Finite Group Databasebeta

← #10826972

Subgroups — #10826972

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×#5627C212yesmaximal, complemented
2C2×#5187C212yesmaximal, complemented
3C2×#4810C212yesmaximal, complemented
4C3×#2001C211yesmaximal, characteristic, complemented
5#71699C212yesmaximal, complemented
6#70269C212yesmaximal, complemented
7#69961C211yesmaximal, characteristic, complemented
8#69955C211yesmaximal, characteristic, complemented
9#68881C212yesmaximal, complemented
10#23923—31nomaximal, Sylow
11C22×#239C2212yes—
12C23×C4×D3C2211yescharacteristic
13C22×C3×Q16C2211yescharacteristic, complemented
14C2×C3×C22⋊Q8C2212yes—
15C2×#946C2212yes—
16C22×#215C2212yescomplemented
17C2×#930C2212yes—
18C22×D24C2211yescharacteristic, complemented
19#5627—28no—
20#5627—28no—
21C2×#895C2212yes—
22C2×#894—22no—
23C2×#893—22no—
24C2×#892C2212yes—
25C2×#882—22no—
26C2×#67⋊C4—22no—
27C2×C3×#103C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×#835C2212yescomplemented
30C2×#830C2211yescharacteristic
31#5187—28no—
32C2×#821C2212yescomplemented
33C2×(C2×D3)⋊C8C2212yes—
34C2×(C2×D3)⋊C8C2212yes—
35C2×#809C2212yescomplemented
36#4810—28no—
37C3×#340C2211yescharacteristic, complemented
38#4551C2211yescharacteristic, complemented
39#4529C2212yescomplemented
40C2×#525—32no—
41C2×#452—32no—
42C2×#449—32no—
43#2001D311yescharacteristic, complemented
44#2000—32no—
45#1999—31no—
46#1875—32no—
47#1873—32no—
48#1779—31no—
49C24×D3D411yescharacteristic
50C22×C3×Q8C2312yes—
51C23×C3×C4C2311yescharacteristic
52C22×#67—22no—
53C23×Dic3D411yescharacteristic
54C2×#239—24no—
55C2×#239—24no—
56C2×#239—24no—
57C2×#239—24no—
58C2×#239—42no—
59C2×#239—42no—
60C22×D12—22no—
61C22×D12—22no—
62C22×D12C2312yes—
63C22×C4×D3—22no—
64C22×C4×D3—24no—
65C22×C4×D3—24no—
66C22×C4×D3—41no—
67C22×C4×D3—41no—
68C22×C4×D3D412yes—
69C22×C4×D3C2312yes—
70C2×C3×Q16—24no—
71C2×C3×Q16—41no—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×C22⋊Q8—24no—
75C3×C22⋊Q8—24no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—22no—
79#946—24no—
80#946D418yescomplemented
81C2×#215—24no—
82C2×#215—24no—
83C2×#215—42no—
84#930—24no—
85#930—24no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3C2312yes—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—22no—
93C22×C3⋊C8C2312yes—
94C2×D24—24no—
95C2×D24—41no—
96#895—24no—
97#895D418yescomplemented
98#894—216no—
99#893—216no—
100#892—24no—
101#892—24no—
102#882—216no—
103#67⋊C4—216no—
104C3×#103—24no—
105C3×C22⋊C8—41no—
106#835—24no—
107#830—41no—
108#821—24no—
109(C2×D3)⋊C8—24no—
110(C2×D3)⋊C8—24no—
111#809—24no—
112C22×#41—32no—
113C24×C4—31no—
114C22×Q16C2×D311yescharacteristic, complemented
115C22×SD16—32no—
116C22×D8—31no—
117#525—68no—
118#525—68no—
119C2×#123—62no—
120C2×C22⋊Q8—32no—
121C2×C22⋊Q8C2×D312yes—
122C2×#121—32no—
123C2×#121—32no—
124C2×#120—62no—
125C2×C4×D4—62no—
126C2×C4×D4—62no—
127C2×#117—32no—
128#452—68no—
129#449—68no—
130C2×#103—32no—
131C2×#103C2×D312yescomplemented
132C2×#102—32no—
133C2×#102—32no—
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×C3C2×D411yescharacteristic
142C23×D3—22no—
143C23×D3—22no—
144C23×D3—24no—
145C23×D3—41no—
146C23×D3—41no—
147C23×D3—42no—
148C23×D3C2×D412yes—
149C2×C3×Q8—22no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8C2×D414yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C2×#67—28no—
164C2×#67—44no—
165C2×#67—44no—
166C22×Dic3—22no—
167C22×Dic3—24no—
168C22×Dic3—41no—
169C22×Dic3C2×D412yes—
170#239—48no—
171#239—48no—
172#239—48no—
