Finite Group Databasebeta

← #10943981

Subgroups — #10943981

Order 768.

2008 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#5217C212yesmaximal, complemented
3C2×#5015C212yesmaximal, complemented
4#82081C211yesmaximal, characteristic, complemented
5#71613C214yesmaximal, complemented
6#71604C212yesmaximal, complemented
7#71560C212yesmaximal, complemented
8#71530C211yesmaximal, characteristic, complemented
9#32902—31nomaximal, Sylow
10C23×C3×D4C2211yescharacteristic
11C24×C3×C4—21no—
12C2×C3×C4×D4—28no—
13C2×C3×C4×D4—21no—
14C2×C3×C4×D4—21no—
15C2×C3×C4×D4C2212yes—
16C2×C3×C4×D4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4—21no—
19C22×C3×C42C2211yescharacteristic
20C22×#213C2214yes—
21C2×#930—21no—
22C2×#930—21no—
23C2×#930C2212yes—
24C2×#874C2211yescharacteristic, complemented
25#5217—28no—
26C2×#832C2214yes—
27C2×#830C2212yes—
28#5015—28no—
29C2×#809C2214yescomplemented
30#4939—21no—
31#4939—21no—
32#4939C2212yescomplemented
33#4935C2211yescharacteristic, complemented
34C2×C4⋊(C3⋊C8)C2212yescomplemented
35#4529C2214yescomplemented
36#4524C2212yescomplemented
37C22×C4×D4D311yescharacteristic, complemented
38C2×#459—32no—
39C2×#447—32no—
40#2856—31no—
41#1879—34no—
42#1874—32no—
43#1802—32no—
44#1797—31no—
45C25×C3—21no—
46C22×C3×D4—24no—
47C22×C3×D4—24no—
48C22×C3×D4—21no—
49C22×C3×D4—22no—
50C22×C3×D4D414yescomplemented
51C22×C3×D4C2312yes—
52C23×C3×C4—28no—
53C23×C3×C4—21no—
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—44no—
57C23×C3×C4—44no—
58C23×C3×C4D411yescharacteristic
59C23×C3×C4D411yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C22×#67—22no—
62C22×D12—21no—
63C22×D12—21no—
64C22×D12C2312yes—
65C3×C4×D4—28no—
66C3×C4×D4—28no—
67C3×C4×D4—24no—
68C3×C4×D4—24no—
69C3×C4×D4—24no—
70C3×C4×D4D418yescomplemented
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—24no—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—28no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C22⋊C4—22no—
77C2×C3×C42—24no—
78C2×C3×C42D411yescharacteristic
79C2×C3×C42D411yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2311yescharacteristic
82C2×#213—28no—
83C2×#213—28no—
84C2×#213—28no—
85#930—24no—
86#930—24no—
87#930—44no—
88#930—44no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—22no—
91C2×(C2×D3)⋊C4—22no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3C2312yes—
95C22×C3⋊C8C2314yes—
96#874—24no—
97#832—28no—
98#830—24no—
99#809—28no—
100C4⋊(C3⋊C8)—24no—
101C23×D4C2×D311yescharacteristic
102C24×C4—21no—
103C22×D8—34no—
104C2×#127—31no—
105C2×#121—32no—
106C2×#121—61no—
107C2×#121—61no—
108C2×C4×D4—28no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4C2×D312yes—
112C2×C4×D4C2×D312yes—
113C22×C4⋊C4C2×D311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C42C2×D311yescharacteristic
116#459—68no—
117#447—68no—
118C2×C4⋊C8—32no—
119C2×#102—34no—
120C2×#102—34no—
121C2×C22⋊C8—32no—
122#390—32no—
123#390—61no—
124#390—61no—
125#384—31no—
126#340—34no—
127#336—32no—
128C24×C3—24no—
129C24×C3—24no—
130C24×C3—24no—
131C24×C3—22no—
132C24×C3—42no—
133C24×C3—42no—
134C24×C3—42no—
135C24×C3—42no—
136C24×C3C2×D411yescharacteristic
137C23×D3—22no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—216no—
141C2×C3×D4—216no—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4C2×D414yes—
147C2×C3×D4C2×D414yes—
148C2×C3×D4#4114yes—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—24no—
154C22×C3×C4—28no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—28no—
159C22×C3×C4—41no—
160C22×C3×C4—44no—
161C22×C3×C4—432no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4—44no—
165C22×C3×C4—44no—
166C22×C3×C4—44no—
167C22×C3×C4D814yescomplemented
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4#4112yes—
175C22×C3×C4C2411yescharacteristic
176C2×#67—44no—
177C2×#67—44no—
178C2×#67—44no—
179C22×Dic3—22no—
180C2×D12—24no—
181C2×D12—22no—
