Finite Group Databasebeta

← #10943852

Subgroups — #10943852

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
2#84031C211yesmaximal, characteristic, complemented
3C2×#4739C211yesmaximal, characteristic, complemented
4C2×#4733C214yesmaximal, complemented
5C2×#4731C211yesmaximal, characteristic, complemented
6C3×#1779C211yesmaximal, characteristic, complemented
7#69963C211yesmaximal, characteristic, complemented
8#69960C214yesmaximal, complemented
9#69959C211yesmaximal, 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×#924C2211yescharacteristic
22C2×#924C2212yescomplemented
23C22×Dic12C2212yescomplemented
24C22×#204C2212yescomplemented
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2214yescomplemented
28#5247—21no—
29#5212C2211yescharacteristic
30#5212C2212yescomplemented
31#5210—22no—
32#5210—21no—
33C2×#821C2212yescomplemented
34C2×#821C2214yescomplemented
35C2×#816C2214yescomplemented
36C2×#816C2212yescomplemented
37#4739—24no—
38#4733—216no—
39#4731—24no—
40C23×#41—31no—
41C2×#452—34no—
42C2×#451—31no—
43C2×#450—31no—
44#2848—31no—
45#1788—34no—
46#1787—31no—
47#1786—31no—
48#1779D311yescharacteristic, complemented
49C25×C3D411yescharacteristic
50C24×D3—21no—
51C23×C3×C4—22no—
52C23×C3×C4—22no—
53C23×C3×C4—28no—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D412yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C22×#67—216no—
59C22×#67—28no—
60C22×#67—24no—
61C23×Dic3—21no—
62C2×#235—216no—
63C2×#235—264no—
64C2×#235—216no—
65C2×#235—216no—
66C2×#235—264no—
67C2×#235—28no—
68C2×#235—28no—
69C2×#235—28no—
70C2×#235—28no—
71C2×#235—28no—
72C2×#235D4116yes—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12—28no—
76C22×D12D414yes—
77C22×D12C2312yes—
78C22×C4×D3—28no—
79C22×C4×D3—216no—
80C22×C4×D3—24no—
81C22×Dic6—21no—
82C22×Dic6—22no—
83C22×Dic6—28no—
84C22×Dic6D414yes—
85C22×Dic6C2311yescharacteristic
86C22×Dic6C2312yes—
87C22×C3×C8C2312yes—
88C22×C3×C8C2312yes—
89C2×#217—21no—
90C2×#217—22no—
91C2×#217—21no—
92C2×#217—22no—
93C2×C4⋊Dic3C2312yes—
94C2×C4⋊Dic3C2311yescharacteristic
95#924—22no—
96#924—28no—
97#924—22no—
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—22no—
100C2×Dic12—28no—
101C2×Dic12—24no—
102C2×#204—24no—
103C2×#204—28no—
104C3×C22⋊C8—22no—
105C3×C22⋊C8—28no—
106C3×C22⋊C8—22no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#821—24no—
110#821—28no—
111#816—28no—
112#816—24no—
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⋊Q8—31no—
126C2×C22⋊Q8—32no—
127#452—616no—
128#451—64no—
129#450—64no—
130C2×#103—32no—
131C2×#103—34no—
132C2×#102—32no—
133C2×#102—34no—
134C2×C22⋊C8C2×D311yescharacteristic
135C2×C22⋊C8C2×D311yescharacteristic
136C2×C22⋊C8C2×D314yescomplemented
137#385—61no—
138#385—62no—
139#380—31no—
140#380—32no—
141#379—61no—
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—
155C22×C3×C4—216no—
156C22×C3×C4—216no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D418yes—
171C22×C3×C4C2411yescharacteristic
172C2×#67—264no—
173C2×#67—216no—
174C2×#67—216no—
175C2×#67—216no—
176C22×Dic3—28no—
177C22×Dic3—24no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—21no—
182#235—2128no—
183#235—232no—
184#235—232no—
185#235—232no—
186#235—232no—
187C2×D12—28no—
188C2×D12—216no—
189C2×D12—24no—
190C2×D12—24no—
191C2×D12—22no—
192C2×D12—216no—
193C2×D12C2×D418yes—
194C2×D12C2×D414yes—
195C2×C4×D3—264no—
196C2×C4×D3—216no—
197C2×C4×D3—216no—
198C2×C4×D3—216no—
199C2×Dic6—28no—
200C2×Dic6—216no—
201C2×Dic6—24no—
202C2×Dic6—22no—
203C2×Dic6—24no—
204C2×Dic6—216no—
205C2×Dic6—21no—
206C2×Dic6—24no—
207C2×Dic6—21no—
208C2×Dic6C2×D418yes—
209C2×Dic6C2×D414yes—
210C2×C3×C8—24no—
211C2×C3×C8—22no—
212C2×C3×C8—22no—
213C2×C3×C8—24no—
214#217—44no—
215#217—44no—
216#217—42no—
217#217—42no—
218C4⋊Dic3—21no—
219C4⋊Dic3—24no—
220C4⋊Dic3—21no—
221Dic3⋊C4—42no—
222Dic3⋊C4—44no—
223Dic12—48no—
224#204—48no—
225C25—61no—
226C25D1211yescharacteristic
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—31no—
241C22×Q8—68no—
242C22×Q8—61no—
243C22×Q8—62no—
244C22×D4—34no—
245C22×D4—32no—
246C22×D4—68no—
247C22×D4—616no—
248C22×D4—68no—
249C22×D4—64no—
250C22×D4—61no—
251C22×D4—62no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—28no—
255C23×C4—68no—
256C23×C4—616no—
257C23×C4—61no—
258C23×C4—64no—
259C23×C4D1211yescharacteristic
260C23×C4D1212yes—
261C23×C4C22×D311yescharacteristic