173#239—48no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—28no—
179C2×D12—28no—
180C2×D12—22no—
181C2×D12—44no—
182C2×D12—44no—
183C2×D12—44no—
184C2×D12—44no—
185C2×D12C2×D414yes—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—28no—
191C2×C4×D3—24no—
192C2×C4×D3—24no—
193C2×C4×D3—44no—
194C2×C4×D3—44no—
195C2×C4×D3—48no—
196C2×C4×D3—42no—
197C2×C4×D3—44no—
198C2×C4×D3—44no—
199C2×C4×D3C2×D414yes—
200C2×C4×D3C2×D414yes—
201C3×Q16—44no—
202C2×C3×C8—22no—
203C2×C3×C8—22no—
204C2×C3×C8—21no—
205C2×C3×C8—21no—
206C3×C4⋊C4—28no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4C2×D414yes—
209C3×C22⋊C4—28no—
210#215—48no—
211(C2×D3)⋊C4—28no—
212(C2×D3)⋊C4—28no—
213(C2×D3)⋊C4—28no—
214(C2×D3)⋊C4—28no—
215C4⋊Dic3—22no—
216C4⋊Dic3C2×D414yes—
217Dic3⋊C4—28no—
218C4×Dic3—28no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222D24—44no—
223C25—31no—
224C2×#41—64no—
225C2×#41—64no—
226C2×#41—64no—
227C2×#41—64no—
228C2×#41—122no—
229C2×#41—122no—
230C22×Q8C22×D312yes—
231C22×D4—32no—
232C22×D4—62no—
233C22×D4—62no—
234C22×D4—62no—
235C23×C4—32no—
236C23×C4—31no—
237C23×C4—32no—
238C23×C4—62no—
239C23×C4—64no—
240C23×C4—64no—
241C23×C4—121no—
242C23×C4—121no—
243C23×C4C22×D311yescharacteristic
244C2×Q16—24no—
245C2×Q16—41no—
246C2×SD16—64no—
247C2×SD16—64no—
248C2×SD16—122no—
249C2×D8—64no—
250C2×D8—121no—
251C22×C8—32no—
252C22×C8C22×D311yescharacteristic
253C22×C8C22×D311yescharacteristic
254#123—616no—
255C22⋊Q8—24no—
256C22⋊Q8—24no—
257C22⋊Q8—38no—
258C22⋊Q8—64no—
259#121—38no—
260#121—64no—
261#121—64no—
262#121—64no—
263#120—616no—
264C4×D4—616no—
265C4×D4—616no—
266#117—64no—
267#117—64no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—32no—
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—24no—
279#103—64no—
280#102—64no—
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#13512yes—
291C23×C3C22×D411yescharacteristic
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—24no—
299C22×D3—44no—
300C22×D3—48no—
301C22×D3—48no—
302C22×D3—44no—
303C22×D3—44no—
304C22×D3—42no—
305C22×D3—42no—
306C22×D3—42no—
307C22×D3—42no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3#12014yes—
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—42no—
314C3×Q8—42no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4C22×D412yes—
327C2×C3×C4C22×D412yes—
328C2×C3×C4H(5,𝔽2)11yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330#67—416no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—24no—
334C2×Dic3—44no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3#12014yes—
338D12—24no—
339D12—24no—
340D12—416no—
341D12—416no—
342D12—42no—
343D12—42no—
344C4×D3—24no—
345C4×D3—24no—
346C4×D3—24no—
347C4×D3—24no—
348C4×D3—48no—
349C4×D3—416no—
350C3×C8—41no—
351C3×C8—41no—
352C3⋊C8—42no—
353C24—32no—
354C24—64no—
355C24—62no—
356C24—62no—
357C24—121no—
358C24—121no—
359C24—122no—
360C24D3×D411yescharacteristic
361#41—128no—
362#41—128no—
363#41—128no—
364#41—128no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8D3×D414yes—
371C2×D4—34no—
372C2×D4—68no—
373C2×D4—68no—
374C2×D4—68no—
375C2×D4—62no—
376C2×D4—62no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—62no—
380C2×D4—124no—
381C2×D4—124no—
382C2×D4—124no—
383C2×D4—124no—
384C2×D4—124no—
385C2×D4—124no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—32no—
393C22×C4—34no—
394C22×C4—34no—
395C22×C4—64no—
396C22×C4—64no—
397C22×C4—68no—
398C22×C4—64no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—122no—
405C22×C4—124no—
406C22×C4—124no—
407C22×C4—124no—
408C22×C4—128no—