182C2×D12—22no—
183C2×D12—24no—
184C2×D12—42no—
185C2×D12—42no—
186C2×D12—42no—
187C2×D12—42no—
188C2×D12—42no—
189C2×D12—42no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4C2×D414yes—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—28no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42C2×D414yes—
199#213—416no—
200(C2×D3)⋊C4—44no—
201(C2×D3)⋊C4—44no—
202(C2×D3)⋊C4—44no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—42no—
206C4⋊Dic3—42no—
207C2×C3⋊C8—24no—
208C2×C3⋊C8—24no—
209C2×C3⋊C8—24no—
210C25—21no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—21no—
215C22×D4—32no—
216C22×D4—62no—
217C22×D4—61no—
218C22×D4—61no—
219C22×D4D1214yescomplemented
220C22×D4C22×D312yes—
221C23×C4—28no—
222C23×C4—21no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4#6711yescharacteristic
228C23×C4#6711yescharacteristic
229C23×C4C22×D311yescharacteristic
230C2×D8—68no—
231C2×D8—68no—
232C2×D8—68no—
233C22×C8—34no—
234#127—64no—
235#121—64no—
236#121—64no—
237#121—124no—
238#121—124no—
239C4×D4—28no—
240C4×D4—28no—
241C4×D4—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4#6718yescomplemented
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—32no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4C22×D312yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C42—24no—
258C2×C42#6711yescharacteristic
259C2×C42#6711yescharacteristic
260C2×C42C22×D311yescharacteristic
261C2×C42C22×D311yescharacteristic
262C4⋊C8—64no—
263#102—68no—
264#102—68no—
265C22⋊C8—64no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—216no—
271C23×C3—24no—
272C23×C3—24no—
273C23×C3—24no—
274C23×C3—48no—
275C23×C3—48no—
276C23×C3—48no—
277C23×C3—48no—
278C23×C3—48no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—48no—
284C23×C3#12112yes—
285C23×C3C2×D812yes—
286C23×C3AGL(1,ℤ/8)12yes—
287C23×C3C22×D411yescharacteristic
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C3×D4—216no—
296C3×D4—24no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4#12118yescomplemented
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—24no—
307C2×C3×C4—24no—
308C2×C3×C4—28no—
309C2×C3×C4—21no—
310C2×C3×C4—24no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—44no—
314C2×C3×C4—416no—
315C2×C3×C4—48no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4—416no—
319C2×C3×C4#12012yes—
320C2×C3×C4#12012yes—
321C2×C3×C4#12112yes—
322C2×C3×C4C2×D812yes—
323C2×C3×C4C2×D812yes—
324C2×C3×C4AGL(1,ℤ/8)12yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D412yes—
327C2×C3×C4C2×#4112yes—
328C2×C3×C4C2×#4111yescharacteristic
329#67—88no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333D12—44no—
334D12—44no—
335D12—84no—
336D12—84no—
337C3⋊C8—44no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24—62no—
347C24C2×#6711yescharacteristic
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—216no—
351C2×D4—216no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—62no—
357C2×D4—62no—
358C2×D4—64no—
359C2×D4—64no—
360C2×D4—122no—
361C2×D4—122no—
362C2×D4—124no—
363C2×D4—124no—
364C2×D4—124no—
365C2×D4—122no—
366C2×D4—122no—
367C2×D4—122no—
368C2×D4—122no—
369C2×D4C2×D1214yes—
370C2×D4#23514yes—
371C2×D4C2×#6714yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—432no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—62no—
391C22×C4#21314yescomplemented
392C22×C4C2×D1212yes—
393C22×C4#23512yes—
394C22×C4C2×#6711yescharacteristic
395C22×C4C2×#6711yescharacteristic
396C22×C4C2×#6711yescharacteristic
397C22×C4C2×#6711yescharacteristic
398C22×C4C2×#6711yescharacteristic
399C22×C4C23×D311yescharacteristic
400D8—1216no—
401C2×C8—64no—
402C2×C8—64no—
403C2×C8—64no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—122no—
409C4⋊C4—122no—
410C4⋊C4C2×#6714yes—
411C22⋊C4—28no—
412C22⋊C4—28no—
413C22⋊C4—124no—
414C22⋊C4—124no—
415C22⋊C4—124no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42C2×#6714yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—28no—
