Finite Group Databasebeta

← #10943830

Subgroups — #10943830

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×C3×#41C211yesmaximal, characteristic, complemented
2C2×#5238C214yesmaximal, complemented
3C2×#5227C211yesmaximal, characteristic, complemented
4C2×#5217C211yesmaximal, characteristic, complemented
5#84032C211yesmaximal, characteristic, complemented
6#72135C214yesmaximal, complemented
7#72132C211yesmaximal, characteristic, complemented
8#72128C211yesmaximal, characteristic, complemented
9#72126C211yesmaximal, characteristic, complemented
10#32880—31nomaximal, Sylow
11C22×C3×#41—216no—
12C22×C3×#41—24no—
13C22×C3×#41—24no—
14C22×C3×#41—22no—
15C22×C3×#41C2214yes—
16C23×C3×Q8C2211yescharacteristic
17C23×C3×D4—21no—
18C23×C3×D4C2211yescharacteristic
19C24×C3×C4—21no—
20C24×C3×C4C2211yescharacteristic
21C2×#953—21no—
22C22×#215C2212yescomplemented
23C22×#213C2212yescomplemented
24C2×#930C2211yescharacteristic
25C2×#930C2212yescomplemented
26#5238—216no—
27#5227—24no—
28C2×#835C2212yescomplemented
29C2×#835C2214yescomplemented
30#5217—24no—
31C2×#832C2212yescomplemented
32C2×#832C2214yescomplemented
33#5213—22no—
34#5213—21no—
35#5212C2211yescharacteristic
36#5212C2212yescomplemented
37C2×#830C2211yescharacteristic
38C2×#830C2211yescharacteristic
39C2×#830C2214yescomplemented
40C23×#41D311yescharacteristic, complemented
41C2×#449—34no—
42C2×#448—31no—
43C2×#447—31no—
44#2847—31no—
45#1788—34no—
46#1787—31no—
47#1786—31no—
48#1779—31no—
49C25×C3—21no—
50C25×C3D411yescharacteristic
51C2×C3×#41—264no—
52C2×C3×#41—264no—
53C2×C3×#41—216no—
54C2×C3×#41—216no—
55C2×C3×#41—216no—
56C2×C3×#41—28no—
57C2×C3×#41—28no—
58C2×C3×#41—28no—
59C2×C3×#41—28no—
60C2×C3×#41—28no—
61C2×C3×#41D4116yescomplemented
62C22×C3×Q8—28no—
63C22×C3×Q8—22no—
64C22×C3×Q8—21no—
65C22×C3×Q8D414yescomplemented
66C22×C3×Q8C2312yes—
67C22×C3×D4—216no—
68C22×C3×D4—28no—
69C22×C3×D4—24no—
70C22×C3×D4—28no—
71C22×C3×D4—21no—
72C22×C3×D4—22no—
73C22×C3×D4D414yescomplemented
74C22×C3×D4C2312yes—
75C23×C3×C4—216no—
76C23×C3×C4—28no—
77C23×C3×C4—21no—
78C23×C3×C4—24no—
79C23×C3×C4—22no—
80C23×C3×C4—22no—
81C23×C3×C4—28no—
82C23×C3×C4D411yescharacteristic
83C23×C3×C4D412yes—
84C23×C3×C4C2311yescharacteristic
85C23×C3×C4C2312yes—
86C22×#67—21no—
87C22×#67—22no—
88C22×D12C2311yescharacteristic
89#953—44no—
90C2×#217—21no—
91C2×#217—22no—
92C2×#215—24no—
93C2×#215—28no—
94C2×#213—24no—
95C2×#213—28no—
96#930—22no—
97#930—22no—
98#930—28no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—22no—
101C2×C4⋊Dic3C2312yes—
102C2×C4⋊Dic3C2311yescharacteristic
103C22×C3⋊C8C2312yes—
104C22×C3⋊C8C2312yes—
105#835—24no—
106#835—28no—
107#832—24no—
108#832—28no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111#830—22no—
112#830—22no—
113#830—28no—
114C22×#41—216no—
115C22×#41—24no—
116C22×#41—24no—
117C22×#41—22no—
118C22×#41C2×D314yes—
119C23×Q8C2×D311yescharacteristic
120C23×D4—21no—
121C23×D4C2×D311yescharacteristic
122C24×C4—21no—
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⋊C8—31no—
137C2×C22⋊C8—31no—
138C2×C22⋊C8—34no—
139#386—61no—
140#386—62no—
141#380—31no—
142#380—32no—
143C24×C3—28no—
144C24×C3—24no—
145C24×C3—24no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—42no—
151C24×C3—44no—
152C24×C3—42no—
153C24×C3—42no—
154C24×C3C2×D412yes—
155C24×C3C2×D411yescharacteristic
156C23×D3—21no—
157C3×#41—2128no—
158C3×#41—232no—
159C3×#41—232no—
160C3×#41—232no—
161C3×#41—232no—
162C2×C3×Q8—216no—
163C2×C3×Q8—28no—
164C2×C3×Q8—216no—
165C2×C3×Q8—24no—
166C2×C3×Q8—24no—
167C2×C3×Q8—22no—
168C2×C3×Q8C2×D418yes—
169C2×C3×Q8C2×D414yes—
170C2×C3×D4—264no—
171C2×C3×D4—216no—