262C23×C4C22×D312yes—
263C2×Q16—64no—
264C2×Q16—68no—
265C2×SD16—64no—
266C2×SD16—68no—
267C22×C8C22×D312yes—
268C22×C8C22×D312yes—
269C22⋊Q8—62no—
270C22⋊Q8—62no—
271C22⋊Q8—68no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—61no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—61no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280#103—64no—
281#103—68no—
282#102—64no—
283#102—68no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—28no—
287#95—62no—
288#95—61no—
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—
317C2×C3×C4—28no—
318C2×C3×C4—216no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—28no—
325C2×C3×C4—28no—
326C2×C3×C4#12012yes—
327C2×C3×C4#12014yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D414yes—
330C2×C3×C4C22×D411yescharacteristic
331#67—264no—
332C2×Dic3—216no—
333C2×Dic3—24no—
334C2×Dic3—28no—
335C2×Dic3—42no—
336C2×Dic3—41no—
337C2×Dic3—41no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—42no—
342D12—216no—
343D12—24no—
344D12—28no—
345D12#12018yes—
346C4×D3—264no—
347Dic6—216no—
348Dic6—28no—
349Dic6—24no—
350Dic6—42no—
351Dic6—42no—
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×D1212yes—
367C24C2×D1211yescharacteristic
368#41—632no—
369#41—632no—
370#41—6128no—
371#41—632no—
372#41—632no—
373C2×Q8—34no—
374C2×Q8—38no—
375C2×Q8—616no—
376C2×Q8—68no—
377C2×Q8—616no—
378C2×Q8—64no—
379C2×Q8—62no—
380C2×Q8—64no—
381C2×Q8—61no—
382C2×Q8—64no—
383C2×Q8—61no—
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—664no—
408C22×C4—616no—
409C22×C4—616no—
410C22×C4—68no—
411C22×C4—64no—
412C22×C4—62no—
413C22×C4—616no—
414C22×C4—62no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4C2×D1212yes—
418C22×C4C2×D1211yescharacteristic
419C22×C4D3×D412yes—
420C22×C4D3×D412yes—
421C22×C4D3×D418yes—
422C22×C4C23×D311yescharacteristic
423Q16—128no—
424SD16—128no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—22no—
429C4⋊C4—61no—
430C4⋊C4—64no—
431C4⋊C4—61no—
432C4⋊C4—124no—
433C4⋊C4—122no—
434C22⋊C4—124no—
435C22⋊C4—124no—
436C22⋊C4—122no—
437C22⋊C4—122no—
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×C4C2×#12012yes—
465C3×C4C2×#12012yes—
466Dic3—28no—
467Dic3—81no—
468Dic3—82no—
469Dic3—81no—
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—128no—
488C23—1216no—
489C23—122no—
490C23—122no—
491C23—124no—
492C23—128no—
493C23#88214yes—
494C23#88212yes—
495C23#90412yes—
496C23#90912yes—
497C23C22×D1211yescharacteristic
498Q8—38no—
499Q8—616no—
500Q8—68no—
501Q8—64no—
502Q8—122no—
503Q8—122no—
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—664no—
519C2×C4—616no—
520C2×C4—64no—
521C2×C4—68no—
522C2×C4—122no—
523C2×C4—122no—
524C2×C4—121no—
525C2×C4—121no—
526C2×C4—122no—
527C2×C4—122no—
528C2×C4—122no—
529C2×C4#88214yes—
530C2×C4#88212yes—
531C2×C4C2×D3×D411yescharacteristic
532C2×C4C2×D3×D414yes—
533C2×C4C2×D3×D411yescharacteristic
534C8—42no—
535C8—42no—
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—124no—
561C22—128no—
562C22#473314yescomplemented
563C22C2×#88211yescharacteristic
564C22C2×#88212yes—
565C22C2×#90411yescharacteristic
566C22C2×#90911yescharacteristic
567C22#574812yes—
568C4—24no—
569C4—22no—
570C4—68no—
571C4—242no—
572C4—241no—
573C4—241no—
574C4D3×#12012yes—
575C4D3×#12012yes—
576C3#3288311yescharacteristic, complemented, Sylow
577C2—24no—
578C2—44no—
579C2—128no—
580C2#7386111yescharacteristic
581C2C2×#473311yescharacteristic
582C2#7861012yes—
583C2#7863811yescharacteristic
584C2#7864012yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1779C2
socleC23×C3C22×D4
prime radicalC23×#235C2
hypercenter#1779D3
hypocenterC3#32883
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #84031C2semidirect
3. C2×#4739C2semidirect
4. C2×#4733C2semidirect
5. C2×#4731C2semidirect
6. C3×#1779C2semidirect
7. #69963C2semidirect
8. #69960C2semidirect
9. #69959C2semidirect
22. C2×#924C22semidirect
23. C22×Dic12C22semidirect
24. C22×#204C22semidirect
27. C2×C3×C22⋊C8C22semidirect
30. #5212C22semidirect
33. C2×#821C22semidirect
34. C2×#821C22semidirect
35. C2×#816C22semidirect
36. C2×#816C22semidirect
48. #1779D3semidirect
136. C2×C22⋊C8C2×D3semidirect
450. C22×C3#452semidirect
562. C22#4733semidirect
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
#840312
C2×#47392
C2×#47332
C2×#47312
C3×#17792
#699632
#699602
#699592
#328833