Finite Group Databasebeta

← #10943851

Subgroups — #10943851

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×#237C211yesmaximal, characteristic
2C2×#5218C211yesmaximal, characteristic, complemented
3C2×#4783C214yesmaximal, complemented
4C2×#4739C211yesmaximal, characteristic, complemented
5#74049C211yesmaximal, characteristic, complemented
6C3×#1786C211yesmaximal, characteristic, complemented
7#70199C214yesmaximal, complemented
8#69955C211yesmaximal, characteristic, complemented
9#69525C211yesmaximal, characteristic, complemented
10#32883—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C23×#67—21no—
13C24×Dic3—21no—
14C22×#237—24no—
15C22×#237—216no—
16C22×#237—22no—
17C22×#237—24no—
18C22×#237C2214yes—
19C23×C4×D3C2211yescharacteristic
20C23×Dic6C2211yescharacteristic
21C22×#214C2212yescomplemented
22C2×#924C2211yescharacteristic
23C22×Dic12C2212yescomplemented
24C2×#897C2212yescomplemented
25C2×C3×#102C2214yescomplemented
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#380C2211yescharacteristic
28#5218—24no—
29C2×#832C2212yescomplemented
30C2×#830C2211yescharacteristic
31C2×(C2×D3)⋊C8C2214yescomplemented
32C2×#816C2212yescomplemented
33#4987C2212yescomplemented
34C2×#810C2214yescomplemented
35#4955—21no—
36#4950—21no—
37#4783—216no—
38#4739—24no—
39#4668—22no—
40C23×#41—31no—
41C2×#452—34no—
42C2×#451—31no—
43C2×#450—31no—
44#2848—31no—
45#1788—34no—
46#1787—31no—
47#1786D311yescharacteristic, complemented
48#1779—31no—
49C25×C3—21no—
50C24×D3D411yescharacteristic
51C22×C3×D4—28no—
52C22×C3×D4—21no—
53C22×C3×D4—22no—
54C22×C3×D4D414yes—
55C22×C3×D4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C22×#67—216no—
58C22×#67—28no—
59C22×#67—24no—
60C23×Dic3—216no—
61C23×Dic3—28no—
62C23×Dic3—21no—
63C23×Dic3—24no—
64C23×Dic3D411yescharacteristic
65C2×#237—264no—
66C2×#237—216no—
67C2×#237—216no—
68C2×#237—216no—
69C2×#237—264no—
70C2×#237—28no—
71C2×#237—28no—
72C2×#237—28no—
73C2×#237—28no—
74C2×#237—28no—
75C2×#237D4116yes—
76C22×C4×D3—22no—
77C22×C4×D3—22no—
78C22×C4×D3—28no—
79C22×C4×D3D412yes—
80C22×C4×D3C2312yes—
81C22×Dic6—22no—
82C22×Dic6—21no—
83C22×Dic6—28no—
84C22×Dic6D414yes—
85C22×Dic6C2311yescharacteristic
86C22×Dic6C2312yes—
87C22×C3×C8C2312yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C22⋊C4—21no—
90C2×#217—21no—
91C2×#214—24no—
92C2×#214—28no—
93C2×(C2×D3)⋊C4—22no—
94C2×(C2×D3)⋊C4—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3C2311yescharacteristic
97#924—22no—
98#924—22no—
99C2×Dic3⋊C4—21no—
100C22×C3⋊C8C2312yes—
101C2×Dic12—24no—
102C2×Dic12—28no—
103#897—28no—
104C3×#102—28no—
105C3×C22⋊C8—22no—
106C3×#95—21no—
107#832—24no—
108C3⋊#95—22no—
109#830—22no—
110(C2×D3)⋊C8—28no—
111#816—24no—
112#810—28no—
113C22×#41—34no—
114C22×#41—62no—
115C22×#41—64no—
116C22×#41—616no—
117C22×#41—64no—
118C23×Q8—31no—
119C23×D4—61no—
120C23×D4C2×D311yescharacteristic
121C24×C4—31no—
122C24×C4—61no—
