Finite Group Databasebeta

← #10795143

Subgroups — #10795143

Order 768.

1694 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×D4C211yesmaximal, characteristic
2#78517C212yesmaximal, complemented
3C3×C4×#100C211yesmaximal, characteristic, complemented
4C4×#833C211yesmaximal, characteristic, complemented
5#70079C212yesmaximal
6#70063C212yesmaximal, complemented
7C4×#823C212yesmaximal, complemented
8#68994C212yesmaximal, complemented
9#68990C212yesmaximal, complemented
10#20726—31nomaximal, Sylow
11C22×D3×D4C2211yescharacteristic
12C23×C4×D3C412yes—
13C2×C3×C4×D4C2211yescharacteristic
14C2×D4×Dic3C2211yescharacteristic
15C2×C4×#67C412yes—
16C4×D3×D4—22no—
17C4×D3×D4—22no—
18C4×D3×D4—22no—
19C4×D3×D4—22no—
20C2×#893C2211yescharacteristic
21C2×D3×C4⋊C4C2211yescharacteristic
22C2×#67⋊C4C412yes—
23C2×D3×C22⋊C4C412yes—
24C2×C4×D12C2211yescharacteristic
25C2×C42×D3C2211yescharacteristic
26C2×C3×#100C2212yes—
27C3×C4×M16C2212yescomplemented
28C2×#833C2212yes—
29C2×#823C2214yes—
30#5126C2214yes—
31#5118C2214yescomplemented
32C4×#208C2212yescomplemented
33#4705—22no—
34#4705—22no—
35#4685—22no—
36#4685—22no—
37C3×#343C2212yescomplemented
38#4554C2214yescomplemented
39#4530C2212yescomplemented
40C22×C4×D4—31no—
41#2920—32no—
42#1821—32no—
43#1800—32no—
44#1800—32no—
45#1783—32no—
46C4×#100—32no—
47C4×#100—31no—
48C4×#100D311yescharacteristic, complemented
49C24×D3C2×C412yes—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4C2×C412yes—
52C22×#67C2×C412yes—
53C23×Dic3C2×C412yes—
54C2×D3×D4—24no—
55C2×D3×D4—24no—
56C2×D3×D4—22no—
57C2×D3×D4—22no—
58C22×D12C2311yescharacteristic
59C22×C4×D3—22no—
60C22×C4×D3—22no—
61C22×C4×D3—22no—
62C22×C4×D3—22no—
63C22×C4×D3—48no—
64C22×C4×D3C2×C411yescharacteristic
65C22×C4×D3C2×C412yes—
66C22×C4×D3C2×C412yes—
67C22×C4×D3C2×C412yes—
68C22×C4×D3C2×C411yescharacteristic
69C22×C4×D3C2311yescharacteristic
70C2×C3×M16C2314yes—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4C2×C412yes—
75C2×C3×C42C2311yescharacteristic
76C2×#217C2×C412yes—
77D4×Dic3—22no—
78D4×Dic3—22no—
79C4×#67—24no—
80C4×#67—24no—
81C2×(C2×D3)⋊C4C2×C412yes—
82C2×(C2×D3)⋊C4C2×C412yes—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4C2×C412yes—
85C2×C4×Dic3C2×C411yescharacteristic
86C2×C4×Dic3C2×C411yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C2×#208C2314yes—
89#893—22no—
90#893—22no—
91D3×C4⋊C4—22no—
92D3×C4⋊C4—22no—
93#67⋊C4—24no—
94#67⋊C4—24no—
95D3×C22⋊C4—24no—
96D3×C22⋊C4—24no—
97C4×D12—22no—
98C4×D12—22no—
99C42×D3D412yes—
100C42×D3D412yes—
101C42×D3D412yes—
102C42×D3D412yes—
103C3×#100—24no—
104C3×#97—22no—
105C3×C4×C8—22no—
106#833—24no—
107#823—28no—
108(C2×D3)⋊C8—24no—
109(C2×D3)⋊C8—24no—
110Dic3⋊C8—24no—
111Dic3⋊C8—24no—
112#803—22no—
113C4×C3⋊C8—22no—
114C23×D4—31no—
115C24×C4—32no—
116C2×C4×D4—31no—
117C2×C4×D4—32no—
118C2×C4×D4—31no—
119C2×C4×D4—32no—
120C2×C4×D4—31no—
121C2×C4×D4—62no—
122C2×C4×D4—62no—
123C2×C4×D4—62no—
124C2×C4×D4—62no—
125C2×C4×D4C2×D311yescharacteristic
126C22×C4⋊C4—31no—
127C22×C22⋊C4—32no—
128C22×C42—31no—
129#432—62no—
