Finite Group Databasebeta

← #10943983

Subgroups — #10943983

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×C4×D12C211yesmaximal, characteristic, complemented
2#82081C211yesmaximal, characteristic, complemented
3C2×#4843C212yesmaximal, complemented
4C2×#4732C212yesmaximal, complemented
5C3×#1797C211yesmaximal, characteristic, complemented
6#70479C212yesmaximal, complemented
7#69932C212yesmaximal, complemented
8#68673C214yesmaximal, complemented
9#32902—31nomaximal, Sylow
10C23×D12C2211yescharacteristic
11C23×C4×D3—21no—
12C22×C3×C42C2211yescharacteristic
13C2×#930—21no—
14C2×#930—21no—
15C2×#930C2212yes—
16C22×(C2×D3)⋊C4—21no—
17C22×C4⋊Dic3C2211yescharacteristic
18C22×D24C2214yes—
19C2×#874C2211yescharacteristic, complemented
20C2×C4×D12—21no—
21C2×C4×D12—21no—
22C2×C4×D12—28no—
23C2×C4×D12C2212yes—
24C2×C4×D12C2212yes—
25C2×C3×C4⋊C8C2212yescomplemented
26C2×C3×C22⋊C8C2212yes—
27C2×#821C2214yes—
28C2×#821C2214yescomplemented
29#4939—21no—
30#4939—21no—
31#4939C2212yescomplemented
32#4935C2211yescharacteristic, complemented
33#4843—28no—
34#4732—28no—
35C3×#336C2212yescomplemented
36#4551C2214yescomplemented
37C22×C4×D4—31no—
38C2×#459—32no—
39C2×#447—32no—
40#2856—31no—
41#1879—34no—
42#1874—32no—
43#1802—32no—
44#1797D311yescharacteristic, complemented
45C24×D3—21no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×#67—22no—
50C23×Dic3—21no—
51C22×D12—21no—
52C22×D12—22no—
53C22×D12—24no—
54C22×D12—24no—
55C22×D12—21no—
56C22×D12—21no—
57C22×D12D414yescomplemented
58C22×D12C2312yes—
59C22×D12C2312yes—
60C22×C4×D3—28no—
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—44no—
64C22×C4×D3—44no—
65C22×C3×C8C2314yes—
66C2×C3×C42—24no—
67C2×C3×C42D411yescharacteristic
68C2×C3×C42D411yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×C3×C42C2311yescharacteristic
71#930—24no—
72#930—24no—
73#930—44no—
74#930—44no—
75C2×(C2×D3)⋊C4—28no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3—21no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2312yes—
87C2×D24—28no—
88C2×D24—28no—
89C2×D24—28no—
90#874—24no—
91C4×D12—28no—
92C4×D12—28no—
93C4×D12—24no—
94C4×D12—24no—
95C4×D12—24no—
96C4×D12D418yescomplemented
97C3×C4⋊C8—24no—
98C3×C22⋊C8—24no—
99#821—28no—
100#821—28no—
101C23×D4—31no—
102C24×C4—61no—
103C22×D8—34no—
104C2×#127—31no—
105C2×#121—32no—
106C2×#121—61no—
107C2×#121—61no—
108C2×C4×D4—32no—
109C2×C4×D4—32no—
110C2×C4×D4—68no—
111C2×C4×D4—61no—
112C2×C4×D4—61no—
113C22×C4⋊C4—31no—
114C22×C22⋊C4—61no—
115C22×C42C2×D311yescharacteristic
116#459—68no—
117#447—68no—
118C2×C4⋊C8C2×D312yescomplemented
119C2×#102—34no—
120C2×#102—34no—
121C2×C22⋊C8C2×D312yes—
122#390—32no—
123#390—61no—
124#390—61no—
125#384—31no—
126#340—34no—
127#336C2×D312yescomplemented
128C24×C3C2×D411yescharacteristic
129C23×D3—24no—
130C23×D3—24no—
131C23×D3—24no—
132C23×D3—22no—
133C23×D3—22no—
134C23×D3—42no—
135C23×D3—42no—
136C23×D3—42no—
137C23×D3—42no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—41no—
145C22×C3×C4D814yescomplemented
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4#4112yes—
153C22×C3×C4C2411yescharacteristic
154C2×#67—44no—
155C2×#67—44no—
156C2×#67—44no—
157C22×Dic3—24no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—42no—
161C22×Dic3—42no—
162C2×D12—24no—
163C2×D12—24no—
164C2×D12—216no—
165C2×D12—24no—
166C2×D12—22no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—216no—
170C2×D12—24no—
171C2×D12—22no—
172C2×D12—24no—
173C2×D12—22no—
174C2×D12—42no—
175C2×D12—42no—
176C2×D12—42no—
177C2×D12—42no—
178C2×D12—42no—
179C2×D12—42no—
180C2×D12C2×D414yes—
181C2×D12C2×D414yes—
182C2×D12#4114yes—
183C2×C4×D3—28no—
184C2×C4×D3—28no—
185C2×C4×D3—44no—
186C2×C4×D3—432no—
187C2×C4×D3—44no—
188C2×C4×D3—44no—
189C2×C4×D3—44no—
190C2×C3×C8—24no—
191C2×C3×C8—24no—
192C2×C3×C8—24no—
