented
57#1973C2214yes—
58#1972C2214yes—
59#1970C2214yes—
60#1968C2214yes—
61#1949C2214yescomplemented
62#1949C2214yescomplemented
63#1936C2214yescomplemented
64#1935C2214yescomplemented
65#1934C2214yescomplemented
66#1933C2214yescomplemented
67#1880C2214yes—
68#1880C2214yes—
69#1879C2214yes—
70#1879C2214yes—
71#1865C2214yescomplemented
72#1864C2214yescomplemented
73#1863C2214yescomplemented
74#1862C2214yescomplemented
75#1804C2212yes—
76#1804C2212yes—
77#1804C2214yes—
78#1768C2214yescomplemented
79#1767C2212yescomplemented
80C4×C4⋊C8C2212yescomplemented
81#1736C2212yes—
82C2×#340C2212yes—
83#1729C2212yes—
84C22×#41C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4—21no—
87C2×AGL(1,ℤ/8)—44no—
88C2×AGL(1,ℤ/8)—44no—
89C2×AGL(1,ℤ/8)C2×C4116yescomplemented
90C22×SD16C2314yes—
91C22×D8C2314yes—
92C22×M16C2312yes—
93C23×C8C2312yes—
94#532—24no—
95#532C2×C418yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#124C2312yes—
99C2×#121—21no—
100C2×#121—21no—
101#520—44no—
102#520—44no—
103C2×C4×Q8C2314yes—
104C2×C4×D4—24no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—21no—
109C2×C4×D4—24no—
110C2×C4×D4C2×C418yescomplemented
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2314yes—
114C2×#117—22no—
115C2×#117—24no—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C42C2311yescharacteristic
125#463C2×C4132yescomplemented
126#461C2×C4116yescomplemented
127#459C2×C4116yescomplemented
128#428—44no—
129C2×#107C2314yes—
130C2×#106C2314yes—
131#423C2×C418yescomplemented
132C2×C4⋊C8—24no—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138#417C2×C4116yescomplemented
139#416—44no—
140#416—44no—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#102—44no—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2312yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154#405—44no—
155C2×#97C2314yes—
156C2×C4×C8C2314yes—
157#391C2314yes—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#390C2314yes—
162#390C2314yes—
163#389—22no—
164#389—22no—
165#389C2314yes—
166#389C2314yes—
167#388—24no—
168#388—24no—
169#388—24no—
170#388—24no—
171#388—24no—
172#388—24no—
173#387—24no—
174#387—24no—
175#386—24no—
176#386—24no—
177#386—24no—
178#382—21no—
179#382—21no—
180#382C2312yes—
181#382C2312yes—
182#381—24no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380—22no—
187#380—24no—
188#379—22no—
189#379—22no—
190C4×C4⋊C4D412yes—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4C2312yes—
193C4×C4⋊C4C2312yes—
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—24no—
197C4×C22⋊C4—24no—
198C4×C22⋊C4—24no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201#376—21no—
202#376C2312yes—
203C2×#95—21no—
204C2×#95—21no—
205C43—21no—
206C43C2312yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2314yes—
212#336C2314yes—
213C25—21no—
214C2×#41—24no—
215C2×#41—24no—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41C22×C418yes—
220C22×Q8C2412yes—
221C22×D4—22no—
222C22×D4—21no—
223C22×D4—21no—
224C22×D4—21no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4C22×C414yes—
231C22×D4C2412yes—
232C22×D4C2412yes—
233C23×C4—21no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—28no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—84no—
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244AGL(1,ℤ/8)—432no—
245C2×SD16—44no—
246C2×SD16—44no—
247C2×SD16C22×C4116yes—
248C2×D8—44no—
249C2×D8—44no—
250C2×D8C22×C4116yes—
251C2×M16—24no—
252C2×M16—24no—
253C2×M16C22×C418yes—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—22no—
261C4⋊Q8C22×C414yes—
262#127—22no—
263#127C22×C414yes—
264#124—24no—
265#124C22×C418yes—
266#121—28no—
267#121—28no—
268C4×Q8—44no—
269C4×Q8—44no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—44no—
277C4×D4—44no—
278C4×D4—416no—
279C4×D4C22×C418yes—
280C4×D4C22×C418yes—
281#117—22no—
282#117—22no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—22no—
289#117—48no—
290#117—88no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—48no—
302C2×C4⋊C4C22×C414yes—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42C22×C414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42#4114yes—
336C2×C42C2412yes—
337C2×C42C2412yes—
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—44no—
341#106—44no—
342C4⋊C8—22no—
343C4⋊C8—22no—
344C4⋊C8—24no—
345C4⋊C8C22×C418yes—
346C4⋊C8C22×C418yes—
347#103—42no—
348#103—42no—
349#103—42no—
350#103—42no—
351#103C22×C4116yes—