130#432—62no—
131#431—62no—
132#431—62no—
133#423—34no—
134C2×#100—32no—
135C2×#100—34no—
136C2×#100C2×D312yes—
137#407—34no—
138C4×M16—32no—
139C4×M16C2×D312yescomplemented
140#343—34no—
141#343—32no—
142#343C2×D312yescomplemented
143C24×C3C22×C412yes—
144C23×D3—24no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—24no—
148C23×D3—44no—
149C23×D3—44no—
150C23×D3C22×C411yescharacteristic
151C23×D3C22×C412yes—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4#4114yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—44no—
158C22×C3×C4C22×C411yescharacteristic
159C22×C3×C4C22×C412yes—
160C22×C3×C4C22×C411yescharacteristic
161C22×C3×C4C2411yescharacteristic
162C2×#67—24no—
163C2×#67—24no—
164C2×#67—24no—
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—44no—
168C22×Dic3C22×C411yescharacteristic
169C22×Dic3C22×C411yescharacteristic
170C22×Dic3C22×C412yes—
171C22×Dic3C22×C411yescharacteristic
172D3×D4—216no—
173D3×D4—216no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12#4114yes—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—24no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—42no—
187C2×C4×D3—48no—
188C2×C4×D3—48no—
189C2×C4×D3—48no—
190C2×C4×D3—48no—
191C2×C4×D3—44no—
192C2×C4×D3—44no—
193C2×C4×D3—44no—
194C2×C4×D3C22⋊C412yes—
195C2×C4×D3C22⋊C411yescharacteristic
196C2×C4×D3C22⋊C411yescharacteristic
197C2×C4×D3C22⋊C411yescharacteristic
198C2×C4×D3C22⋊C412yes—
199C2×C4×D3C22⋊C411yescharacteristic
200C2×C4×D3C2×D412yes—
201C2×C4×D3C2×D412yes—
202C3×M16—24no—
203C3×M16—24no—
204C2×C3×C8—24no—
205C3×C4⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C42C2×D412yes—
208C3×C42C2×D412yes—
209#217—24no—
210(C2×D3)⋊C4—24no—
211(C2×D3)⋊C4—24no—
212C4⋊Dic3—22no—
213Dic3⋊C4—24no—
214C4×Dic3C22⋊C412yes—
215C4×Dic3C22⋊C412yes—
216C4×Dic3C22⋊C412yes—
217C4×Dic3C22⋊C412yes—
218C4×Dic3C2×D412yes—
219C4×Dic3C2×D412yes—
220#208—24no—
221#208—24no—
222C2×C3⋊C8—24no—
223C25—32no—
224C22×D4—32no—
225C22×D4—31no—
226C22×D4—62no—
227C22×D4—62no—
228C22×D4—64no—
229C22×D4—64no—
230C22×D4C22×D311yescharacteristic
231C23×C4—31no—
232C23×C4—31no—
233C23×C4—31no—
234C23×C4—32no—
235C23×C4—32no—
236C23×C4—32no—
237C23×C4—32no—
238C23×C4—62no—
239C23×C4—62no—
240C23×C4—62no—
241C23×C4—62no—
242C23×C4—128no—
243C23×C4C4×D312yes—
244C2×M16—34no—
245C2×M16C22×D314yes—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—64no—
249C4×D4—64no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—64no—
253C4×D4—64no—
254C4×D4—62no—
255C4×D4—62no—
256C4×D4—62no—
257C4×D4—62no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4C22×D311yescharacteristic
263C2×C22⋊C4—32no—
264C2×C22⋊C4—32no—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—64no—
267C2×C22⋊C4—64no—
268C2×C22⋊C4C4×D312yes—
269C2×C42—31no—
270C2×C42—31no—
271C2×C42—31no—
272C2×C42—32no—
273C2×C42—32no—
274C2×C42—32no—
275C2×C42—32no—
276C2×C42C22×D311yescharacteristic
277C4⋊C8—64no—
278C4⋊C8—64no—
279#100—24no—
280#100—68no—
281#100—64no—
282C22⋊C8—64no—
283C22⋊C8—64no—
284#97—22no—
285#97—62no—
286C4×C8—22no—