193C3×C42—22no—
194C3×C42—22no—
195C3×C42—22no—
196C3×C42C2×D414yes—
197(C2×D3)⋊C4—28no—
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4—44no—
200(C2×D3)⋊C4—44no—
201(C2×D3)⋊C4—44no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3—22no—
206C4⋊Dic3—42no—
207C4⋊Dic3—42no—
208C4⋊Dic3C2×D414yes—
209D24—416no—
210C25—61no—
211C22×D4—34no—
212C22×D4—32no—
213C22×D4—32no—
214C22×D4—64no—
215C22×D4—64no—
216C22×D4—62no—
217C22×D4—61no—
218C22×D4—61no—
219C22×D4—61no—
220C22×D4—62no—
221C23×C4—68no—
222C23×C4—61no—
223C23×C4—62no—
224C23×C4—62no—
225C23×C4—124no—
226C23×C4—124no—
227C23×C4D1211yescharacteristic
228C23×C4D1211yescharacteristic
229C23×C4C22×D311yescharacteristic
230C2×D8—68no—
231C2×D8—68no—
232C2×D8—68no—
233C22×C8C22×D314yes—
234#127—64no—
235#121—64no—
236#121—64no—
237#121—124no—
238#121—124no—
239C4×D4—38no—
240C4×D4—64no—
241C4×D4—64no—
242C4×D4—68no—
243C4×D4—68no—
244C4×D4—64no—
245C2×C4⋊C4—32no—
246C2×C4⋊C4—32no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C22⋊C4—68no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C42—24no—
258C2×C42D1211yescharacteristic
259C2×C42D1211yescharacteristic
260C2×C42C22×D311yescharacteristic
261C2×C42C22×D311yescharacteristic
262C4⋊C8—24no—
263#102—68no—
264#102—68no—
265C22⋊C8—24no—
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3#12112yes—
271C23×C3C2×D812yes—
272C23×C3AGL(1,ℤ/8)12yes—
273C23×C3C22×D411yescharacteristic
274C22×D3—216no—
275C22×D3—24no—
276C22×D3—24no—
277C22×D3—24no—
278C22×D3—48no—
279C22×D3—48no—
280C22×D3—48no—
281C22×D3—48no—
282C22×D3—48no—
283C22×D3—42no—
284C22×D3—42no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—48no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—28no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—44no—
308C2×C3×C4#12012yes—
309C2×C3×C4#12012yes—
310C2×C3×C4#12112yes—
311C2×C3×C4C2×D812yes—
312C2×C3×C4C2×D812yes—
313C2×C3×C4AGL(1,ℤ/8)12yes—
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C22×D412yes—
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4C2×#4112yes—
318#67—88no—
319C2×Dic3—24no—
320C2×Dic3—48no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326D12—216no—
327D12—24no—
328D12—24no—
329D12—24no—
330D12—44no—
331D12—44no—
332D12—84no—
333D12—84no—
334D12#12118yescomplemented
335C4×D3—416no—
336C4×D3—416no—
337C3×C8—44no—
338C24—64no—
339C24—64no—
340C24—64no—
341C24—62no—
342C24—62no—
343C24—122no—
344C24—122no—
345C24—122no—
346C24—122no—
347C24C2×D1211yescharacteristic
348C2×D4—34no—
349C2×D4—34no—
350C2×D4—34no—
351C2×D4—616no—
352C2×D4—64no—
353C2×D4—64no—
354C2×D4—616no—
355C2×D4—64no—
356C2×D4—62no—
357C2×D4—64no—
358C2×D4—62no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—64no—
363C2×D4—122no—
364C2×D4—122no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—124no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—122no—
371C2×D4—122no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—41no—
379C22×C4—68no—
380C22×C4—64no—
381C22×C4—62no—
382C22×C4—68no—
383C22×C4—62no—
384C22×C4—124no—
385C22×C4—1232no—
386C22×C4—122no—
387C22×C4—122no—
388C22×C4—124no—
389C22×C4—124no—
390C22×C4—124no—
391C22×C4D2414yescomplemented
392C22×C4C2×D1211yescharacteristic
393C22×C4C2×D1211yescharacteristic
394C22×C4C2×D1211yescharacteristic
395C22×C4C2×D1211yescharacteristic
396C22×C4C2×D1211yescharacteristic
397C22×C4D3×D412yes—
398C22×C4#23912yes—
399C22×C4C23×D311yescharacteristic
400D8—1216no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C4⋊C4—34no—
405C4⋊C4—64no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—62no—
409C4⋊C4—122no—
410C4⋊C4—122no—
411C22⋊C4—68no—
412C22⋊C4—68no—
413C22⋊C4—124no—
414C22⋊C4—124no—
415C22⋊C4—124no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42D3×D414yes—
