Finite Group Databasebeta

← #10943847

Subgroups — #10943847

Order 768.

3926 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
1C23×#235C211yesmaximal, characteristic
2C3×#2848C211yesmaximal, characteristic, complemented
3C2×#5050C211yesmaximal, characteristic, complemented
4C2×#5047C214yesmaximal, complemented
5C2×#5046C211yesmaximal, characteristic, complemented
6#72126C211yesmaximal, characteristic, complemented
7#69525C211yesmaximal, characteristic, complemented
8#69522C214yesmaximal, complemented
9#69521C211yesmaximal, characteristic, complemented
10#32883—31nomaximal, Sylow
11C24×C3×C4C2211yescharacteristic
12C23×#67—21no—
13C22×#235—22no—
14C22×#235—216no—
15C22×#235—24no—
16C22×#235—24no—
17C22×#235C2214yes—
18C23×D12C2211yescharacteristic
19C23×C4×D3—21no—
20C23×Dic6C2211yescharacteristic
21C2×C3×C22⋊Q8C2211yescharacteristic
22C2×C3×C22⋊Q8C2212yescomplemented
23C22×#216C2212yescomplemented
24C22×#215C2212yescomplemented
25C3×#385—22no—
26C3×#385—21no—
27C3×#380C2211yescharacteristic
28C3×#380C2212yescomplemented
29C3×#379—21no—
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2214yescomplemented
33#5050—24no—
34#5047—216no—
35#5046—24no—
36C2×#810C2214yescomplemented
37C2×#810C2212yescomplemented
38C2×#809C2212yescomplemented
39C2×#809C2214yescomplemented
40C23×#41—31no—
41C2×#452—34no—
42C2×#451—31no—
43C2×#450—31no—
44#2848D311yescharacteristic, complemented
45#1788—34no—
46#1787—31no—
47#1786—31no—
48#1779—31no—
49C25×C3D411yescharacteristic
50C24×D3—21no—
51C22×C3×Q8C2311yescharacteristic
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—28no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—216no—
60C22×#67—28no—
61C22×#67—24no—
62C23×Dic3—21no—
63C2×#235—216no—
64C2×#235—264no—
65C2×#235—216no—
66C2×#235—216no—
67C2×#235—264no—
68C2×#235—28no—
69C2×#235—28no—
70C2×#235—28no—
71C2×#235—28no—
72C2×#235—28no—
73C2×#235D4116yes—
74C22×D12—22no—
75C22×D12—21no—
76C22×D12—28no—
77C22×D12D414yes—
78C22×D12C2312yes—
79C22×C4×D3—28no—
80C22×C4×D3—216no—
81C22×C4×D3—24no—
82C22×Dic6—22no—
83C22×Dic6—21no—
84C22×Dic6—28no—
85C22×Dic6D414yes—
86C22×Dic6C2312yes—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—28no—
89C3×C22⋊Q8—22no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4C2312yes—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—22no—
98C2×#216—28no—
99C2×#216—24no—
100C2×#215—24no—
101C2×#215—28no—
102C22×C3⋊C8C2312yes—
103C22×C3⋊C8C2312yes—
104C3×#95—21no—
105C3×#95—22no—
106#830—22no—
107#830—22no—
108#830—28no—
109#810—28no—
110#810—24no—
111#809—24no—
112#809—28no—
113C22×#41—34no—
114C22×#41—64no—
115C22×#41—64no—
116C22×#41—616no—
117C22×#41—62no—
118C23×Q8—31no—
119C23×D4—31no—
120C23×D4—61no—
121C24×C4—61no—
122C24×C4C2×D311yescharacteristic
123C22×Q16—32no—
124C22×SD16—32no—
125C2×C22⋊Q8C2×D311yescharacteristic
126C2×C22⋊Q8C2×D312yescomplemented
127#452—616no—
128#451—64no—
129#450—64no—
130C2×#103—32no—
131C2×#103—34no—
132C2×#102—32no—
133C2×#102—34no—
134C2×C22⋊C8—31no—
135C2×C22⋊C8—31no—
136C2×C22⋊C8—34no—
137#385—22no—
138#385—21no—
139#380C2×D311yescharacteristic
140#380C2×D312yescomplemented
141#379—21no—
142C24×C3—24no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—42no—
147C24×C3C2×D412yes—
148C24×C3C2×D411yescharacteristic
149C23×D3—28no—
150C23×D3—24no—
151C23×D3—22no—
152C23×D3—44no—
153C23×D3—42no—
154C23×D3—42no—
155C2×C3×Q8—21no—
156C2×C3×Q8—24no—
157C2×C3×Q8—21no—
158C22×C3×C4—216no—
159C22×C3×C4—216no—
160C22×C3×C4—22no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—24no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D418yes—
175C22×C3×C4C2×D411yescharacteristic
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2411yescharacteristic
