Finite Group Databasebeta

← #10826965

Subgroups — #10826965

Order 768.

1948 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5606C212yesmaximal, complemented
2C2×#5158C212yesmaximal, complemented
3C2×#4783C212yesmaximal, complemented
4C3×#1999C211yesmaximal, characteristic, complemented
5#71685C212yesmaximal, complemented
6#70201C212yesmaximal, complemented
7#69964C211yesmaximal, characteristic, complemented
8#69955C211yesmaximal, characteristic, complemented
9#68879C212yesmaximal, complemented
10#23923—31nomaximal, Sylow
11C22×#237C2212yes—
12C23×C4×D3C2211yescharacteristic
13C22×C3×D8C2211yescharacteristic, complemented
14C2×C3×#121C2212yes—
15C2×#938C2212yes—
16C2×#937—22no—
17C2×D4×Dic3—22no—
18C22×#214C2212yescomplemented
19C2×#924C2212yes—
20C22×Dic12C2211yescharacteristic, complemented
21#5606—28no—
22#5606—28no—
23C2×#897C2212yes—
24C2×#892C2212yes—
25C2×#883—22no—
26C2×#67⋊C4—22no—
27C2×C3×#102C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#832C2212yescomplemented
30C2×#830C2211yescharacteristic
31#5158—28no—
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yescomplemented
35C2×#810C2212yescomplemented
36#4783—28no—
37C3×#340C2211yescharacteristic, complemented
38#4551C2211yescharacteristic, complemented
39#4529C2212yescomplemented
40C2×#525—32no—
41C2×#452—32no—
42C2×#449—32no—
43#2001—31no—
44#2000—32no—
45#1999D311yescharacteristic, complemented
46#1875—32no—
47#1873—32no—
48#1779—31no—
49C24×D3D411yescharacteristic
50C22×C3×D4—22no—
51C22×C3×D4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C22×#67—22no—
54C22×#67—22no—
55C23×Dic3—22no—
56C23×Dic3D411yescharacteristic
57C2×#237—24no—
58C2×#237—24no—
59C2×#237—24no—
60C2×#237—24no—
61C2×#237—42no—
62C2×#237—42no—
63C22×C4×D3—24no—
64C22×C4×D3—24no—
65C22×C4×D3—41no—
66C22×C4×D3—41no—
67C22×C4×D3D412yes—
68C22×C4×D3C2312yes—
69C22×Dic6C2312yes—
70C2×C3×D8—24no—
71C2×C3×D8—41no—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#121—24no—
75C3×#121—24no—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×#217—22no—
79C2×#217—22no—
80#938—24no—
81#938D418yescomplemented
82#937—216no—
83D4×Dic3—216no—
84C2×#214—24no—
85C2×#214—24no—
86C2×#214—42no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3C2312yes—
91#924—24no—
92#924—24no—
93C2×Dic3⋊C4—22no—
94C2×C4×Dic3—22no—
95C22×C3⋊C8C2312yes—
96C2×Dic12—24no—
97C2×Dic12—41no—
98#897—24no—
99#897D418yescomplemented
100#892—24no—
101#892—24no—
102#883—216no—
103#67⋊C4—216no—
104C3×#102—24no—
105C3×C22⋊C8—41no—
106#832—24no—
107#830—41no—
108(C2×D3)⋊C8—24no—
109(C2×D3)⋊C8—24no—
110#816—24no—
111#810—24no—
112C22×#41—32no—
113C24×C4—31no—
114C22×Q16—31no—
115C22×SD16—32no—
116C22×D8C2×D311yescharacteristic, complemented
117#525—68no—
118#525—68no—
119C2×#123—62no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8—32no—
122C2×#121—32no—
123C2×#121C2×D312yes—
124C2×#120—62no—
125C2×C4×D4—62no—
126C2×C4×D4—62no—
127C2×#117—32no—
128#452—68no—
129#449—68no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#102—32no—
133C2×#102C2×D312yescomplemented
134C2×C22⋊C8—32no—
135C2×C22⋊C8—32no—
136C2×C22⋊C8—31no—
137C2×C22⋊C8C2×D311yescharacteristic
138#340—31no—
139#340—32no—
140#340C2×D311yescharacteristic, complemented
141C24×C3—22no—
142C24×C3C2×D411yescharacteristic
143C23×D3—22no—
144C23×D3—24no—
145C23×D3—41no—
146C23×D3—41no—
147C23×D3—42no—
148C23×D3C2×D412yes—
149C2×C3×D4—28no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—44no—
156C2×C3×D4—44no—
157C2×C3×D4C2×D414yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—28no—
166C2×#67—28no—
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—44no—
170C2×#67—44no—
171C22×Dic3—28no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—24no—