287C4×C8—62no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3#11714yes—
293C23×C3C23×C411yescharacteristic
294C22×D3—21no—
295C22×D3—21no—
296C22×D3—22no—
297C22×D3—22no—
298C22×D3—28no—
299C22×D3—28no—
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—44no—
303C22×D3—416no—
304C22×D3—416no—
305C22×D3—44no—
306C22×D3—44no—
307C22×D3—44no—
308C22×D3—42no—
309C22×D3—42no—
310C22×D3C2×C22⋊C411yescharacteristic
311C22×D3C2×C22⋊C411yescharacteristic
312C22×D3#11714yes—
313C3×D4—28no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—41no—
317C2×C3×C4—44no—
318C2×C3×C4—44no—
319C2×C3×C4—42no—
320C2×C3×C4C2×C22⋊C412yes—
321C2×C3×C4C2×C22⋊C411yescharacteristic
322C2×C3×C4C2×C22⋊C411yescharacteristic
323C2×C3×C4C22×D412yes—
324C2×C3×C4C2×#4112yes—
325C2×C3×C4C2×#4112yes—
326#67—216no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—41no—
332C2×Dic3—44no—
333C2×Dic3—44no—
334C2×Dic3—42no—
335C2×Dic3C2×C22⋊C412yes—
336C2×Dic3C2×C22⋊C412yes—
337C2×Dic3C2×C22⋊C411yescharacteristic
338C2×Dic3C2×C22⋊C411yescharacteristic
339D12—28no—
340C4×D3—22no—
341C4×D3—24no—
342C4×D3—24no—
343C4×D3—24no—
344C4×D3—22no—
345C4×D3—22no—
346C4×D3—42no—
347C4×D3—44no—
348C4×D3—44no—
349C4×D3—44no—
350C4×D3—44no—
351C4×D3—44no—
352C4×D3—44no—
353C4×D3#10014yescomplemented
354C3×C8—44no—
355C3⋊C8—44no—
356C24—31no—
357C24—32no—
358C24—62no—
359C24—62no—
360C24—64no—
361C24—64no—
362C24—124no—
363C24—124no—
364C24C2×C4×D312yes—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—34no—
368C2×D4—64no—
369C2×D4—616no—
370C2×D4—616no—
371C2×D4—64no—
372C2×D4—64no—
373C2×D4—62no—
374C2×D4—62no—
375C2×D4#23514yes—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—32no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—32no—
387C22×C4—32no—
388C22×C4—32no—
389C22×C4—32no—
390C22×C4—44no—
391C22×C4—64no—
392C22×C4—62no—
393C22×C4—62no—
394C22×C4—62no—
395C22×C4—62no—
396C22×C4—62no—
397C22×C4—62no—
398C22×C4—62no—
399C22×C4—62no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—124no—
403C22×C4—128no—
404C22×C4—122no—
405C22×C4—128no—
406C22×C4—128no—
407C22×C4—128no—
408C22×C4—124no—
409C22×C4—124no—
410C22×C4—124no—
411C22×C4C2×C4×D311yescharacteristic
412C22×C4C2×C4×D312yes—
413C22×C4C2×C4×D311yescharacteristic
414C22×C4C23×D311yescharacteristic
415M16—24no—
416M16—24no—
417M16—64no—
418M16—64no—
419C2×C8—24no—
420C2×C8—64no—
421C4⋊C4—22no—
422C4⋊C4—64no—
423C4⋊C4—62no—
424C22⋊C4—24no—
425C22⋊C4—64no—
426C22⋊C4—64no—
427C22⋊C4—64no—
428C42—32no—
429C42—32no—
430C42—32no—
431C42—32no—
432C42—32no—
433C42—32no—
434C42D3×D412yes—
435C42D3×D412yes—
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—24no—
439C22×C3—44no—
440C22×C3—44no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3C22×C22⋊C411yescharacteristic
444C22×C3C2×#11711yescharacteristic
445C22×C3C2×#11711yescharacteristic
446C2×D3—24no—
447C2×D3—24no—
448C2×D3—21no—
449C2×D3—21no—
450C2×D3—21no—
451C2×D3—21no—
452C2×D3—24no—
453C2×D3—416no—
454C2×D3—416no—
455C2×D3—44no—
456C2×D3—42no—
457C2×D3—42no—
458C2×D3—44no—
459C2×D3C2×#10012yes—
