Finite Group Databasebeta

← #10943828

Subgroups — #10943828

Order 768.

4004 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#235C211yesmaximal, characteristic, complemented
2C3×#2847C211yesmaximal, characteristic, complemented
3C2×#5040C211yesmaximal, characteristic, complemented
4C2×#5037C214yesmaximal, complemented
5C2×#5036C211yesmaximal, characteristic, complemented
6#72126C211yesmaximal, characteristic, complemented
7#69525C211yesmaximal, characteristic, complemented
8#69522C214yesmaximal, complemented
9#69521C211yesmaximal, characteristic, complemented
10#32880—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×#121C2211yescharacteristic
22C2×C3×#121C2212yescomplemented
23C2×C3×#120—21no—
24C22×#214C2212yescomplemented
25C22×#213C2212yescomplemented
26C3×#386—22no—
27C3×#386—21no—
28C3×#380C2211yescharacteristic
29C3×#380C2212yescomplemented
30C2×#830C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#830C2214yescomplemented
33#5040—24no—
34#5037—216no—
35#5036—24no—
36C2×#810C2212yescomplemented
37C2×#810C2214yescomplemented
38C2×#809C2214yescomplemented
39C2×#809C2212yescomplemented
40C23×#41—31no—
41C2×#449—34no—
42C2×#448—31no—
43C2×#447—31no—
44#2847D311yescharacteristic, complemented
45#1788—34no—
46#1787—31no—
47#1786—31no—
48#1779—31no—
49C25×C3D411yescharacteristic
50C24×D3—21no—
51C22×C3×D4—21no—
52C22×C3×D4—22no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—22no—
55C23×C3×C4—22no—
56C23×C3×C4—28no—
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D412yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2312yes—
61C22×#67—216no—
62C22×#67—28no—
63C22×#67—24no—
64C23×Dic3—21no—
65C2×#235—216no—
66C2×#235—264no—
67C2×#235—216no—
68C2×#235—216no—
69C2×#235—264no—
70C2×#235—28no—
71C2×#235—28no—
72C2×#235—28no—
73C2×#235—28no—
74C2×#235—28no—
75C2×#235D4116yescomplemented
76C22×D12—22no—
77C22×D12—21no—
78C22×D12—28no—
79C22×D12D414yescomplemented
80C22×D12C2312yes—
81C22×C4×D3—28no—
82C22×C4×D3—216no—
83C22×C4×D3—24no—
84C22×Dic6—22no—
85C22×Dic6—21no—
86C22×Dic6—28no—
87C22×Dic6D414yescomplemented
88C22×Dic6C2312yes—
89C3×#121—22no—
90C3×#121—28no—
91C3×#121—22no—
92C3×#120—44no—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—22no—
99C2×#214—28no—
100C2×#214—24no—
101C2×#213—24no—
102C2×#213—28no—
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105C3×#95—21no—
106C3×#95—22no—
107#830—22no—
108#830—22no—
109#830—28no—
110#810—28no—
111#810—24no—
112#809—24no—
113#809—28no—
114C22×#41—34no—
115C22×#41—64no—
116C22×#41—64no—
117C22×#41—616no—
118C22×#41—62no—
119C23×Q8—31no—
120C23×D4—31no—
121C23×D4—61no—
122C24×C4—61no—
123C24×C4C2×D311yescharacteristic
124C22×SD16—32no—
125C22×D8—32no—
126C2×#121C2×D311yescharacteristic
127C2×#121C2×D312yescomplemented
128C2×#120—21no—
129#449—616no—
130#448—64no—
131#447—64no—
132C2×#103—32no—
133C2×#103—34no—
134C2×#102—32no—
135C2×#102—34no—
136C2×C22⋊C8—31no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8—34no—
139#386—22no—
140#386—21no—
141#380C2×D311yescharacteristic
142#380C2×D312yescomplemented
143C24×C3—24no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—21no—
148C24×C3—42no—
149C24×C3C2×D412yes—
150C24×C3C2×D411yescharacteristic
151C23×D3—28no—
152C23×D3—24no—
153C23×D3—22no—
154C23×D3—44no—
155C23×D3—42no—
156C23×D3—42no—
157C2×C3×D4—21no—
158C2×C3×D4—24no—
159C2×C3×D4—21no—
160C2×C3×D4—44no—
161C2×C3×D4—44no—
162C2×C3×D4—44no—
163C2×C3×D4—42no—
164C2×C3×D4—42no—
165C2×C3×D4—42no—
166C22×C3×C4—216no—
167C22×C3×C4—216no—
168C22×C3×C4—22no—
169C22×C3×C4—24no—
170C22×C3×C4—24no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—24no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D418yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2411yescharacteristic
185C2×#67—264no—
186C2×#67—216no—
187C2×#67—216no—
188C2×#67—216no—
189C22×Dic3—28no—
190C22×Dic3—24no—
191C22×Dic3—22no—
192#235—2128no—