175C22×Dic3—41no—
176C22×Dic3—44no—
177C22×Dic3—44no—
178C22×Dic3C2×D412yes—
179#237—48no—
180#237—48no—
181#237—48no—
182#237—48no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—24no—
188C2×C4×D3—24no—
189C2×C4×D3—44no—
190C2×C4×D3—44no—
191C2×C4×D3—48no—
192C2×C4×D3—42no—
193C2×C4×D3C2×D414yes—
194C2×C4×D3C2×D414yes—
195C2×Dic6—22no—
196C2×Dic6—22no—
197C2×Dic6—22no—
198C2×Dic6—22no—
199C2×Dic6—22no—
200C2×Dic6C2×D414yes—
201C3×D8—44no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4C2×D414yes—
208C3×C22⋊C4—28no—
209#217—28no—
210#217—28no—
211#214—48no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—28no—
214C4⋊Dic3—22no—
215C4⋊Dic3—28no—
216C4⋊Dic3C2×D414yes—
217Dic3⋊C4—28no—
218C4×Dic3—28no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222Dic12—44no—
223C25—31no—
224C2×#41—64no—
225C2×#41—64no—
226C2×#41—64no—
227C2×#41—64no—
228C2×#41—122no—
229C2×#41—122no—
230C22×Q8—32no—
231C22×D4—22no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4C22×D312yes—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—64no—
239C23×C4—64no—
240C23×C4—62no—
241C23×C4—121no—
242C23×C4—121no—
243C23×C4C22×D311yescharacteristic
244C2×Q16—64no—
245C2×Q16—121no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C2×D8—24no—
250C2×D8—41no—
251C22×C8—32no—
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254#123—616no—
255C22⋊Q8—38no—
256C22⋊Q8—64no—
257C22⋊Q8—64no—
258C22⋊Q8—64no—
259#121—24no—
260#121—24no—
261#121—38no—
262#121—64no—
263#120—616no—
264C4×D4—616no—
265C4×D4—616no—
266#117—64no—
267#117—64no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4C22×D312yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—62no—
275C2×C22⋊C4—62no—
276C2×C22⋊C4—62no—
277C2×C42—62no—
278#103—64no—
279#103—64no—
280#102—24no—
281#102—64no—
282C22⋊C8—41no—
283C22⋊C8—64no—
284C22⋊C8—64no—
285C22⋊C8—121no—
286C23×C3—22no—
287C23×C3—21no—
288C23×C3—22no—
289C23×C3—21no—
290C23×C3—24no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3#13512yes—
298C23×C3C22×D411yescharacteristic
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—22no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—44no—
306C22×D3—48no—
307C22×D3—48no—
308C22×D3—44no—
309C22×D3—44no—
310C22×D3#12014yes—
311C3×D4—24no—
312C3×D4—24no—
313C3×D4—416no—
314C3×D4—42no—
315C3×D4—42no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4C22×D412yes—
325C2×C3×C4C22×D412yes—
326C2×C3×C4H(5,𝔽2)11yescharacteristic
327C2×C3×C4H(5,𝔽2)11yescharacteristic
328#67—416no—
329#67—416no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—44no—
335C2×Dic3—416no—
336C2×Dic3—42no—
337C2×Dic3—42no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3#12014yes—
341C4×D3—24no—
342C4×D3—24no—
343C4×D3—24no—
344C4×D3—24no—
345C4×D3—48no—
346Dic6—24no—
347Dic6—24no—
348Dic6—42no—
349Dic6—42no—
350C3×C8—41no—
351C3×C8—41no—
352C3⋊C8—42no—
353C24—22no—
354C24—32no—
355C24—64no—
356C24—62no—
357C24—121no—
358C24—121no—
359C24—122no—
360C24D3×D411yescharacteristic
361#41—128no—
362#41—128no—
363#41—128no—
364#41—128no—
365C2×Q8—34no—
366C2×Q8—62no—
367C2×Q8—62no—
368C2×Q8—62no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×D4—28no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—68no—
380C2×D4—68no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—124no—
384C2×D4—124no—
385C2×D4D3×D414yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—32no—
392C22×C4—34no—
393C22×C4—34no—
394C22×C4—64no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—68no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—122no—
405C22×C4—124no—
406C22×C4—124no—
407C22×C4—124no—
408C22×C4—128no—
409C22×C4—121no—
410C22×C4—124no—
411C22×C4D3×D411yescharacteristic
412C22×C4C23×D311yescharacteristic