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×#102C2×D314yescomplemented
134C2×C22⋊C8—31no—
135C2×C22⋊C8—34no—
136C2×C22⋊C8C2×D311yescharacteristic
137#385—62no—
138#385—61no—
139#380—32no—
140#380C2×D311yescharacteristic
141#379—61no—
142C24×C3—28no—
143C24×C3—24no—
144C24×C3—22no—
145C24×C3—44no—
146C24×C3—42no—
147C24×C3—42no—
148C24×C3C2×D411yescharacteristic
149C23×D3—22no—
150C23×D3—24no—
151C23×D3—22no—
152C23×D3—22no—
153C23×D3—42no—
154C23×D3C2×D412yes—
155C2×C3×D4—28no—
156C2×C3×D4—216no—
157C2×C3×D4—216no—
158C2×C3×D4—24no—
159C2×C3×D4—24no—
160C2×C3×D4—22no—
161C2×C3×D4C2×D414yes—
162C2×C3×D4C2×D418yes—
163C22×C3×C4—24no—
164C22×C3×C4—22no—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2411yescharacteristic
169C2×#67—264no—
170C2×#67—216no—
171C2×#67—216no—
172C2×#67—216no—
173C22×Dic3—28no—
174C22×Dic3—264no—
175C22×Dic3—216no—
176C22×Dic3—216no—
177C22×Dic3—24no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—24no—
181C22×Dic3—216no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3C2×D412yes—
185#237—2128no—
186#237—232no—
187#237—232no—
188#237—232no—
189#237—232no—
190C2×C4×D3—216no—
191C2×C4×D3—216no—
192C2×C4×D3—24no—
193C2×C4×D3—24no—
194C2×C4×D3—22no—
195C2×C4×D3—24no—
196C2×C4×D3—22no—
197C2×C4×D3—24no—
198C2×C4×D3C2×D418yes—
199C2×Dic6—28no—
200C2×Dic6—216no—
201C2×Dic6—24no—
202C2×Dic6—22no—
203C2×Dic6—24no—
204C2×Dic6—216no—
205C2×Dic6—24no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6C2×D418yes—
209C2×Dic6C2×D414yes—
210C2×C3×C8—22no—
211C2×C3×C8—24no—
212C3×C4⋊C4—24no—
213C3×C22⋊C4—42no—
214#217—42no—
215#214—48no—
216(C2×D3)⋊C4—44no—
217(C2×D3)⋊C4—44no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—44no—
221Dic3⋊C4—42no—
222C2×C3⋊C8—24no—
223C2×C3⋊C8—22no—
224Dic12—48no—
225C25—21no—
226C25—31no—
227C2×#41—316no—
228C2×#41—68no—
229C2×#41—68no—
230C2×#41—68no—
231C2×#41—616no—
232C2×#41—616no—
233C2×#41—664no—
234C2×#41—664no—
235C2×#41—616no—
236C2×#41—68no—
237C2×#41—68no—
238C22×Q8—34no—
239C22×Q8—32no—
240C22×Q8—31no—
241C22×Q8—68no—
242C22×Q8—62no—
243C22×Q8—61no—
244C22×D4—28no—
245C22×D4—21no—
246C22×D4—22no—
247C22×D4—64no—
248C22×D4—616no—
249C22×D4—68no—
250C22×D4D1214yes—
251C22×D4C22×D312yes—
252C23×C4—32no—
253C23×C4—31no—
254C23×C4—32no—
255C23×C4—68no—
256C23×C4—64no—
257C23×C4—616no—
258C23×C4—68no—
259C23×C4—61no—
260C23×C4—62no—
261C23×C4—62no—
262C23×C4C22×D311yescharacteristic
263C2×Q16—64no—
264C2×Q16—68no—
265C2×SD16—64no—
266C2×SD16—68no—
267C22×C8—32no—
268C22×C8C22×D312yes—
269C22⋊Q8—68no—
270C22⋊Q8—62no—
271C22⋊Q8—62no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—61no—
275C2×C4⋊C4C22×D312yes—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—62no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—61no—
280#103—68no—
281#103—64no—
282#102—28no—
283#102—64no—
284C22⋊C8—22no—
285C22⋊C8—68no—
286C22⋊C8—62no—
287#95—21no—
