Finite Group Databasebeta

← #10943834

Subgroups — #10943834

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×#239C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic, complemented
3C2×#4810C214yesmaximal, complemented
4C2×#4732C211yesmaximal, characteristic, complemented
5#74048C211yesmaximal, characteristic, complemented
6C3×#1787C211yesmaximal, characteristic, complemented
7#70267C214yesmaximal, complemented
8#69955C211yesmaximal, characteristic, complemented
9#69521C211yesmaximal, characteristic, complemented
10#32880—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×#239—24no—
13C22×#239—216no—
14C22×#239—24no—
15C22×#239—22no—
16C22×#239C2214yes—
17C23×D12—21no—
18C23×D12C2211yescharacteristic
19C23×C4×D3—21no—
20C23×C4×D3C2211yescharacteristic
21C22×#215C2212yescomplemented
22C2×#930C2211yescharacteristic
23C22×D24C2212yescomplemented
24C2×#895C2212yescomplemented
25C2×#882—21no—
26C2×C3×#103C2214yescomplemented
27C2×C3×C22⋊C8C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5227—24no—
30C2×#835C2212yescomplemented
31C2×#830C2211yescharacteristic
32C2×#821C2212yescomplemented
33C2×(C2×D3)⋊C8C2214yescomplemented
34#4987C2212yescomplemented
35C2×#809C2214yescomplemented
36#4958—21no—
37#4810—216no—
38#4732—24no—
39#4665—22no—
40C23×#41—31no—
41C2×#449—34no—
42C2×#448—31no—
43C2×#447—31no—
44#2847—31no—
45#1788—34no—
46#1787D311yescharacteristic, complemented
47#1786—31no—
48#1779—31no—
49C24×D3—21no—
50C24×D3D411yescharacteristic
51C22×C3×Q8—28no—
52C22×C3×Q8—22no—
53C22×C3×Q8—21no—
54C22×C3×Q8D414yescomplemented
55C22×C3×Q8C2312yes—
56C23×C3×C4—21no—
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C23×Dic3D411yescharacteristic
60C2×#239—264no—
61C2×#239—264no—
62C2×#239—216no—
63C2×#239—216no—
64C2×#239—216no—
65C2×#239—28no—
66C2×#239—28no—
67C2×#239—28no—
68C2×#239—28no—
69C2×#239—28no—
70C2×#239D4116yescomplemented
71C22×D12—28no—
72C22×D12—216no—
73C22×D12—24no—
74C22×D12—22no—
75C22×D12—21no—
76C22×D12—28no—
77C22×D12—22no—
78C22×D12D414yescomplemented
79C22×D12C2311yescharacteristic
80C22×D12C2312yes—
81C22×C4×D3—28no—
82C22×C4×D3—216no—
83C22×C4×D3—22no—
84C22×C4×D3—22no—
85C22×C4×D3—24no—
86C22×C4×D3—28no—
87C22×C4×D3D412yes—
88C22×C4×D3C2312yes—
89C22×C3×C8C2312yes—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—21no—
92C2×#215—24no—
93C2×#215—28no—
94#930—22no—
95#930—22no—
96C2×(C2×D3)⋊C4—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3C2311yescharacteristic
100C22×C3⋊C8C2312yes—
101C2×D24—24no—
102C2×D24—28no—
103#895—28no—
104#882—44no—
105C3×#103—28no—
106C3×C22⋊C8—22no—
107C3×#95—21no—
108#835—24no—
109C3⋊#95—22no—
110#830—22no—
111#821—24no—
112(C2×D3)⋊C8—28no—
113#809—28no—
114C22×#41—34no—
115C22×#41—62no—
116C22×#41—64no—
117C22×#41—616no—
118C22×#41—64no—
119C23×Q8C2×D311yescharacteristic
120C23×D4—31no—
121C23×D4—61no—
122C24×C4—31no—
123C24×C4—61no—
124C22×SD16—32no—
125C22×D8—32no—
126C2×#121—31no—
127C2×#121—32no—
128C2×#120—61no—
129#449—616no—
130#448—64no—
131#447—64no—
132C2×#103—32no—
133C2×#103C2×D314yescomplemented
134C2×#102—32no—
135C2×#102—34no—
136C2×C22⋊C8—31no—