413Q16—124no—
414SD16—128no—
415D8—44no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C4⋊C4—22no—
424C4⋊C4—34no—
425C4⋊C4—68no—
426C4⋊C4—68no—
427C4⋊C4—62no—
428C4⋊C4D3×D414yes—
429C22⋊C4—28no—
430C22⋊C4—68no—
431C22⋊C4—68no—
432C22⋊C4—68no—
433C22⋊C4—68no—
434C42—68no—
435C22×C3—22no—
436C22×C3—24no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—42no—
440C22×C3—44no—
441C22×C3—44no—
442C22×C3—44no—
443C22×C3—82no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3—82no—
447C22×C3C2×#12012yes—
448C22×C3#53411yescharacteristic
449C22×C3C2×#13511yescharacteristic
450C22×C3#57511yescharacteristic
451C22×C3#57512yes—
452C2×D3—24no—
453C2×D3—24no—
454C2×D3—24no—
455C2×D3—416no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—42no—
459C2×D3—42no—
460C3×C4—41no—
461C3×C4—44no—
462C3×C4D4211yescharacteristic
463C3×C4D4211yescharacteristic
464C3×C4D4212yes—
465Dic3—24no—
466Dic3—44no—
467Dic3—44no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—24no—
473C23—34no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—62no—
481C23—62no—
482C23—62no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—128no—
487C23—124no—
488C23—124no—
489C23—128no—
490C23—124no—
491C23#91212yes—
492C23C2×D3×D411yescharacteristic
493Q8—64no—
494Q8—64no—
495Q8—122no—
496Q8—122no—
497D4—24no—
498D4—24no—
499D4—416no—
500D4—42no—
501D4—42no—
502D4—1216no—
503D4—1216no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—34no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—62no—
514C2×C4—62no—
515C2×C4—64no—
516C2×C4—64no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—128no—
522C2×C4—124no—
523C2×C4—1216no—
524C2×C4—122no—
525C2×C4—122no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4C2×D3×D412yes—
529C2×C4C2×D3×D412yes—
530C2×C4#100411yescharacteristic
531C2×C4#100911yescharacteristic
532C8—41no—
533C8—41no—
534C8—122no—
535C2×C3—42no—
536C2×C3—84no—
537C2×C3#240311yescharacteristic
538C2×C3#300212yes—
539C2×C3#300512yes—
540C2×C3#319111yescharacteristic
541C2×C3#319711yescharacteristic
542D3—44no—
543C22—22no—
544C22—24no—
545C22—41no—
546C22—42no—
547C22—41no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—64no—
552C22—64no—
553C22—64no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—1216no—
559C22—122no—
560C22—122no—
561C22—122no—
562C22—122no—
563C22D3×#12012yes—
564C22#560511yescharacteristic
565C22C2×#91211yescharacteristic
566C22#576811yescharacteristic
567C22#577312yes—
568C4—44no—
569C4—41no—
570C4—64no—
571C4—124no—
572C4—124no—
573C4#559011yescharacteristic
574C4#560912yes—
575C4#571711yescharacteristic
576C3#2392311yescharacteristic, complemented, Sylow
577C2—42no—
578C2—84no—
579C2—124no—
580C2#7675311yescharacteristic
581C2#8000112yes—
582C2#8008312yes—
583C2#8072111yescharacteristic
584C2#8087011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#23923
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C2×#5606C2semidirect
2. C2×#5158C2semidirect
3. C2×#4783C2semidirect
4. C3×#1999C2semidirect
5. #71685C2semidirect
6. #70201C2semidirect
7. #69964C2semidirect
8. #69955C2semidirect
9. #68879C2semidirect
13. C22×C3×D8C22semidirect
18. C22×#214C22semidirect
20. C22×Dic12C22semidirect
27. C2×C3×#102C22semidirect
29. C2×#832C22semidirect
34. C2×#816C22semidirect
35. C2×#810C22semidirect
37. C3×#340C22semidirect
38. #4551C22semidirect
39. #4529C22semidirect
45. #1999D3semidirect
81. #938D4semidirect
99. #897D4semidirect
116. C22×D8C2×D3semidirect
133. C2×#102C2×D3semidirect
140. #340C2×D3semidirect
576. C3#23923semidirect

Maximal subgroups

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

subgroupindex
C2×#56062
C2×#51582
C2×#47832
C3×#19992
#716852
#702012
#699642
#699552
#688792
#239233