Finite Group Databasebeta

← #6376008

Subgroups — #6376008

Order 512.

2044 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3109C211yesmaximal, characteristic
2#34226C211yesmaximal, characteristic, complemented
3#34058C211yesmaximal, characteristic, complemented
4#22151C212yesmaximal, complemented
5#22145C212yesmaximal, complemented
6#22117C212yesmaximal, complemented
7#22111C212yesmaximal, complemented
8#22091C214yesmaximal, complemented
9#22056C214yesmaximal, complemented
10#21390C211yesmaximal, characteristic, complemented
11#21364C211yesmaximal, characteristic, complemented
12#21290C214yesmaximal, complemented
13#21126C214yesmaximal, complemented
14#20732C211yesmaximal, characteristic, complemented
15#20470C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3109C4116yescomplemented
19C2×#481C2212yes—
20C2×#476C2212yes—
21C2×#459C2212yes—
22C2×#447C2212yes—
23C2×#442C2214yes—
24C2×#432C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2367—22no—
27#2363—22no—
28#2357—21no—
29#2349—21no—
30#2341—21no—
31#2339—21no—
32#2324C2214yes—
33#2319—22no—
34#2319C2212yes—
35#2319C2212yes—
36#2319C2212yescomplemented
37#2316C2212yes—
38#2312—21no—
39#2312—21no—
40#2309C2212yes—
41C4×C4⋊Q8C2212yescomplemented
42C4×#127C2212yescomplemented
43#2293—21no—
44#2293—21no—
45#2293C2212yes—
46#2293C2212yes—
47#2285C2211yescharacteristic
48#2280—22no—
49#2275C2212yescomplemented
50C4×#117C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#1987C2211yescharacteristic, complemented
53#1987C2211yescharacteristic, complemented
54#1982C2211yescharacteristic, complemented
55#1982C2211yescharacteristic, complemented
56#1972C2214yes—
57#1970C2214yes—
58#1948C2212yescomplemented
59#1947C2212yescomplemented
60#1944C2214yescomplemented
61#1941C2214yescomplemented
62#1936C2214yes—
63#1936C2214yescomplemented
64#1934C2214yes—
65#1934C2214yescomplemented
66#1926C2214yes—
67#1925C2214yes—
68#1920C2214yescomplemented
69#1919C2214yescomplemented
70#1879C2214yes—
71#1879C2214yes—
72#1875C2214yes—
73#1864C2214yescomplemented
74#1862C2214yescomplemented
75#1854C2214yes—
76#1823C2212yes—
77#1802C2212yes—
78#1802C2212yes—
79#1786C2212yes—
80#1776C2214yescomplemented
81#1767C2212yescomplemented
82C4×C4⋊C8C2212yescomplemented
83C4×#102C2214yes—
84C4×#102C2214yescomplemented
85#1750C2214yes—
86#1746C2212yescomplemented
87#1726C2212yescomplemented
88C22×#41—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×D8C2314yes—
92#532—24no—
93#532C2×C418yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124—21no—
97C2×#121—21no—
98C2×#121C2312yes—
99C2×C4×Q8—22no—
100C2×C4×D4—21no—
101C2×C4×D4—21no—
102C2×C4×D4—21no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4—24no—
106C2×C4×D4C2×C418yescomplemented
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×C4×D4C2312yes—
110C2×C4×D4C2314yes—
111C2×#117—21no—
112C2×#117—22no—
113C2×#117—22no—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4—21no—
118C22×C42C2311yescharacteristic
119#481C2×C4116yescomplemented
120#476C2×C4116yescomplemented
121#459C2×C4116yes—
122#447C2×C4116yescomplemented
123#442C2×C4132yescomplemented
124#432C2×C418yescomplemented
125#431—24no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×C22⋊C8C2312yes—
139C2×C22⋊C8C2312yes—
140C2×#97C2314yes—
141C2×#97C2312yes—
142C2×C4×C8C2312yes—
143C2×C4×C8C2314yes—
144#391—22no—
145#390—22no—
146#390—22no—
147#390—24no—
148#390—22no—
149#390—22no—