409C22×C4—124no—
410C22×C4—121no—
411C22×C4D3×D411yescharacteristic
412C22×C4C23×D311yescharacteristic
413Q16—44no—
414SD16—128no—
415D8—124no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—62no—
421C2×C8—62no—
422C2×C8—62no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4—34no—
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×C3C2×#12012yes—
441C22×C3#53411yescharacteristic
442C22×C3C2×#13511yescharacteristic
443C22×C3#57511yescharacteristic
444C22×C3#57512yes—
445C2×D3—24no—
446C2×D3—24no—
447C2×D3—24no—
448C2×D3—416no—
449C2×D3—42no—
450C2×D3—42no—
451C2×D3—42no—
452C2×D3—42no—
453C2×D3—44no—
454C2×D3—44no—
455C2×D3—44no—
456C2×D3—82no—
457C2×D3—82no—
458C2×D3—82no—
459C2×D3—82no—
460C3×C4—41no—
461C3×C4—44no—
462C3×C4—44no—
463C3×C4D4211yescharacteristic
464C3×C4D4211yescharacteristic
465C3×C4D4212yes—
466Dic3—24no—
467Dic3—44no—
468C23—21no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—34no—
473C23—62no—
474C23—62no—
475C23—62no—
476C23—62no—
477C23—62no—
478C23—62no—
479C23—64no—
480C23—128no—
481C23—124no—
482C23—124no—
483C23—128no—
484C23—124no—
485C23—122no—
486C23—122no—
487C23—122no—
488C23—122no—
489C23—122no—
490C23—122no—
491C23#91912yes—
492C23C2×D3×D411yescharacteristic
493Q8—24no—
494Q8—24no—
495Q8—42no—
496Q8—42no—
497D4—64no—
498D4—64no—
499D4—1216no—
500D4—1216no—
501D4—1216no—
502D4—122no—
503D4—122no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—34no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—64no—
519C2×C4—64no—
520C2×C4—64no—
521C2×C4—64no—
522C2×C4—64no—
523C2×C4—128no—
524C2×C4—1216no—
525C2×C4—124no—
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#240311yescharacteristic
537C2×C3#300212yes—
538C2×C3#300512yes—
539C2×C3#319111yescharacteristic
540C2×C3#319711yescharacteristic
541D3—44no—
542D3—84no—
543C22—24no—
544C22—22no—
545C22—41no—
546C22—42no—
547C22—41no—
548C22—64no—
549C22—64no—
550C22—64no—
551C22—1216no—
552C22—122no—
553C22—122no—
554C22—122no—
555C22—122no—
556C22—124no—
557C22—124no—
558C22—124no—
559C22—242no—
560C22—242no—
561C22—242no—
562C22—242no—
563C22D3×#12012yes—
564C22#560511yescharacteristic
565C22C2×#91911yescharacteristic
566C22#576811yescharacteristic
567C22#577712yes—
568C4—44no—
569C4—41no—
570C4—44no—
571C4—64no—
572C4—124no—
573C4#559011yescharacteristic
574C4#560912yes—
575C4#571711yescharacteristic
576C3#2392311yescharacteristic, complemented, Sylow
577C2—42no—
578C2—124no—
579C2—244no—
580C2#7675311yescharacteristic
581C2#8018212yes—
582C2#8022712yes—
583C2#8076511yescharacteristic
584C2#8091011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2001C2
socleC23×C3C22×D4
hypercenter#2001D3
hypocenterC3#23923
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2001

Complements

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

NH ≅ G/Ntype
1. C2×#5627C2semidirect
2. C2×#5187C2semidirect
3. C2×#4810C2semidirect
4. C3×#2001C2semidirect
5. #71699C2semidirect
6. #70269C2semidirect
7. #69961C2semidirect
8. #69955C2semidirect
9. #68881C2semidirect
13. C22×C3×Q16C22semidirect
16. C22×#215C22semidirect
18. C22×D24C22semidirect
27. C2×C3×#103C22semidirect
29. C2×#835C22semidirect
32. C2×#821C22semidirect
35. C2×#809C22semidirect
37. C3×#340C22semidirect
38. #4551C22semidirect
39. #4529C22semidirect
43. #2001D3semidirect
80. #946D4semidirect
97. #895D4semidirect
114. C22×Q16C2×D3semidirect
131. C2×#103C2×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×#56272
C2×#51872
C2×#48102
C3×#20012
#716992
#702692
#699612
#699552
#688812
#239233