178C2×#67—264no—
179C2×#67—216no—
180C2×#67—216no—
181C2×#67—216no—
182C22×Dic3—28no—
183C22×Dic3—24no—
184C22×Dic3—22no—
185#235—2128no—
186#235—232no—
187#235—232no—
188#235—232no—
189#235—232no—
190C2×D12—28no—
191C2×D12—216no—
192C2×D12—24no—
193C2×D12—24no—
194C2×D12—22no—
195C2×D12—216no—
196C2×D12C2×D418yes—
197C2×D12C2×D414yes—
198C2×C4×D3—264no—
199C2×C4×D3—216no—
200C2×C4×D3—216no—
201C2×C4×D3—216no—
202C2×Dic6—28no—
203C2×Dic6—216no—
204C2×Dic6—24no—
205C2×Dic6—22no—
206C2×Dic6—24no—
207C2×Dic6—216no—
208C2×Dic6C2×D418yes—
209C2×Dic6C2×D414yes—
210C3×C4⋊C4—21no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—42no—
214C3×C4⋊C4—44no—
215C3×C22⋊C4—44no—
216C3×C22⋊C4—44no—
217C3×C22⋊C4—42no—
218C3×C22⋊C4—42no—
219#216—48no—
220#215—48no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—24no—
223C2×C3⋊C8—24no—
224C2×C3⋊C8—22no—
225C25—61no—
226C25#6711yescharacteristic
227C2×#41—316no—
228C2×#41—68no—
229C2×#41—68no—
230C2×#41—68no—
231C2×#41—68no—
232C2×#41—664no—
233C2×#41—664no—
234C2×#41—616no—
235C2×#41—616no—
236C2×#41—616no—
237C2×#41—68no—
238C22×Q8—34no—
239C22×Q8—32no—
240C22×Q8—68no—
241C22×Q8—62no—
242C22×Q8—61no—
243C22×Q8C22×D311yescharacteristic
244C22×D4—34no—
245C22×D4—32no—
246C22×D4—68no—
247C22×D4—616no—
248C22×D4—68no—
249C22×D4—64no—
250C22×D4—62no—
251C22×D4—61no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—28no—
255C23×C4—68no—
256C23×C4—616no—
257C23×C4—61no—
258C23×C4—64no—
259C23×C4#6711yescharacteristic
260C23×C4#6712yes—
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D312yes—
263C2×Q16—64no—
264C2×Q16—68no—
265C2×SD16—64no—
266C2×SD16—68no—
267C22×C8—32no—
268C22×C8—32no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—28no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4C22×D311yescharacteristic
275C2×C4⋊C4C22×D312yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280#103—64no—
281#103—68no—
282#102—64no—
283#102—68no—
284C22⋊C8—62no—
285C22⋊C8—62no—
286C22⋊C8—68no—
287#95—21no—
288#95—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—28no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—48no—
299C23×C3—48no—
300C23×C3—44no—
301C23×C3—44no—
302C23×C3#12014yes—
303C23×C3#12012yes—
304C23×C3#13512yes—
305C23×C3#13712yes—
306C23×C3C22×D411yescharacteristic
307C22×D3—216no—
308C22×D3—24no—
309C22×D3—28no—
310C22×D3—416no—
311C22×D3—48no—
312C22×D3—416no—
313C22×D3—42no—
314C22×D3—42no—
315C22×D3—44no—
316C22×D3—48no—
317C3×Q8—42no—
318C3×Q8—42no—
319C2×C3×C4—28no—
320C2×C3×C4—216no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—28no—
327C2×C3×C4—28no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4#12012yes—
336C2×C3×C4#12014yes—
337C2×C3×C4C22×D411yescharacteristic
338C2×C3×C4C22×D414yes—
339C2×C3×C4C22×D411yescharacteristic
340#67—264no—
341C2×Dic3—216no—
342C2×Dic3—24no—
343C2×Dic3—28no—
344D12—216no—
345D12—24no—
346D12—28no—
347D12#12018yes—
348C4×D3—264no—
349Dic6—216no—
350Dic6—28no—
351Dic6—24no—
352Dic6#12018yes—
353C3⋊C8—42no—
354C3⋊C8—42no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—42no—
360C24—68no—
361C24—64no—
362C24—62no—
363C24—124no—
364C24—122no—
365C24—122no—
366C24C2×#6712yes—
367C24C2×#6711yescharacteristic
368#41—632no—
369#41—632no—
370#41—6128no—
371#41—632no—
372#41—632no—
373C2×Q8—21no—
374C2×Q8—24no—
375C2×Q8—21no—
376C2×Q8—34no—
377C2×Q8—38no—
378C2×Q8—616no—
379C2×Q8—68no—
380C2×Q8—616no—
381C2×Q8—64no—
382C2×Q8—62no—
383C2×Q8—64no—
384C2×D4—34no—
385C2×D4—38no—
386C2×D4—616no—
387C2×D4—68no—
388C2×D4—616no—
389C2×D4—664no—
390C2×D4—616no—
391C2×D4—616no—
392C2×D4—616no—
393C2×D4—64no—
394C2×D4—64no—
395C2×D4—62no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—664no—