150#390C2314yes—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#388—24no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—24no—
173#386—24no—
174#385—24no—
175#384C2312yes—
176#382—21no—
177#381—22no—
178#381—22no—
179#381—22no—
180#380—21no—
181#380—22no—
182#380—22no—
183#380—22no—
184#380—22no—
185#380—22no—
186#380C2312yes—
187#379—22no—
188#379—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4D412yes—
192C4×C4⋊C4D411yescharacteristic
193C4×C4⋊C4D411yescharacteristic
194C4×C4⋊C4C2314yes—
195C4×C4⋊C4C2312yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2311yescharacteristic
198C4×C4⋊C4C2311yescharacteristic
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—24no—
205C4×C22⋊C4D411yescharacteristic
206C4×C22⋊C4D411yescharacteristic
207C4×C22⋊C4D412yes—
208#376—21no—
209#376C2311yescharacteristic
210#376C2311yescharacteristic
211C2×#95—22no—
212C43—21no—
213C43C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2314yes—
222C25—21no—
223C2×#41—28no—
224C2×#41—84no—
225C22×Q8—21no—
226C22×D4—22no—
227C22×D4—21no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—22no—
231C22×D4—42no—
232C22×D4—42no—
233C22×D4C22×C414yes—
234C22×D4C2412yes—
235C22×D4C2412yes—
236C23×C4—21no—
237C23×C4—22no—
238C23×C4—28no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—24no—
242C23×C4—82no—
243C23×C4—82no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×D8—44no—
248C2×D8—44no—
249C2×D8C22×C4116yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C4⋊Q8—22no—
254C4⋊Q8C22×C414yes—
255#127—22no—
256#127C22×C414yes—
257#124—28no—
258#121—28no—
259#121—24no—
260#121C22×C418yes—
261C4×Q8—84no—
262C4×D4—28no—
263C4×D4—28no—
264C4×D4—28no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—24no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—84no—
271C4×D4—84no—
272C4×D4C22×C418yes—
273C4×D4C22×C418yes—
274C4×D4C22×C418yes—
275#117—42no—
276#117—42no—
277#117—42no—
278#117—42no—
279#117—84no—
280#117—84no—
281#117—82no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
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—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4C22×C414yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42C22×C414yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42#4114yes—
334C2×C42C2412yes—
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—44no—
339#106—44no—
340#106C22×C4116yes—
341C4⋊C8—22no—
342C4⋊C8—22no—
343C4⋊C8C22×C418yes—
344#102—44no—
345#102—44no—
346#102—44no—
347#102—44no—
348#102C22×C4116yes—
349#102C22×C4116yes—
350C22⋊C8—24no—
351C22⋊C8C22×C418yes—
352#97—44no—
353#97C22×C418yes—
354C4×C8—24no—
355C4×C8—44no—
356#95—22no—
357#95—24no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95C2×D411yescharacteristic
368#95C2×D412yes—
369#95C2×D411yescharacteristic
370#95C2×D411yescharacteristic
371#95C2×D412yes—
372#95C2×D411yescharacteristic
373C24—22no—
374C24—22no—
375C24—24no—
376C24—44no—
377C24—81no—
378C24—81no—
379C24—82no—
380C24C22×D411yescharacteristic
381#41—816no—
382C2×Q8—24no—
383C2×Q8—44no—
384C2×D4—28no—
385C2×D4—24no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—48no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—84no—
404C2×D4C23×C414yes—
405C2×D4C23×C414yes—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—42no—
420C22×C4—48no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—816no—
424C22×C4—82no—
425C22×C4—82no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4C4×D414yes—
432C22×C4C4×D414yes—
433C22×C4C23×C412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C2×#4112yes—