137C2×C22⋊C8—34no—
138C2×C22⋊C8C2×D311yescharacteristic
139#386—62no—
140#386—61no—
141#380—32no—
142#380C2×D311yescharacteristic
143C24×C3C2×D411yescharacteristic
144C23×D3—28no—
145C23×D3—22no—
146C23×D3—24no—
147C23×D3—22no—
148C23×D3—24no—
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—21no—
152C23×D3—42no—
153C23×D3—44no—
154C23×D3—42no—
155C23×D3—42no—
156C23×D3C2×D412yes—
157C2×C3×Q8—216no—
158C2×C3×Q8—28no—
159C2×C3×Q8—216no—
160C2×C3×Q8—24no—
161C2×C3×Q8—24no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D414yes—
164C2×C3×Q8C2×D418yes—
165C22×C3×C4—28no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2411yescharacteristic
174C2×#67—42no—
175C2×#67—42no—
176C2×#67—42no—
177C22×Dic3—22no—
178C22×Dic3—24no—
179C22×Dic3—21no—
180C22×Dic3C2×D412yes—
181#239—2128no—
182#239—232no—
183#239—232no—
184#239—232no—
185#239—232no—
186C2×D12—264no—
187C2×D12—216no—
188C2×D12—216no—
189C2×D12—28no—
190C2×D12—216no—
191C2×D12—216no—
192C2×D12—24no—
193C2×D12—22no—
194C2×D12—24no—
195C2×D12—216no—
196C2×D12—24no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—44no—
200C2×D12—44no—
201C2×D12—44no—
202C2×D12C2×D418yes—
203C2×D12C2×D414yes—
204C2×C4×D3—264no—
205C2×C4×D3—216no—
206C2×C4×D3—216no—
207C2×C4×D3—216no—
208C2×C4×D3—216no—
209C2×C4×D3—24no—
210C2×C4×D3—24no—
211C2×C4×D3—22no—
212C2×C4×D3—24no—
213C2×C4×D3—22no—
214C2×C4×D3—216no—
215C2×C4×D3—24no—
216C2×C4×D3C2×D418yes—
217C2×C3×C8—24no—
218C2×C3×C8—22no—
219C3×C4⋊C4—24no—
220C3×C22⋊C4—42no—
221#215—48no—
222(C2×D3)⋊C4—44no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—44no—
225C4⋊Dic3—21no—
226C4⋊Dic3—21no—
227C2×C3⋊C8—24no—
228C2×C3⋊C8—22no—
229D24—48no—
230C25—31no—
231C25—61no—
232C2×#41—316no—
233C2×#41—68no—
234C2×#41—68no—
235C2×#41—68no—
236C2×#41—616no—
237C2×#41—616no—
238C2×#41—664no—
239C2×#41—664no—
240C2×#41—616no—
241C2×#41—68no—
242C2×#41—68no—
243C22×Q8—28no—
244C22×Q8—22no—
245C22×Q8—21no—
246C22×Q8D1214yescomplemented
247C22×Q8C22×D312yes—
248C22×D4—34no—
249C22×D4—32no—
250C22×D4—31no—
251C22×D4—68no—
252C22×D4—64no—
253C22×D4—616no—
254C22×D4—68no—
255C22×D4—62no—
256C22×D4—61no—
257C22×D4—62no—
258C22×D4—61no—
259C23×C4—21no—
260C23×C4—32no—
261C23×C4—31no—
262C23×C4—32no—
263C23×C4—68no—
264C23×C4—64no—
265C23×C4—616no—
266C23×C4—68no—
267C23×C4—62no—
268C23×C4—62no—
269C23×C4C22×D311yescharacteristic
270C2×SD16—64no—
271C2×SD16—68no—
272C2×D8—64no—
273C2×D8—68no—
274C22×C8—32no—
275C22×C8C22×D312yes—
276#121—68no—
277#121—62no—
278#121—62no—
279#120—124no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4C22×D312yes—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—62no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—61no—
286#103—28no—
287#103—64no—
288#102—68no—
289#102—64no—
290C22⋊C8—22no—
291C22⋊C8—68no—
292C22⋊C8—62no—
293#95—21no—
294#95—62no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3#12012yes—
298C23×C3#13512yes—
299C23×C3AGL(1,ℤ/8)12yes—
300C23×C3C22×D411yescharacteristic
301C22×D3—28no—
302C22×D3—216no—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—24no—
306C22×D3—28no—
307C22×D3—22no—
308C22×D3—22no—
