Finite Group Databasebeta

← #10943833

Subgroups — #10943833

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
2#84032C211yesmaximal, characteristic, complemented
3C2×#4738C211yesmaximal, characteristic, complemented
4C2×#4734C214yesmaximal, complemented
5C2×#4732C211yesmaximal, characteristic, complemented
6C3×#1779C211yesmaximal, characteristic, complemented
7#69963C211yesmaximal, characteristic, complemented
8#69960C214yesmaximal, complemented
9#69959C211yesmaximal, 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×#953—21no—
22C2×#930C2211yescharacteristic
23C2×#930C2212yescomplemented
24C22×D24C2212yescomplemented
25C22×#204C2212yescomplemented
26C2×C3×C22⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2211yescharacteristic
28C2×C3×C22⋊C8C2214yescomplemented
29#5213—22no—
30#5213—21no—
31#5212C2211yescharacteristic
32#5212C2212yescomplemented
33C2×#821C2212yescomplemented
34C2×#821C2214yescomplemented
35C2×#816C2214yescomplemented
36C2×#816C2212yescomplemented
37#4738—24no—
38#4734—216no—
39#4732—24no—
40C23×#41—31no—
41C2×#449—34no—
42C2×#448—31no—
43C2×#447—31no—
44#2847—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—
61C22×#67—21no—
62C22×#67—22no—
63C23×Dic3—21no—
64C2×#235—216no—
65C2×#235—264no—
66C2×#235—216no—
67C2×#235—216no—
68C2×#235—264no—
69C2×#235—28no—
70C2×#235—28no—
71C2×#235—28no—
72C2×#235—28no—
73C2×#235—28no—
74C2×#235D4116yescomplemented
75C22×D12—21no—
76C22×D12—22no—
77C22×D12—28no—
78C22×D12D414yescomplemented
79C22×D12C2311yescharacteristic
80C22×D12C2312yes—
81C22×C4×D3—28no—
82C22×C4×D3—216no—
83C22×C4×D3—24no—
84C22×Dic6—21no—
85C22×Dic6—22no—
86C22×Dic6—28no—
87C22×Dic6D414yescomplemented
88C22×Dic6C2312yes—
89C22×C3×C8C2312yes—
90C22×C3×C8C2312yes—
91#953—44no—
92C2×#217—21no—
93C2×#217—22no—
94#930—22no—
95#930—28no—
96#930—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—22no—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×D24—24no—
102C2×D24—28no—
103C2×#204—28no—
104C2×#204—24no—
105C3×C22⋊C8—22no—
106C3×C22⋊C8—28no—
107C3×C22⋊C8—22no—
108C3⋊#95—21no—
109C3⋊#95—22no—
110#821—24no—
111#821—28no—
112#816—28no—
113#816—24no—
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×#121—31no—
127C2×#121—32no—
128C2×#120—61no—
129#449—616no—
130#448—64no—
131#447—64no—
132C2×#103—32no—
133C2×#103—34no—
134C2×#102—32no—
135C2×#102—34no—
136C2×C22⋊C8C2×D311yescharacteristic
137C2×C22⋊C8C2×D311yescharacteristic
138C2×C22⋊C8C2×D314yescomplemented
139#386—61no—
140#386—62no—
141#380—31no—
142#380—32no—
143C24×C3—24no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—22no—
147C24×C3—42no—
148C24×C3C2×D412yes—
149C24×C3C2×D411yescharacteristic
150C23×D3—28no—
151C23×D3—24no—
152C23×D3—22no—
153C23×D3—21no—
154C23×D3—44no—
155C23×D3—42no—
156C23×D3—42no—
157C22×C3×C4—216no—
158C22×C3×C4—216no—
159C22×C3×C4—22no—
160C22×C3×C4—24no—
161C22×C3×C4—24no—
162C22×C3×C4—24no—
163C22×C3×C4—24no—
164C22×C3×C4—24no—
165C22×C3×C4—22no—
166C22×C3×C4—24no—
167C22×C3×C4—22no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D418yes—
173C22×C3×C4C2411yescharacteristic
174C2×#67—264no—
175C2×#67—216no—
176C2×#67—216no—
177C2×#67—216no—
178C2×#67—44no—
179C2×#67—44no—
180C2×#67—44no—
181C2×#67—42no—
182C2×#67—42no—
183C2×#67—42no—
184C22×Dic3—28no—
185C22×Dic3—24no—
186C22×Dic3—22no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189#235—2128no—
190#235—232no—
191#235—232no—
192#235—232no—