193#235—232no—
194#235—232no—
195#235—232no—
196#235—232no—
197C2×D12—28no—
198C2×D12—216no—
199C2×D12—24no—
200C2×D12—24no—
201C2×D12—22no—
202C2×D12—216no—
203C2×D12C2×D418yes—
204C2×D12C2×D414yes—
205C2×C4×D3—264no—
206C2×C4×D3—216no—
207C2×C4×D3—216no—
208C2×C4×D3—216no—
209C2×Dic6—28no—
210C2×Dic6—216no—
211C2×Dic6—24no—
212C2×Dic6—22no—
213C2×Dic6—24no—
214C2×Dic6—216no—
215C2×Dic6C2×D418yes—
216C2×Dic6C2×D414yes—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—21no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—42no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—44no—
224#214—48no—
225#213—48no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—24no—
228C2×C3⋊C8—24no—
229C2×C3⋊C8—22no—
230C25—61no—
231C25#6711yescharacteristic
232C2×#41—316no—
233C2×#41—68no—
234C2×#41—68no—
235C2×#41—68no—
236C2×#41—68no—
237C2×#41—664no—
238C2×#41—664no—
239C2×#41—616no—
240C2×#41—616no—
241C2×#41—616no—
242C2×#41—68no—
243C22×Q8—34no—
244C22×Q8—32no—
245C22×Q8—68no—
246C22×Q8—62no—
247C22×Q8—61no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—34no—
251C22×D4—32no—
252C22×D4—68no—
253C22×D4—616no—
254C22×D4—68no—
255C22×D4—64no—
256C22×D4—62no—
257C22×D4—61no—
258C22×D4C22×D311yescharacteristic
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—28no—
262C23×C4—68no—
263C23×C4—616no—
264C23×C4—61no—
265C23×C4—64no—
266C23×C4#6711yescharacteristic
267C23×C4#6712yes—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D312yes—
270C2×SD16—64no—
271C2×SD16—68no—
272C2×D8—64no—
273C2×D8—68no—
274C22×C8—32no—
275C22×C8—32no—
276#121—22no—
277#121—22no—
278#121—28no—
279#120—44no—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C4⋊C4C22×D312yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286#103—64no—
287#103—68no—
288#102—64no—
289#102—68no—
290C22⋊C8—62no—
291C22⋊C8—62no—
292C22⋊C8—68no—
293#95—21no—
294#95—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—28no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—48no—
305C23×C3—48no—
306C23×C3—44no—
307C23×C3—44no—
308C23×C3—41no—
309C23×C3—42no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—42no—
313C23×C3#12014yes—
314C23×C3#12012yes—
315C23×C3#13512yes—
316C23×C3AGL(1,ℤ/8)12yes—
317C23×C3C22×D411yescharacteristic
318C22×D3—216no—
319C22×D3—24no—
320C22×D3—28no—
321C22×D3—416no—
322C22×D3—48no—
323C22×D3—416no—
324C22×D3—42no—
325C22×D3—42no—
326C22×D3—44no—
327C22×D3—48no—
328C3×D4—42no—
329C3×D4—42no—
330C3×D4—88no—
331C3×D4—84no—
332C2×C3×C4—28no—
333C2×C3×C4—216no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—28no—
340C2×C3×C4—28no—
341C2×C3×C4—41no—
342C2×C3×C4—42no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4#12012yes—
348C2×C3×C4#12014yes—
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D414yes—
351C2×C3×C4C22×D411yescharacteristic
352#67—264no—
353C2×Dic3—216no—
354C2×Dic3—24no—
355C2×Dic3—28no—
356D12—216no—
357D12—24no—
358D12—28no—
359D12#12018yescomplemented
360C4×D3—264no—
361Dic6—216no—
362Dic6—28no—
363Dic6—24no—
364Dic6#12018yescomplemented
365C3⋊C8—42no—
366C3⋊C8—42no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—42no—
373C24—68no—
374C24—64no—
375C24—62no—
376C24—124no—
377C24—122no—
378C24—122no—
379C24C2×#6712yes—
380C24C2×#6711yescharacteristic
381#41—632no—
382#41—632no—
383#41—6128no—
384#41—632no—
385#41—632no—
386C2×Q8—34no—
387C2×Q8—38no—
388C2×Q8—616no—
389C2×Q8—68no—
390C2×Q8—616no—
391C2×Q8—64no—
392C2×Q8—62no—
393C2×Q8—64no—
394C2×D4—21no—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—34no—
398C2×D4—38no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—616no—
406C2×D4—68no—
407C2×D4—616no—
408C2×D4—664no—
409C2×D4—616no—
410C2×D4—616no—
411C2×D4—616no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—62no—
415C22×C4—216no—
416C22×C4—216no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—664no—
429C22×C4—616no—
430C22×C4—616no—