309C22×D3—22no—
310C22×D3—22no—
311C22×D3—48no—
312C22×D3—48no—
313C22×D3—416no—
314C22×D3—44no—
315C22×D3—48no—
316C22×D3—416no—
317C22×D3—42no—
318C22×D3—42no—
319C22×D3—44no—
320C22×D3—44no—
321C22×D3—48no—
322C22×D3—42no—
323C22×D3—41no—
324C22×D3—41no—
325C22×D3—41no—
326C22×D3—42no—
327C22×D3#12014yes—
328C3×Q8—216no—
329C3×Q8—24no—
330C3×Q8—28no—
331C3×Q8#12018yescomplemented
332C2×C3×C4—216no—
333C2×C3×C4—24no—
334C2×C3×C4—28no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4#12012yes—
342C2×C3×C4C22×D411yescharacteristic
343C2×C3×C4C22×D414yes—
344C2×C3×C4C22×D411yescharacteristic
345#67—84no—
346C2×Dic3—28no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3#12014yes—
353D12—264no—
354D12—216no—
355D12—28no—
356D12—24no—
357D12—42no—
358D12—42no—
359D12—88no—
360D12#12018yescomplemented
361C4×D3—264no—
362C4×D3—216no—
363C4×D3—28no—
364C4×D3—28no—
365C3×C8—42no—
366C3⋊C8—42no—
367C24—32no—
368C24—64no—
369C24—68no—
370C24—64no—
371C24—62no—
372C24—62no—
373C24—62no—
374C24—62no—
375C24—61no—
376C24—122no—
377C24—124no—
378C24—122no—
379C24—122no—
380C24D3×D411yescharacteristic
381#41—632no—
382#41—6128no—
383#41—632no—
384#41—632no—
385#41—632no—
386C2×Q8—216no—
387C2×Q8—28no—
388C2×Q8—216no—
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—24no—
392C2×Q8C2×D1214yes—
393C2×Q8D3×D418yes—
394C2×D4—34no—
395C2×D4—38no—
396C2×D4—616no—
397C2×D4—616no—
398C2×D4—664no—
399C2×D4—616no—
400C2×D4—616no—
401C2×D4—68no—
402C2×D4—616no—
403C2×D4—64no—
404C2×D4—62no—
405C2×D4—61no—
406C2×D4—64no—
407C2×D4—61no—
408C2×D4—64no—
409C2×D4—124no—
410C2×D4—124no—
411C2×D4—124no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4—122no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—32no—
421C22×C4—38no—
422C22×C4—64no—
423C22×C4—616no—
424C22×C4—64no—
425C22×C4—664no—
426C22×C4—616no—
427C22×C4—616no—
428C22×C4—616no—
429C22×C4—616no—
430C22×C4—64no—
431C22×C4—62no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—64no—
436C22×C4—61no—
437C22×C4C2×D1212yes—
438C22×C4D3×D411yescharacteristic
439C22×C4D3×D412yes—
440C22×C4C23×D311yescharacteristic
441SD16—128no—
442D8—128no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—62no—
446C2×C8—64no—
447C4⋊C4—24no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C22⋊C4—42no—
451C22⋊C4—124no—
452C22⋊C4—124no—
453C22⋊C4—122no—
454C22×C3—22no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—24no—
458C22×C3#44914yescomplemented
459C22×C3C2×#12011yescharacteristic
460C22×C3C2×#12012yes—
461C22×C3C2×#13511yescharacteristic
462C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
463C22×C3#57512yes—
464C2×D3—24no—
465C2×D3—28no—
466C2×D3—24no—
467C2×D3—24no—
468C2×D3—416no—
469C2×D3—432no—
470C2×D3—42no—
471C2×D3—44no—
472C2×D3—42no—
473C2×D3—48no—
474C2×D3—42no—
475C2×D3—44no—
476C2×D3—42no—
477C2×D3—48no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—82no—
481C2×D3—81no—
482C2×D3—82no—
483C3×C4—28no—
484C3×C4—22no—
485C3×C4—82no—
486C3×C4C2×#12012yes—
487C3×C4C2×#12012yes—
488Dic3—24no—
489Dic3—81no—
490C23—22no—
491C23—22no—
492C23—34no—
493C23—616no—
494C23—68no—