193#235—232no—
194C2×D12—28no—
195C2×D12—216no—
196C2×D12—24no—
197C2×D12—24no—
198C2×D12—22no—
199C2×D12—216no—
200C2×D12—21no—
201C2×D12—24no—
202C2×D12—21no—
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—
217C2×C3×C8—22no—
218C2×C3×C8—24no—
219C2×C3×C8—24no—
220C2×C3×C8—22no—
221#217—44no—
222#217—42no—
223(C2×D3)⋊C4—42no—
224(C2×D3)⋊C4—44no—
225C4⋊Dic3—21no—
226C4⋊Dic3—24no—
227C4⋊Dic3—21no—
228D24—48no—
229#204—48no—
230C25—61no—
231C25D1211yescharacteristic
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—61no—
247C22×Q8—62no—
248C22×D4—34no—
249C22×D4—32no—
250C22×D4—31no—
251C22×D4—68no—
252C22×D4—616no—
253C22×D4—68no—
254C22×D4—64no—
255C22×D4—62no—
256C22×D4—61no—
257C22×D4—61no—
258C22×D4—62no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—28no—
262C23×C4—68no—
263C23×C4—616no—
264C23×C4—61no—
265C23×C4—64no—
266C23×C4D1211yescharacteristic
267C23×C4D1212yes—
268C23×C4C22×D311yescharacteristic
269C23×C4C22×D312yes—
270C2×SD16—64no—
271C2×SD16—68no—
272C2×D8—64no—
273C2×D8—68no—
274C22×C8C22×D312yes—
275C22×C8C22×D312yes—
276#121—62no—
277#121—62no—
278#121—68no—
279#120—124no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—32no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—62no—
285C2×C22⋊C4—61no—
286#103—64no—
287#103—68no—
288#102—64no—
289#102—68no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—28no—
293#95—62no—
294#95—61no—
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#12014yes—
309C23×C3#12012yes—
310C23×C3#13512yes—
311C23×C3AGL(1,ℤ/8)12yes—
312C23×C3C22×D411yescharacteristic
313C22×D3—216no—
314C22×D3—24no—
315C22×D3—28no—
316C22×D3—416no—
317C22×D3—48no—
318C22×D3—416no—
319C22×D3—42no—
320C22×D3—42no—
321C22×D3—44no—
322C22×D3—48no—
323C22×D3—41no—
324C22×D3—42no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—42no—
328C2×C3×C4—28no—
329C2×C3×C4—216no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—28no—
336C2×C3×C4—28no—
337C2×C3×C4#12012yes—
338C2×C3×C4#12014yes—
339C2×C3×C4C22×D411yescharacteristic
340C2×C3×C4C22×D414yes—
341C2×C3×C4C22×D411yescharacteristic
342#67—264no—
343#67—88no—
344#67—84no—
345C2×Dic3—216no—
346C2×Dic3—24no—
347C2×Dic3—28no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354D12—216no—
355D12—24no—
356D12—28no—
357D12—42no—
358D12—42no—
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—42no—
372C24—68no—
373C24—64no—
374C24—62no—
375C24—61no—
376C24—124no—
377C24—122no—
378C24—122no—
379C24C2×D1212yes—
380C24C2×D1211yescharacteristic
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—34no—
395C2×D4—38no—
396C2×D4—616no—
397C2×D4—68no—
398C2×D4—616no—
399C2×D4—664no—
400C2×D4—616no—
401C2×D4—616no—
402C2×D4—616no—
403C2×D4—64no—
404C2×D4—64no—
405C2×D4—62no—
406C2×D4—61no—
407C2×D4—64no—
408C2×D4—61no—
409C2×D4—124no—
410C2×D4—124no—
411C2×D4—124no—
412C2×D4—122no—
413C2×D4—122no—
414C2×D4—122no—
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—664no—
427C22×C4—616no—
428C22×C4—616no—
429C22×C4—68no—
430C22×C4—64no—
431C22×C4—62no—
432C22×C4—616no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4C2×D1212yes—
436C22×C4C2×D1211yescharacteristic
437C22×C4D3×D412yes—
438C22×C4D3×D412yes—
439C22×C4D3×D418yes—
440C22×C4C23×D311yescharacteristic
441SD16—128no—
442D8—128no—
443C2×C8—22no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—22no—
447C4⋊C4—61no—
448C4⋊C4—64no—
449C4⋊C4—61no—
450C22⋊C4—124no—
451C22⋊C4—122no—