172C2×C3×D4—216no—
173C2×C3×D4—28no—
174C2×C3×D4—216no—
175C2×C3×D4—216no—
176C2×C3×D4—216no—
177C2×C3×D4—24no—
178C2×C3×D4—22no—
179C2×C3×D4—24no—
180C2×C3×D4C2×D418yes—
181C2×C3×D4C2×D414yes—
182C22×C3×C4—28no—
183C22×C3×C4—264no—
184C22×C3×C4—216no—
185C22×C3×C4—216no—
186C22×C3×C4—216no—
187C22×C3×C4—216no—
188C22×C3×C4—22no—
189C22×C3×C4—24no—
190C22×C3×C4—24no—
191C22×C3×C4—22no—
192C22×C3×C4—24no—
193C22×C3×C4—216no—
194C22×C3×C4—24no—
195C22×C3×C4—24no—
196C22×C3×C4—24no—
197C22×C3×C4—22no—
198C22×C3×C4—24no—
199C22×C3×C4—22no—
200C22×C3×C4C2×D412yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D412yes—
203C22×C3×C4C2×D412yes—
204C22×C3×C4C2×D418yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—44no—
207C2×#67—44no—
208C2×#67—44no—
209C2×#67—42no—
210C2×#67—42no—
211C2×#67—42no—
212C22×Dic3—21no—
213C22×Dic3—22no—
214C2×D12—21no—
215C2×D12—24no—
216C2×D12—21no—
217#217—44no—
218#217—42no—
219#215—48no—
220#213—48no—
221(C2×D3)⋊C4—44no—
222(C2×D3)⋊C4—42no—
223C4⋊Dic3—21no—
224C4⋊Dic3—24no—
225C4⋊Dic3—21no—
226C2×C3⋊C8—22no—
227C2×C3⋊C8—24no—
228C2×C3⋊C8—24no—
229C2×C3⋊C8—22no—
230C25—21no—
231C25#6711yescharacteristic
232C2×#41—264no—
233C2×#41—264no—
234C2×#41—216no—
235C2×#41—216no—
236C2×#41—216no—
237C2×#41—28no—
238C2×#41—28no—
239C2×#41—28no—
240C2×#41—28no—
241C2×#41—28no—
242C2×#41#67116yescomplemented
243C22×Q8—28no—
244C22×Q8—22no—
245C22×Q8—21no—
246C22×Q8#6714yescomplemented
247C22×Q8C22×D312yes—
248C22×D4—216no—
249C22×D4—28no—
250C22×D4—24no—
251C22×D4—28no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—31no—
255C22×D4—62no—
256C22×D4—61no—
257C22×D4#6714yescomplemented
258C22×D4C22×D312yes—
259C23×C4—216no—
260C23×C4—28no—
261C23×C4—21no—
262C23×C4—24no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—28no—
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—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—62no—
291C22⋊C8—62no—
292C22⋊C8—68no—
293#95—62no—
294#95—61no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—28no—
298C23×C3—216no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—28no—
303C23×C3—22no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—48no—
308C23×C3—48no—
309C23×C3—416no—
310C23×C3—44no—
311C23×C3—48no—
312C23×C3—416no—
313C23×C3—42no—
314C23×C3—42no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3—48no—
318C23×C3#12014yes—
319C23×C3#12012yes—
320C23×C3#13512yes—
321C23×C3AGL(1,ℤ/8)12yes—
322C23×C3C22×D411yescharacteristic
323C22×D3—41no—
324C22×D3—42no—
325C22×D3—41no—
326C22×D3—41no—
327C22×D3—42no—
328C3×Q8—216no—
329C3×Q8—24no—
330C3×Q8—28no—
331C3×Q8#12018yescomplemented
332C3×D4—264no—
333C3×D4—216no—
334C3×D4—28no—
335C3×D4—24no—
336C3×D4#12018yescomplemented
337C2×C3×C4—28no—
338C2×C3×C4—216no—
339C2×C3×C4—264no—
340C2×C3×C4—216no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—28no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—28no—
349C2×C3×C4—28no—
350C2×C3×C4#12014yes—
351C2×C3×C4#12012yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D414yes—
354C2×C3×C4C22×D411yescharacteristic
355#67—88no—
356#67—84no—
357C2×Dic3—41no—
358C2×Dic3—42no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363D12—42no—
364D12—42no—
365C3⋊C8—42no—
366C3⋊C8—42no—
367C24—28no—
368C24—24no—
369C24—22no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—42no—
375C24—44no—
376C24—42no—
377C24—42no—
378C24—61no—
379C24C2×#6712yes—
380C24C2×#6711yescharacteristic
381#41—2128no—
382#41—232no—
383#41—232no—
384#41—232no—
385#41—232no—
386C2×Q8—216no—
387C2×Q8—28no—
388C2×Q8—216no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8C2×#6718yes—
393C2×Q8C2×#6714yes—