420C22×C3—22no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3#45914yescomplemented
427C22×C3C2×#12011yescharacteristic
428C22×C3C2×#12111yescharacteristic
429C22×C3#52511yescharacteristic
430C22×C3C22×D811yescharacteristic
431C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
432C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
433C22×C3#57511yescharacteristic
434C2×D3—28no—
435C2×D3—432no—
436C2×D3—44no—
437C2×D3—44no—
438C2×D3—44no—
439C2×D3—44no—
440C2×D3—44no—
441C2×D3—44no—
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—41no—
453C3×C4#44714yescomplemented
454C3×C4C2×#12112yes—
455C3×C4#52512yes—
456Dic3—44no—
457Dic3—82no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—616no—
463C23—64no—
464C23—64no—
465C23—64no—
466C23—128no—
467C23—128no—
468C23—128no—
469C23—128no—
470C23—128no—
471C23—122no—
472C23—122no—
473C23—122no—
474C23—122no—
475C23—128no—
476C23—122no—
477C23—122no—
478C23—122no—
479C23—122no—
480C23—122no—
481C23—122no—
482C23—122no—
483C23#89512yes—
484C23C2×D2412yes—
485C23#90812yes—
486C23C22×D1211yescharacteristic
487D4—38no—
488D4—616no—
489D4—64no—
490D4—64no—
491D4—64no—
492D4—124no—
493D4—124no—
494D4—244no—
495D4—248no—
496D4—244no—
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—44no—
510C2×C4—64no—
511C2×C4—1216no—
512C2×C4—128no—
513C2×C4—122no—
514C2×C4—122no—
515C2×C4—1216no—
516C2×C4—122no—
517C2×C4—122no—
518C2×C4—122no—
519C2×C4#88212yes—
520C2×C4#88212yes—
521C2×C4#89512yes—
522C2×C4C2×D2412yes—
523C2×C4C2×D2412yes—
524C2×C4#90812yes—
525C2×C4C2×D3×D411yescharacteristic
526C2×C4C2×D3×D412yes—
527C2×C4C2×#23911yescharacteristic
528C2×C4D3×#4112yes—
529C8—44no—
530C2×C3—24no—
531C2×C3#284711yescharacteristic
532C2×C3C2×#44711yescharacteristic
533C2×C3#300611yescharacteristic
534C2×C3C2×#45911yescharacteristic
535C2×C3#303611yescharacteristic
536C2×C3#310011yescharacteristic
537C2×C3#310911yescharacteristic
538D3—48no—
539D3—162no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—68no—
547C22—1232no—
548C22—124no—
549C22—124no—
550C22—124no—
551C22—124no—
552C22—124no—
553C22—124no—
554C22—242no—
555C22—242no—
556C22—242no—
557C22—242no—
558C22—242no—
559C22—242no—
560C22—242no—
561C22#484314yescomplemented
562C22C2×#88211yescharacteristic
563C22C2×#89511yescharacteristic
564C22#552411yescharacteristic
565C22C22×D2411yescharacteristic
566C22C2×#90811yescharacteristic
567C22C2×#90811yescharacteristic
568C22#574811yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—41no—
573C4—124no—
574C4—242no—
575C4#473214yescomplemented
576C4D3×#12112yes—
577C4#562712yes—
578C3#3290211yescharacteristic, complemented, Sylow
579C2—24no—
580C2—128no—
581C2—482no—
582C2#7404811yescharacteristic
583C2C2×#473211yescharacteristic
584C2#7863911yescharacteristic
585C2C2×#484311yescharacteristic
586C2#7905511yescharacteristic
587C2#7929711yescharacteristic
588C2#7934311yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
hypercenter#1797D3
hypocenterC3#32902
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
1. C22×C4×D12C2semidirect
2. #82081C2semidirect
3. C2×#4843C2semidirect
4. C2×#4732C2semidirect
5. C3×#1797C2semidirect
6. #70479C2semidirect
7. #69932C2semidirect
8. #68673C2semidirect
19. C2×#874C22semidirect
25. C2×C3×C4⋊C8C22semidirect
28. C2×#821C22semidirect
31. #4939C22semidirect
32. #4935C22semidirect
35. C3×#336C22semidirect
36. #4551C22semidirect
44. #1797D3semidirect
57. C22×D12D4semidirect
96. C4×D12D4semidirect
118. C2×C4⋊C8C2×D3semidirect
127. #336C2×D3semidirect
145. C22×C3×C4D8semidirect
334. D12#121semidirect
391. C22×C4D24semidirect
426. C22×C3#459semidirect
453. C3×C4#447semidirect
561. C22#4843semidirect
575. C4#4732semidirect
578. C3#32902semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D122
#820812
C2×#48432
C2×#47322
C3×#17972
#704792
#699322
#686732
#329023