452C22⋊C4—124no—
453C22⋊C4—122no—
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#44914yescomplemented
467C22×C3C2×#12011yescharacteristic
468C22×C3C2×#12012yes—
469C22×C3C2×#13511yescharacteristic
470C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
471C22×C3#57512yes—
472C2×D3—28no—
473C2×D3—432no—
474C2×D3—44no—
475C2×D3—48no—
476C2×D3—44no—
477C2×D3—48no—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—82no—
481C2×D3—81no—
482C2×D3—82no—
483C3×C4—22no—
484C3×C4—24no—
485C3×C4C2×#12012yes—
486C3×C4C2×#12012yes—
487Dic3—28no—
488Dic3—81no—
489Dic3—82no—
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—616no—
504C23—64no—
505C23—68no—
506C23—1216no—
507C23—128no—
508C23—1216no—
509C23—122no—
510C23—122no—
511C23—124no—
512C23—128no—
513C23—121no—
514C23—121no—
515C23—122no—
516C23—121no—
517C23—122no—
518C23#88214yes—
519C23#88212yes—
520C23#90412yes—
521C23#90812yes—
522C23C22×D1211yescharacteristic
523Q8—38no—
524Q8—616no—
525Q8—68no—
526Q8—64no—
527D4—38no—
528D4—616no—
529D4—664no—
530D4—64no—
531D4—68no—
532D4—122no—
533D4—122no—
534D4—248no—
535D4—244no—
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—664no—
546C2×C4—616no—
547C2×C4—64no—
548C2×C4—68no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—122no—
552C2×C4—121no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4#88214yes—
556C2×C4#88212yes—
557C2×C4C2×D3×D411yescharacteristic
558C2×C4C2×D3×D414yes—
559C2×C4C2×D3×D411yescharacteristic
560C8—42no—
561C8—42no—
562C2×C3—24no—
563C2×C3—44no—
564C2×C3#284611yescharacteristic
565C2×C3C2×#44911yescharacteristic
566C2×C3#300212yes—
567C2×C3#300511yescharacteristic
568C2×C3#300612yes—
569D3—48no—
570D3—161no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—416no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—68no—
584C22—1232no—
585C22—124no—
586C22—128no—
587C22—124no—
588C22—128no—
589C22—241no—
590C22—242no—
591C22—241no—
592C22—242no—
593C22—241no—
594C22#473414yescomplemented
595C22C2×#88211yescharacteristic
596C22C2×#88212yes—
597C22C2×#90411yescharacteristic
598C22C2×#90811yescharacteristic
599C22#574812yes—
600C4—24no—
601C4—22no—
602C4—68no—
603C4—241no—
604C4—242no—
605C4D3×#12012yes—
606C4D3×#12012yes—
607C3#3288011yescharacteristic, complemented, Sylow
608C2—24no—
609C2—44no—
610C2—128no—
611C2—481no—
612C2#7386111yescharacteristic
613C2C2×#473411yescharacteristic
614C2#7861012yes—
615C2#7863811yescharacteristic
616C2#7863912yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1779C2
socleC23×C3C22×D4
hypercenter#1779D3
hypocenterC3#32880
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
1. C23×#235C2semidirect
2. #84032C2semidirect
3. C2×#4738C2semidirect
4. C2×#4734C2semidirect
5. C2×#4732C2semidirect
6. C3×#1779C2semidirect
7. #69963C2semidirect
8. #69960C2semidirect
9. #69959C2semidirect
23. C2×#930C22semidirect
24. C22×D24C22semidirect
25. C22×#204C22semidirect
28. C2×C3×C22⋊C8C22semidirect
32. #5212C22semidirect
33. C2×#821C22semidirect
34. C2×#821C22semidirect
35. C2×#816C22semidirect
36. C2×#816C22semidirect
48. #1779D3semidirect
74. C2×#235D4semidirect
78. C22×D12D4semidirect
87. C22×Dic6D4semidirect
138. C2×C22⋊C8C2×D3semidirect
359. D12#120semidirect
364. Dic6#120semidirect
466. C22×C3#449semidirect
594. C22#4734semidirect
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
#840322
C2×#47382
C2×#47342
C2×#47322
C3×#17792
#699632
#699602
#699592
#328803