394C2×D4—264no—
395C2×D4—216no—
396C2×D4—216no—
397C2×D4—28no—
398C2×D4—216no—
399C2×D4—216no—
400C2×D4—216no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—24no—
404C2×D4—61no—
405C2×D4—64no—
406C2×D4—61no—
407C2×D4—124no—
408C2×D4—124no—
409C2×D4—122no—
410C2×D4—122no—
411C2×D4—124no—
412C2×D4—122no—
413C2×D4C2×#6718yes—
414C2×D4C2×#6714yes—
415C22×C4—28no—
416C22×C4—264no—
417C22×C4—216no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—216no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—61no—
434C22×C4—62no—
435C22×C4C2×#6712yes—
436C22×C4C2×#6711yescharacteristic
437C22×C4C2×#6712yes—
438C22×C4C2×#6712yes—
439C22×C4C2×#6718yes—
440C22×C4C23×D311yescharacteristic
441SD16—128no—
442D8—128no—
443C2×C8—62no—
444C2×C8—64no—
445C2×C8—64no—
446C2×C8—62no—
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—28no—
460C22×C3—24no—
461C22×C3—24no—
462C22×C3—416no—
463C22×C3—432no—
464C22×C3—42no—
465C22×C3—44no—
466C22×C3—42no—
467C22×C3—48no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—42no—
471C22×C3—48no—
472C22×C3#44914yescomplemented
473C22×C3C2×#12011yescharacteristic
474C22×C3C2×#12012yes—
475C22×C3C2×#13511yescharacteristic
476C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
477C22×C3#57512yes—
478C2×D3—81no—
479C2×D3—81no—
480C2×D3—82no—
481C2×D3—81no—
482C2×D3—82no—
483C3×C4—24no—
484C3×C4—28no—
485C3×C4—22no—
486C3×C4C2×#12012yes—
487C3×C4C2×#12012yes—
488Dic3—81no—
489Dic3—82no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—216no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—28no—
502C23—48no—
503C23—48no—
504C23—416no—
505C23—44no—
506C23—48no—
507C23—416no—
508C23—44no—
509C23—48no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—121no—
514C23—121no—
515C23—122no—
516C23—121no—
517C23—122no—
518C23#94912yes—
519C23#95012yes—
520C23#95314yes—
521C23#95312yes—
522C23C22×#6711yescharacteristic
523Q8—216no—
524Q8—24no—
525Q8—28no—
526Q8#95318yescomplemented
527D4—264no—
528D4—216no—
529D4—28no—
530D4—24no—
531D4—122no—
532D4—122no—
533D4—248no—
534D4—244no—
535D4#95318yescomplemented
536C2×C4—28no—
537C2×C4—216no—
538C2×C4—264no—
539C2×C4—216no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—121no—
550C2×C4—122no—
551C2×C4—121no—
552C2×C4—122no—
553C2×C4—121no—
554C2×C4—122no—
555C2×C4#95314yes—
556C2×C4#95312yes—
557C2×C4C22×#6711yescharacteristic
558C2×C4C22×#6714yes—
559C2×C4C22×#6711yescharacteristic
560C8—122no—
561C8—122no—
562C2×C3—24no—
563C2×C3—44no—
564C2×C3—48no—
565C2×C3#284611yescharacteristic
566C2×C3C2×#44911yescharacteristic
567C2×C3#300212yes—
568C2×C3#300511yescharacteristic
569C2×C3#300612yes—
570D3—161no—
571C22—24no—
572C22—22no—
573C22—24no—
574C22—22no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—28no—
579C22—416no—
580C22—432no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22—48no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—48no—
589C22—241no—
590C22—242no—
591C22—241no—
592C22—242no—
593C22—241no—
594C22#523814yescomplemented
595C22C2×#94911yescharacteristic
596C22C2×#95011yescharacteristic
597C22#582212yes—
598C22C2×#95311yescharacteristic
599C22C2×#95312yes—
600C4—24no—
601C4—28no—
602C4—22no—
603C4—241no—
604C4—242no—
605C4C2×#95312yes—
606C4C2×#95312yes—
607C3#3288011yescharacteristic, complemented, Sylow
608C2—24no—
609C2—44no—
610C2—48no—
611C2—481no—
612C2C2×#523811yescharacteristic
613C2#8408012yes—
614C2#8408111yescharacteristic
615C2#8408212yes—
616C2#8412811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC23×C3×#41C2
socleC23×C3C22×D4
hypercenterC23×#41D3
hypocenterC3#32880
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×#41