460C2×D3#41312yes—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—42no—
464C3×C4C2×#10012yes—
465C3×C4#52514yes—
466Dic3—21no—
467Dic3—22no—
468Dic3—22no—
469Dic3—42no—
470Dic3C2×#10012yes—
471C23—22no—
472C23—22no—
473C23—31no—
474C23—31no—
475C23—34no—
476C23—42no—
477C23—42no—
478C23—62no—
479C23—62no—
480C23—68no—
481C23—68no—
482C23—61no—
483C23—61no—
484C23—62no—
485C23—62no—
486C23—122no—
487C23—122no—
488C23—124no—
489C23—124no—
490C23—124no—
491C23—1216no—
492C23—1216no—
493C23—124no—
494C23#87214yes—
495C23C22×C4×D311yescharacteristic
496D4—28no—
497D4—616no—
498D4—68no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—31no—
502C2×C4—34no—
503C2×C4—32no—
504C2×C4—32no—
505C2×C4—31no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—62no—
511C2×C4—62no—
512C2×C4—64no—
513C2×C4—62no—
514C2×C4—61no—
515C2×C4—61no—
516C2×C4—62no—
517C2×C4—64no—
518C2×C4—64no—
519C2×C4—62no—
520C2×C4—122no—
521C2×C4—124no—
522C2×C4—124no—
523C2×C4—124no—
524C2×C4—124no—
525C2×C4—124no—
526C2×C4—124no—
527C2×C4—121no—
528C2×C4—124no—
529C2×C4—124no—
530C2×C4—122no—
531C2×C4D3×C22⋊C412yes—
532C2×C4D3×C22⋊C411yescharacteristic
533C2×C4D3×C22⋊C411yescharacteristic
534C2×C4C2×#23512yes—
535C2×C4C2×D3×D412yes—
536C2×C4D3×#4112yes—
537C8—44no—
538C8—124no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—44no—
542C2×C3#228011yescharacteristic
543C2×C3C22×#10011yescharacteristic
544C2×C3C2×#41311yescharacteristic
545D3—22no—
546D3—22no—
547D3—44no—
548C22—21no—
549C22—21no—
550C22—24no—
551C22—32no—
552C22—32no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22—64no—
558C22—64no—
559C22—61no—
560C22—61no—
561C22—61no—
562C22—61no—
563C22—64no—
564C22—124no—
565C22—124no—
566C22—1216no—
567C22—1216no—
568C22—122no—
569C22—122no—
570C22C2×#87211yescharacteristic
571C22C2×D3×C22⋊C411yescharacteristic
572C22D3×#11711yescharacteristic
573C4—21no—
574C4—22no—
575C4—32no—
576C4—42no—
577C4—62no—
578C4—61no—
579C4—62no—
580C4—122no—
581C4D3×#10012yes—
582C4#554714yes—
583C3#2072611yescharacteristic, complemented, Sylow
584C2—21no—
585C2—21no—
586C2—44no—
587C2—62no—
588C2—62no—
589C2—124no—
590C2#7356111yescharacteristic
591C2C2×D3×#10011yescharacteristic
592C2D3×#41311yescharacteristic
subgroupG/H
centerC2×C4D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×C4×#100C2
socleC22×C3C22×C22⋊C4
prime radicalC22×D3×D4C22
cosocleC22×C3×C4C24
hypercenterC4×#100D3
hypocenterC3#20726
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
2. #78517C2semidirect
3. C3×C4×#100C2semidirect
4. C4×#833C2semidirect
6. #70063C2semidirect
7. C4×#823C2semidirect
8. #68994C2semidirect
9. #68990C2semidirect
27. C3×C4×M16C22semidirect
31. #5118C22semidirect
32. C4×#208C22semidirect
37. C3×#343C22semidirect
38. #4554C22semidirect
39. #4530C22semidirect
48. C4×#100D3semidirect
139. C4×M16C2×D3semidirect
142. #343C2×D3semidirect
353. C4×D3#100semidirect
583. C3#20726semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×D42
#785172
C3×C4×#1002
C4×#8332
#700792
#700632
C4×#8232
#689942
#689902
#207263