427C22×C3—432no—
428C22×C3—44no—
429C22×C3—44no—
430C22×C3—44no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3—44no—
434C22×C3#45914yescomplemented
435C22×C3C2×#12011yescharacteristic
436C22×C3C2×#12111yescharacteristic
437C22×C3#52511yescharacteristic
438C22×C3C22×D811yescharacteristic
439C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
440C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
441C22×C3#57511yescharacteristic
442C2×D3—82no—
443C2×D3—82no—
444C2×D3—82no—
445C2×D3—82no—
446C2×D3—82no—
447C2×D3—82no—
448C2×D3—82no—
449C3×C4—22no—
450C3×C4—22no—
451C3×C4—22no—
452C3×C4—44no—
453C3×C4—41no—
454C3×C4#44714yescomplemented
455C3×C4C2×#12112yes—
456C3×C4#52512yes—
457Dic3—82no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—216no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—48no—
467C23—48no—
468C23—48no—
469C23—48no—
470C23—48no—
471C23—48no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23—122no—
483C23#93012yes—
484C23C2×#21312yes—
485C23#94912yes—
486C23C22×#6711yescharacteristic
487D4—216no—
488D4—24no—
489D4—24no—
490D4—24no—
491D4—124no—
492D4—124no—
493D4—244no—
494D4—248no—
495D4—244no—
496D4#93018yescomplemented
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—44no—
511C2×C4—416no—
512C2×C4—48no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—416no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4#93012yes—
520C2×C4C2×#21312yes—
521C2×C4C2×#21312yes—
522C2×C4#93212yes—
523C2×C4#95312yes—
524C2×C4#95312yes—
525C2×C4C22×D1211yescharacteristic
526C2×C4C2×#23512yes—
527C2×C4C2×#23511yescharacteristic
528C2×C4C22×#6712yes—
529C8—124no—
530C2×C3—24no—
531C2×C3—48no—
532C2×C3#284711yescharacteristic
533C2×C3C2×#44711yescharacteristic
534C2×C3#300611yescharacteristic
535C2×C3C2×#45911yescharacteristic
536C2×C3#303611yescharacteristic
537C2×C3#310011yescharacteristic
538C2×C3#310911yescharacteristic
539D3—162no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—28no—
547C22—432no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—242no—
555C22—242no—
556C22—242no—
557C22—242no—
558C22—242no—
559C22—242no—
560C22—242no—
561C22#501514yescomplemented
562C22C2×#93011yescharacteristic
563C22#579111yescharacteristic
564C22C22×#21311yescharacteristic
565C22C2×#93211yescharacteristic
566C22C2×#94911yescharacteristic
567C22#582211yescharacteristic
568C22C2×#95311yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—44no—
573C4—41no—
574C4—242no—
575C4#521714yescomplemented
576C4C2×#93012yes—
577C4#579112yes—
578C3#3290211yescharacteristic, complemented, Sylow
579C2—24no—
580C2—48no—
581C2—482no—
582C2C2×#501511yescharacteristic
583C2#8230711yescharacteristic
584C2#8230811yescharacteristic
585C2#8231011yescharacteristic
586C2#8403211yescharacteristic
587C2C2×#521711yescharacteristic
588C2#8408211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C4×D4C2
socleC23×C3C22×D4
hypercenterC22×C4×D4D3
hypocenterC3#32902
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×D4C2semidirect
2. C2×#5217C2semidirect
3. C2×#5015C2semidirect
4. #82081C2semidirect
5. #71613C2semidirect
6. #71604C2semidirect
7. #71560C2semidirect
8. #71530C2semidirect
24. C2×#874C22semidirect
29. C2×#809C22semidirect
32. #4939C22semidirect
33. #4935C22semidirect
34. C2×C4⋊(C3⋊C8)C22semidirect
35. #4529C22semidirect
36. #4524C22semidirect
37. C22×C4×D4D3semidirect
50. C22×C3×D4D4semidirect
70. C3×C4×D4D4semidirect
167. C22×C3×C4D8semidirect
219. C22×D4D12semidirect
244. C4×D4#67semidirect
299. C3×D4#121semidirect
391. C22×C4#213semidirect
434. C22×C3#459semidirect
454. C3×C4#447semidirect
496. D4#930semidirect
561. C22#5015semidirect
575. C4#5217semidirect
578. C3#32902semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D42
C2×#52172
C2×#50152
#820812
#716132
#716042
#715602
#715302
#329023