495C23—64no—
496C23—62no—
497C23—62no—
498C23—62no—
499C23—62no—
500C23—62no—
501C23—68no—
502C23—62no—
503C23—128no—
504C23—128no—
505C23—1216no—
506C23—124no—
507C23—128no—
508C23—1216no—
509C23—122no—
510C23—122no—
511C23—124no—
512C23—124no—
513C23—128no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23—122no—
518C23—122no—
519C23#88212yes—
520C23#91412yes—
521C23#91912yes—
522C23C2×D3×D411yescharacteristic
523Q8—216no—
524Q8—24no—
525Q8—28no—
526Q8#88218yescomplemented
527D4—38no—
528D4—664no—
529D4—616no—
530D4—68no—
531D4—64no—
532D4—122no—
533D4—122no—
534D4—248no—
535D4—244no—
536C2×C4—216no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—34no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—664no—
547C2×C4—616no—
548C2×C4—68no—
549C2×C4—62no—
550C2×C4—62no—
551C2×C4—68no—
552C2×C4—68no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
556C2×C4#88212yes—
557C2×C4C22×D1211yescharacteristic
558C2×C4C2×D3×D414yes—
559C2×C4C2×D3×D411yescharacteristic
560C8—42no—
561C8—122no—
562C2×C3—24no—
563C2×C3#284611yescharacteristic
564C2×C3C2×#44911yescharacteristic
565C2×C3#300212yes—
566C2×C3#300511yescharacteristic
567C2×C3#300612yes—
568D3—44no—
569D3—48no—
570D3—161no—
571C22—24no—
572C22—22no—
573C22—22no—
574C22—24no—
575C22—64no—
576C22—68no—
577C22—64no—
578C22—64no—
579C22—1216no—
580C22—1232no—
581C22—122no—
582C22—128no—
583C22—122no—
584C22—124no—
585C22—122no—
586C22—128no—
587C22—122no—
588C22—124no—
589C22—242no—
590C22—241no—
591C22—241no—
592C22—242no—
593C22—241no—
594C22#481014yescomplemented
595C22C2×#88211yescharacteristic
596C22D3×#12012yes—
597C22C2×#91411yescharacteristic
598C22C2×#91911yescharacteristic
599C22#577712yes—
600C4—28no—
601C4—22no—
602C4—64no—
603C4—82no—
604C4—241no—
605C4C2×#88212yes—
606C4D3×#12012yes—
607C3#3288011yescharacteristic, complemented, Sylow
608C2—24no—
609C2—124no—
610C2—128no—
611C2—481no—
612C2#7386111yescharacteristic
613C2C2×#481011yescharacteristic
614C2#7895911yescharacteristic
615C2#8004012yes—
616C2#8018212yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1787C2
socleC23×C3C22×D4
hypercenter#1787D3
hypocenterC3#32880
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C23×#239C2semidirect
2. C2×#5227C2semidirect
3. C2×#4810C2semidirect
4. C2×#4732C2semidirect
5. #74048C2semidirect
6. C3×#1787C2semidirect
7. #70267C2semidirect
8. #69955C2semidirect
9. #69521C2semidirect
21. C22×#215C22semidirect
23. C22×D24C22semidirect
24. C2×#895C22semidirect
26. C2×C3×#103C22semidirect
30. C2×#835C22semidirect
32. C2×#821C22semidirect
33. C2×(C2×D3)⋊C8C22semidirect
34. #4987C22semidirect
35. C2×#809C22semidirect
46. #1787D3semidirect
54. C22×C3×Q8D4semidirect
70. C2×#239D4semidirect
78. C22×D12D4semidirect
133. C2×#103C2×D3semidirect
246. C22×Q8D12semidirect
331. C3×Q8#120semidirect
360. D12#120semidirect
458. C22×C3#449semidirect
526. Q8#882semidirect
594. C22#4810semidirect
607. C3#32880semidirect

Maximal subgroups

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

subgroupindex
C23×#2392
C2×#52272
C2×#48102
C2×#47322
#740482
C3×#17872
#702672
#699552
#695212
#328803