411C22×C4—616no—
412C22×C4—616no—
413C22×C4—68no—
414C22×C4—64no—
415C22×C4—62no—
416C22×C4—616no—
417C22×C4D3×D412yes—
418C22×C4D3×D412yes—
419C22×C4D3×D418yes—
420C22×C4C2×#6712yes—
421C22×C4C2×#6711yescharacteristic
422C22×C4C23×D311yescharacteristic
423Q16—128no—
424SD16—128no—
425C2×C8—62no—
426C2×C8—64no—
427C2×C8—64no—
428C2×C8—62no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22×C3—22no—
439C22×C3—24no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3—24no—
443C22×C3—24no—
444C22×C3—24no—
445C22×C3—416no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3#45214yescomplemented
451C22×C3C2×#12011yescharacteristic
452C22×C3C2×#12012yes—
453C22×C3C2×#13511yescharacteristic
454C22×C3C2×#13711yescharacteristic
455C22×C3#57512yes—
456C2×D3—28no—
457C2×D3—432no—
458C2×D3—44no—
459C2×D3—48no—
460C2×D3—44no—
461C2×D3—48no—
462C3×C4—22no—
463C3×C4—24no—
464C3×C4—81no—
465C3×C4—82no—
466C3×C4—81no—
467C3×C4C2×#12012yes—
468C3×C4C2×#12012yes—
469Dic3—28no—
470C23—22no—
471C23—22no—
472C23—28no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—48no—
480C23—48no—
481C23—44no—
482C23—44no—
483C23—616no—
484C23—64no—
485C23—68no—
486C23—1216no—
487C23—1216no—
488C23—128no—
489C23—122no—
490C23—128no—
491C23—122no—
492C23—124no—
493C23#93714yes—
494C23#93712yes—
495C23#94212yes—
496C23#95012yes—
497C23C22×#6711yescharacteristic
498Q8—38no—
499Q8—42no—
500Q8—42no—
501Q8—616no—
502Q8—68no—
503Q8—64no—
504D4—38no—
505D4—616no—
506D4—664no—
507D4—64no—
508D4—68no—
509C2×C4—28no—
510C2×C4—216no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—664no—
526C2×C4—616no—
527C2×C4—64no—
528C2×C4—68no—
529C2×C4#93714yes—
530C2×C4#93712yes—
531C2×C4C2×D3×D411yescharacteristic
532C2×C4C2×D3×D414yes—
533C2×C4C2×D3×D411yescharacteristic
534C8—122no—
535C8—122no—
536C2×C3—24no—
537C2×C3—44no—
538C2×C3#284611yescharacteristic
539C2×C3C2×#45211yescharacteristic
540C2×C3#300212yes—
541C2×C3#300511yescharacteristic
542C2×C3#300712yes—
543D3—48no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—22no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—416no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—68no—
557C22—1232no—
558C22—124no—
559C22—128no—
560C22—128no—
561C22—124no—
562C22#504714yescomplemented
563C22C2×#93711yescharacteristic
564C22C2×#93712yes—
565C22C2×#94211yescharacteristic
566C22C2×#95011yescharacteristic
567C22#582212yes—
568C4—24no—
569C4—22no—
570C4—68no—
571C4—81no—
572C4—82no—
573C4—81no—
574C4D3×#12012yes—
575C4D3×#12012yes—
576C3#3288311yescharacteristic, complemented, Sylow
577C2—24no—
578C2—44no—
579C2—128no—
580C2#8261211yescharacteristic
581C2C2×#504711yescharacteristic
582C2#8273512yes—
583C2#8292612yes—
584C2#8292811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2848C2
socleC23×C3C22×D4
prime radicalC23×#235C2
hypercenter#2848D3
hypocenterC3#32883
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
2. C3×#2848C2semidirect
3. C2×#5050C2semidirect
4. C2×#5047C2semidirect
5. C2×#5046C2semidirect
6. #72126C2semidirect
7. #69525C2semidirect
8. #69522C2semidirect
9. #69521C2semidirect
22. C2×C3×C22⋊Q8C22semidirect
23. C22×#216C22semidirect
24. C22×#215C22semidirect
28. C3×#380C22semidirect
32. C2×#830C22semidirect
36. C2×#810C22semidirect
37. C2×#810C22semidirect
38. C2×#809C22semidirect
39. C2×#809C22semidirect
44. #2848D3semidirect
126. C2×C22⋊Q8C2×D3semidirect
140. #380C2×D3semidirect
450. C22×C3#452semidirect
562. C22#5047semidirect
576. C3#32883semidirect

Maximal subgroups

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

subgroupindex
C23×#2352
C3×#28482
C2×#50502
C2×#50472
C2×#50462
#721262
#695252
#695222
#695212
#328833