Complements

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

NH ≅ G/Ntype
1. C23×C3×#41C2semidirect
2. C2×#5238C2semidirect
3. C2×#5227C2semidirect
4. C2×#5217C2semidirect
5. #84032C2semidirect
6. #72135C2semidirect
7. #72132C2semidirect
8. #72128C2semidirect
9. #72126C2semidirect
22. C22×#215C22semidirect
23. C22×#213C22semidirect
25. C2×#930C22semidirect
28. C2×#835C22semidirect
29. C2×#835C22semidirect
31. C2×#832C22semidirect
32. C2×#832C22semidirect
36. #5212C22semidirect
39. C2×#830C22semidirect
40. C23×#41D3semidirect
61. C2×C3×#41D4semidirect
65. C22×C3×Q8D4semidirect
73. C22×C3×D4D4semidirect
242. C2×#41#67semidirect
246. C22×Q8#67semidirect
257. C22×D4#67semidirect
331. C3×Q8#120semidirect
336. C3×D4#120semidirect
472. C22×C3#449semidirect
526. Q8#953semidirect
535. D4#953semidirect
594. C22#5238semidirect
607. C3#32880semidirect

Maximal subgroups

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

subgroupindex
C23×C3×#412
C2×#52382
C2×#52272
C2×#52172
#840322
#721352
#721322
#721282
#721262
#328803