444C22×C4C2511yescharacteristic
445D8—416no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8C23×C418yes—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4C23×C414yes—
473C4⋊C4C23×C414yes—
474C22⋊C4—28no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C22⋊C4—82no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—82no—
493C42—24no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—42no—
500C42—42no—
501C42—44no—
502C42—42no—
503C42—42no—
504C42—82no—
505C42—82no—
506C42—84no—
507C42C4×D414yes—
508C42C4×D414yes—
509C42C23×C412yes—
510C42C23×C412yes—
511C42C2×#4112yes—
512C42C2×#4112yes—
513C23—22no—
514C23—22no—
515C23—21no—
516C23—21no—
517C23—24no—
518C23—24no—
519C23—48no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—84no—
531C23—84no—
532C23—88no—
533C23—88no—
534C23C2×C4×D412yes—
535C23C23×D411yescharacteristic
536Q8—48no—
537D4—24no—
538D4—28no—
539D4—24no—
540D4—48no—
541D4—48no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546D4—48no—
547D4—88no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
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—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—88no—
586C2×C4—88no—
587C2×C4—84no—
588C2×C4—168no—
589C2×C4C2×C4×D412yes—
590C2×C4C2×C4×D412yes—
591C2×C4C2×C4×D412yes—
592C2×C4C2×C4×D412yes—
593C2×C4C2×C4×D412yes—
594C2×C4#57612yes—
595C2×C4#57712yes—
596C2×C4C24×C411yescharacteristic
597C2×C4C22×#4111yescharacteristic
598C2×C4#58512yes—
599C2×C4#58512yes—
600C8—28no—
601C8—42no—
602C8—42no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—816no—
615C22—84no—
616C22—82no—
617C22—82no—
618C22—84no—
619C22—82no—
620C22—82no—
621C22C22×C4×D411yescharacteristic
622C22#344912yes—
623C22C2×#57611yescharacteristic
624C22C2×#57711yescharacteristic
625C22#358512yes—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—44no—
636C4—82no—
637C4—82no—
638C4—84no—
639C4#294614yes—
640C4#342812yes—
641C4#345511yescharacteristic
642C4#345511yescharacteristic
643C2—42no—
644C2—84no—
645C2—84no—
646C2#3406511yescharacteristic
647C2C2×#294611yescharacteristic
648C2#6016811yescharacteristic
649C2#6098912yes—
650C2#6099012yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
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. #34226C2semidirect
3. #34058C2semidirect
4. #22151C2semidirect
5. #22145C2semidirect
6. #22117C2semidirect
7. #22111C2semidirect
8. #22091C2semidirect
9. #22056C2semidirect
10. #21390C2semidirect
11. #21364C2semidirect
12. #21290C2semidirect
13. #21126C2semidirect
14. #20732C2semidirect
15. #20470C2semidirect
18. #3109C4semidirect
36. #2319C22semidirect
41. C4×C4⋊Q8C22semidirect
42. C4×#127C22semidirect
49. #2275C22semidirect
52. #1987C22semidirect
53. #1987C22semidirect
54. #1982C22semidirect
55. #1982C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1944C22semidirect
61. #1941C22semidirect
63. #1936C22semidirect
65. #1934C22semidirect
68. #1920C22semidirect
69. #1919C22semidirect
73. #1864C22semidirect
74. #1862C22semidirect
80. #1776C22semidirect
81. #1767C22semidirect
82. C4×C4⋊C8C22semidirect
84. C4×#102C22semidirect
86. #1746C22semidirect
87. #1726C22semidirect
106. C2×C4×D4C2×C4semidirect
119. #481C2×C4semidirect
120. #476C2×C4semidirect
122. #447C2×C4semidirect
123. #442C2×C4semidirect
124. #432C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
#342262
#340582
#221512
#221452
#221172
#221112
#220912
#220562
#213902
#213642
#212902
#211262
#207322
#204702