288#95—62no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—216no—
292C23×C3—24no—
293C23×C3—28no—
294C23×C3—416no—
295C23×C3—48no—
296C23×C3—416no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3—44no—
300C23×C3—48no—
301C23×C3#12012yes—
302C23×C3#13512yes—
303C23×C3#13712yes—
304C23×C3C22×D411yescharacteristic
305C22×D3—28no—
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—22no—
312C22×D3—48no—
313C22×D3—48no—
314C22×D3—44no—
315C22×D3—44no—
316C22×D3#12014yes—
317C3×D4—216no—
318C3×D4—24no—
319C3×D4—28no—
320C3×D4#12018yes—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4#12012yes—
328C2×C3×C4C22×D411yescharacteristic
329C2×C3×C4C22×D414yes—
330C2×C3×C4C22×D411yescharacteristic
331#67—264no—
332C2×Dic3—216no—
333C2×Dic3—28no—
334C2×Dic3—264no—
335C2×Dic3—24no—
336C2×Dic3—28no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3#12014yes—
344C4×D3—216no—
345C4×D3—28no—
346C4×D3—28no—
347Dic6—216no—
348Dic6—28no—
349Dic6—24no—
350Dic6—42no—
351Dic6—42no—
352Dic6#12018yes—
353C3×C8—42no—
354C3⋊C8—42no—
355C24—28no—
356C24—24no—
357C24—22no—
358C24—32no—
359C24—44no—
360C24—42no—
361C24—42no—
362C24—64no—
363C24—62no—
364C24—62no—
365C24—62no—
366C24—122no—
367C24D3×D411yescharacteristic
368#41—632no—
369#41—6128no—
370#41—632no—
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—28no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—22no—
390C2×D4—616no—
391C2×D4—664no—
392C2×D4—616no—
393C2×D4—616no—
394C2×D4C2×D1214yes—
395C2×D4D3×D418yes—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—32no—
399C22×C4—38no—
400C22×C4—64no—
401C22×C4—64no—
402C22×C4—616no—
403C22×C4—616no—
404C22×C4—64no—
405C22×C4—616no—
406C22×C4—68no—
407C22×C4—664no—
408C22×C4—616no—
409C22×C4—616no—
410C22×C4—62no—
411C22×C4—64no—
412C22×C4—62no—
413C22×C4—64no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—64no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4C2×D1212yes—
420C22×C4D3×D411yescharacteristic
421C22×C4D3×D412yes—
422C22×C4C23×D311yescharacteristic
423Q16—128no—
424SD16—128no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—62no—
428C2×C8—64no—
429C4⋊C4—24no—
430C4⋊C4—61no—
431C4⋊C4—61no—
432C4⋊C4—124no—
433C4⋊C4—122no—
434C22⋊C4—42no—
435C22⋊C4—124no—
436C22⋊C4—124no—
437C22⋊C4—122no—
438C22×C3—22no—
439C22×C3—24no—
440C22×C3—22no—
441C22×C3—24no—
442C22×C3—28no—
443C22×C3—432no—
444C22×C3—44no—
445C22×C3—48no—
446C22×C3—44no—
447C22×C3—48no—
448C22×C3#45214yescomplemented
449C22×C3C2×#12011yescharacteristic
450C22×C3C2×#12012yes—
451C22×C3C2×#13511yescharacteristic
452C22×C3C2×#13711yescharacteristic
453C22×C3#57512yes—
454C2×D3—24no—
455C2×D3—24no—
456C2×D3—24no—
457C2×D3—416no—
458C2×D3—42no—
459C2×D3—42no—
460C2×D3—42no—
461C2×D3—42no—
462C3×C4—22no—
463C3×C4—82no—
464C3×C4C2×#12012yes—
465C3×C4C2×#12012yes—
466Dic3—28no—
467Dic3—24no—
468Dic3—81no—
469Dic3—81no—
470C23—22no—
471C23—22no—
472C23—216no—