352#102—44no—
353#102—44no—
354#102—42no—
355#102—42no—
356#102—42no—
357#102—42no—
358#102C22×C4116yes—
359#97—44no—
360C4×C8—44no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D414yes—
372#95C2×D414yes—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—42no—
377C24—42no—
378C24—82no—
379C24—82no—
380C24C22×D411yescharacteristic
381#41—216no—
382#41—216no—
383#41—48no—
384#41—48no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8C23×C414yes—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—28no—
396C2×D4—28no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4—82no—
415C2×D4C23×C414yes—
416C2×D4C23×C414yes—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4—816no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4—82no—
440C22×C4C4×D414yes—
441C22×C4C4×D414yes—
442C22×C4C23×C412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C4C2511yescharacteristic
456SD16—416no—
457D8—416no—
458M16—216no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—48no—
468C2×C8C23×C418yes—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4C23×C414yes—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—48no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—44no—
506C42—44no—
507C42—48no—
508C42—44no—
509C42—42no—
510C42—42no—
511C42—82no—
512C42C4×D414yes—
513C42C4×D414yes—
514C42C23×C412yes—
515C42C23×C412yes—
516C42C2×#4112yes—
517C42C2×#4112yes—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—84no—
534C23—84no—
535C23—88no—
536C23—88no—
537C23C2×C4×D412yes—
538C23#53512yes—
539C23#53712yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—42no—
544Q8—42no—
545D4—24no—
546D4—24no—
547D4—24no—
548D4—24no—
549D4—416no—
550D4—48no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—84no—
556D4—84no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—416no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—416no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—88no—
586C2×C4—84no—
587C2×C4—168no—
588C2×C4—164no—
589C2×C4C2×C4×D412yes—
590C2×C4C2×C4×D412yes—
591C2×C4C2×C4×D412yes—
592C2×C4C2×C4×D412yes—
593C2×C4C2×C4×D412yes—
594C2×C4#51814yes—
595C2×C4#52014yes—
596C2×C4#53512yes—
597C2×C4#53712yes—
598C2×C4#54014yes—
599C2×C4#54214yes—
600C2×C4C24×C411yescharacteristic
601C2×C4C22×#4111yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×H(5,𝔽2)11yescharacteristic
605C2×C4C2×2−1+411yescharacteristic
606C8—28no—
607C8—42no—
608C8—42no—
609C22—22no—
610C22—24no—
611C22—22no—
612C22—24no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—816no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22#232414yes—
627C22C22×C4×D411yescharacteristic
628C22C2×#53511yescharacteristic
629C22C2×#53711yescharacteristic
630C22#358412yes—
631C22#358512yes—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—82no—
641C4—82no—
642C4—84no—
643C4#232414yes—
644C4C2×#51811yescharacteristic
645C4C2×#52011yescharacteristic
646C4C2×#54011yescharacteristic
647C4C2×#54211yescharacteristic
648C2—24no—
649C2—84no—
650C2—84no—
651C2C2×#232411yescharacteristic
652C2#6016111yescharacteristic
653C2#6016811yescharacteristic
654C2#6083412yes—
655C2#6093312yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3036C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34208C2semidirect
3. #34058C2semidirect
4. #22162C2semidirect
5. #22158C2semidirect
6. #22155C2semidirect
7. #22149C2semidirect
8. #21257C2semidirect
9. #21235C2semidirect
10. #21172C2semidirect
11. #21092C2semidirect
12. #20932C2semidirect
13. #20878C2semidirect
14. C2×#1804C2semidirect
15. #20431C2semidirect
20. #3036C4semidirect
36. #2328C22semidirect
37. #2327C22semidirect
46. C4×#124C22semidirect
53. #1983C22semidirect
54. #1983C22semidirect
55. #1982C22semidirect
56. #1981C22semidirect
61. #1949C22semidirect
62. #1949C22semidirect
63. #1936C22semidirect
64. #1935C22semidirect
65. #1934C22semidirect
66. #1933C22semidirect
71. #1865C22semidirect
72. #1864C22semidirect
73. #1863C22semidirect
74. #1862C22semidirect
78. #1768C22semidirect
79. #1767C22semidirect
80. C4×C4⋊C8C22semidirect
89. C2×AGL(1,ℤ/8)C2×C4semidirect
110. C2×C4×D4C2×C4semidirect
125. #463C2×C4semidirect
126. #461C2×C4semidirect
127. #459C2×C4semidirect
131. #423C2×C4semidirect
138. #417C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30362
#342082
#340582
#221622
#221582
#221552
#221492
#212572
#212352
#211722
#210922
#209322
#208782
C2×#18042
#204312