431C22×C4—68no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—616no—
435C22×C4D3×D412yes—
436C22×C4D3×D412yes—
437C22×C4D3×D418yes—
438C22×C4C2×#6712yes—
439C22×C4C2×#6711yescharacteristic
440C22×C4C23×D311yescharacteristic
441SD16—128no—
442D8—128no—
443C2×C8—62no—
444C2×C8—64no—
445C2×C8—64no—
446C2×C8—62no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—24no—
458C22×C3—24no—
459C22×C3—24no—
460C22×C3—24no—
461C22×C3—416no—
462C22×C3—42no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—82no—
469C22×C3—81no—
470C22×C3—82no—
471C22×C3#44914yescomplemented
472C22×C3C2×#12011yescharacteristic
473C22×C3C2×#12012yes—
474C22×C3C2×#13511yescharacteristic
475C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
476C22×C3#57512yes—
477C2×D3—28no—
478C2×D3—432no—
479C2×D3—44no—
480C2×D3—48no—
481C2×D3—44no—
482C2×D3—48no—
483C3×C4—22no—
484C3×C4—24no—
485C3×C4—81no—
486C3×C4—82no—
487C3×C4C2×#12012yes—
488C3×C4C2×#12012yes—
489Dic3—28no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—48no—
500C23—48no—
501C23—44no—
502C23—44no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—616no—
509C23—64no—
510C23—68no—
511C23—1216no—
512C23—1216no—
513C23—128no—
514C23—122no—
515C23—128no—
516C23—122no—
517C23—124no—
518C23#93212yes—
519C23#93714yes—
520C23#93712yes—
521C23#95012yes—
522C23C22×#6711yescharacteristic
523Q8—38no—
524Q8—616no—
525Q8—68no—
526Q8—64no—
527D4—38no—
528D4—42no—
529D4—42no—
530D4—616no—
531D4—664no—
532D4—64no—
533D4—68no—
534D4—88no—
535D4—84no—
536C2×C4—28no—
537C2×C4—216no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—664no—
552C2×C4—616no—
553C2×C4—64no—
554C2×C4—68no—
555C2×C4#93714yes—
556C2×C4#93712yes—
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×D3×D414yes—
559C2×C4C2×D3×D411yescharacteristic
560C8—122no—
561C8—122no—
562C2×C3—24no—
563C2×C3—44no—
564C2×C3—161no—
565C2×C3#284611yescharacteristic
566C2×C3C2×#44911yescharacteristic
567C2×C3#300212yes—
568C2×C3#300511yescharacteristic
569C2×C3#300612yes—
570D3—48no—
571C22—22no—
572C22—24no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—416no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—68no—
584C22—81no—
585C22—81no—
586C22—82no—
587C22—82no—
588C22—81no—
589C22—1232no—
590C22—124no—
591C22—128no—
592C22—128no—
593C22—124no—
594C22#503714yescomplemented
595C22C2×#93211yescharacteristic
596C22C2×#93711yescharacteristic
597C22C2×#93712yes—
598C22C2×#95011yescharacteristic
599C22#582212yes—
600C4—24no—
601C4—22no—
602C4—68no—
603C4—81no—
604C4—82no—
605C4D3×#12012yes—
606C4D3×#12012yes—
607C3#3288011yescharacteristic, complemented, Sylow
608C2—24no—
609C2—44no—
610C2—128no—
611C2—161no—
612C2#8261211yescharacteristic
613C2C2×#503711yescharacteristic
614C2#8262412yes—
615C2#8292612yes—
616C2#8292811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2847C2
socleC23×C3C22×D4
hypercenter#2847D3
hypocenterC3#32880
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2847

Complements

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

NH ≅ G/Ntype
1. C23×#235C2semidirect
2. C3×#2847C2semidirect
3. C2×#5040C2semidirect
4. C2×#5037C2semidirect
5. C2×#5036C2semidirect
6. #72126C2semidirect
7. #69525C2semidirect
8. #69522C2semidirect
9. #69521C2semidirect
22. C2×C3×#121C22semidirect
24. C22×#214C22semidirect
25. C22×#213C22semidirect
29. C3×#380C22semidirect
32. C2×#830C22semidirect
36. C2×#810C22semidirect
37. C2×#810C22semidirect
38. C2×#809C22semidirect
39. C2×#809C22semidirect
44. #2847D3semidirect
75. C2×#235D4semidirect
79. C22×D12D4semidirect
87. C22×Dic6D4semidirect
127. C2×#121C2×D3semidirect
142. #380C2×D3semidirect
359. D12#120semidirect
364. Dic6#120semidirect
471. C22×C3#449semidirect
594. C22#5037semidirect
607. C3#32880semidirect

Maximal subgroups

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

subgroupindex
C23×#2352
C3×#28472
C2×#50402
C2×#50372
C2×#50362
#721262
#695252
#695222
#695212
#328803