473C23—24no—
474C23—28no—
475C23—34no—
476C23—416no—
477C23—48no—
478C23—416no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—48no—
483C23—68no—
484C23—62no—
485C23—62no—
486C23—62no—
487C23—62no—
488C23—62no—
489C23—62no—
490C23—128no—
491C23—128no—
492C23—124no—
493C23—124no—
494C23#88212yes—
495C23#91212yes—
496C23#91512yes—
497C23C2×D3×D411yescharacteristic
498Q8—38no—
499Q8—616no—
500Q8—68no—
501Q8—64no—
502Q8—122no—
503Q8—122no—
504D4—216no—
505D4—24no—
506D4—28no—
507D4—664no—
508D4#88218yes—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—34no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—616no—
517C2×C4—68no—
518C2×C4—616no—
519C2×C4—664no—
520C2×C4—62no—
521C2×C4—64no—
522C2×C4—62no—
523C2×C4—68no—
524C2×C4—68no—
525C2×C4—68no—
526C2×C4—121no—
527C2×C4—121no—
528C2×C4—122no—
529C2×C4—122no—
530C2×C4#88212yes—
531C2×C4C22×D1211yescharacteristic
532C2×C4C2×D3×D414yes—
533C2×C4C2×D3×D411yescharacteristic
534C8—42no—
535C8—122no—
536C2×C3—24no—
537C2×C3—48no—
538C2×C3#284611yescharacteristic
539C2×C3C2×#45211yescharacteristic
540C2×C3#300212yes—
541C2×C3#300511yescharacteristic
542C2×C3#300712yes—
543D3—44no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—22no—
548C22—28no—
549C22—432no—
550C22—48no—
551C22—44no—
552C22—48no—
553C22—44no—
554C22—64no—
555C22—64no—
556C22—64no—
557C22—1216no—
558C22—122no—
559C22—122no—
560C22—122no—
561C22—122no—
562C22#478314yescomplemented
563C22C2×#88211yescharacteristic
564C22D3×#12012yes—
565C22C2×#91211yescharacteristic
566C22C2×#91511yescharacteristic
567C22#577312yes—
568C4—22no—
569C4—64no—
570C4—68no—
571C4—82no—
572C4—241no—
573C4—241no—
574C4C2×#88212yes—
575C4D3×#12012yes—
576C3#3288311yescharacteristic, complemented, Sylow
577C2—24no—
578C2—48no—
579C2—124no—
580C2#7386111yescharacteristic
581C2C2×#478311yescharacteristic
582C2#7892711yescharacteristic
583C2#8000112yes—
584C2#8014712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1786C2
socleC23×C3C22×D4
prime radicalC23×#237C2
hypercenter#1786D3
hypocenterC3#32883
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1786

Complements

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

NH ≅ G/Ntype
2. C2×#5218C2semidirect
3. C2×#4783C2semidirect
4. C2×#4739C2semidirect
5. #74049C2semidirect
6. C3×#1786C2semidirect
7. #70199C2semidirect
8. #69955C2semidirect
9. #69525C2semidirect
21. C22×#214C22semidirect
23. C22×Dic12C22semidirect
24. C2×#897C22semidirect
25. C2×C3×#102C22semidirect
29. C2×#832C22semidirect
31. C2×(C2×D3)⋊C8C22semidirect
32. C2×#816C22semidirect
33. #4987C22semidirect
34. C2×#810C22semidirect
47. #1786D3semidirect
133. C2×#102C2×D3semidirect
448. C22×C3#452semidirect
562. C22#4783semidirect
576. C3#32883semidirect

Maximal subgroups

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

subgroupindex
C23×#2372
C2×#52182
C2×#47832
C2×#47392
#740492
C3×#17862
#701992
#699552
#695252
#328833