Finite Group Databasebeta

← #11633947

Subgroups — #11633947

Order 768.

3322 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#532C211yesmaximal, characteristic, complemented
2C2×#5717C211yesmaximal, characteristic, complemented
3#84096C214yesmaximal, complemented
4#83260C212yesmaximal, complemented
5#83173C211yesmaximal, characteristic, complemented
6#83005C214yesmaximal, complemented
7#82666C214yesmaximal, complemented
8#82640C214yesmaximal, complemented
9#82405C214yesmaximal, complemented
10D4×(C2×D3)⋊C4C214yesmaximal, complemented
11C4×#930C212yesmaximal, complemented
12#59041—31nomaximal, Sylow
13C22×C3×#41C2212yescomplemented
14C22×D3×D4C2214yes—
15C3×#532—28no—
16C2×C3×C4⋊Q8C2211yescharacteristic, complemented
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124C2212yescomplemented
19C2×C3×#121C2214yescomplemented
20C2×C3×C4×D4C2214yescomplemented
21C22×C3×C42C2211yescharacteristic, complemented
22C2×#938C2214yes—
23C2×#937C2214yes—
24C2×#937C2214yescomplemented
25C2×D4×Dic3C2214yescomplemented
26C2×#930C2212yes—
27C2×C4×#67C2214yescomplemented
28C22×(C2×D3)⋊C4C2214yescomplemented
29#5717—28no—
30C2×C4×D12C2212yes—
31#5231C2218yescomplemented
32#5223C2218yescomplemented
33#5221C2214yescomplemented
34#5220C2214yescomplemented
35#5220C2214yescomplemented
36#5213C2214yescomplemented
37#5213C2218yescomplemented
38#5212C2212yescomplemented
39#5211C2214yescomplemented
40#4988C2214yes—
41#4987C2214yescomplemented
42#4987C2214yescomplemented
43#4977C2214yescomplemented
44#4959C2218yescomplemented
45#4958C2218yescomplemented
46#4955C2218yescomplemented
47#4952C2218yescomplemented
48#4940C2214yescomplemented
49#4939C2212yes—
50C4×(C2×D3)⋊C4C2214yescomplemented
51C4×C4⋊Dic3C2212yescomplemented
52#4928C2212yescomplemented
53C2×D42—31no—
54C2×#532D311yescharacteristic, complemented
55#2684—34no—
56#2563—32no—
57#2535—34no—
58#2420—34no—
59#2408—34no—
60#2404—34no—
61#2366—31no—
62D4×C22⋊C4—34no—
63C4×#121—32no—
64C24×D3C2314yes—
65C2×C3×#41—28no—
66C2×C3×#41—44no—
67C2×C3×#41—44no—
68C22×C3×Q8C2312yes—
69C22×C3×D4C2314yes—
70C22×C3×D4C2314yes—
71C22×C3×D4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2312yes—
74C23×C3×C4C2314yes—
75C22×#67C2318yes—
76C22×#67C2314yes—
77C23×Dic3C2314yes—
78C2×D3×D4—216no—
79C2×D3×D4—216no—
80C2×D3×D4—48no—
81C22×D12C2312yes—
82C22×C4×D3C2314yes—
83C3×C4⋊Q8—24no—
84C3×#127D418yescomplemented
85C3×#124—28no—
86C3×#121—216no—
87C3×C4×D4—216no—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C22⋊C4C2318yes—
90C2×C3×C42—24no—
91C2×C3×C42C2312yes—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C2×#217C2314yes—
95C2×#217C2318yes—
96C2×#217C2314yes—
97C2×#217C2314yes—
98#938—216no—
99#937—216no—
100#937—48no—
101D4×Dic3—48no—
102#930—28no—
103C4×#67—48no—
104C2×(C2×D3)⋊C4—48no—
105C2×(C2×D3)⋊C4C2314yes—
106C2×(C2×D3)⋊C4C2314yes—
107C2×(C2×D3)⋊C4C2314yes—
108C2×(C2×D3)⋊C4C2314yes—
109C2×(C2×D3)⋊C4C2318yes—
110C2×(C2×D3)⋊C4C2318yes—
111C2×C4⋊Dic3C2312yes—
112C2×C4⋊Dic3C2312yes—
113C2×C4⋊Dic3C2312yes—
114C2×Dic3⋊C4C2314yes—
115C2×C4×Dic3C2314yes—
116C4×D12—28no—
117C3⋊#95C2314yes—
118C3⋊#95C2314yes—
119C3⋊#95C2314yes—
120C3⋊#95C2318yes—
121C3⋊#95C2318yes—
122C22×#41C2×D312yescomplemented
123C23×D4—34no—
124D42—68no—
125#532—28no—
126C2×C4⋊Q8C2×D311yescharacteristic, complemented
127C2×#127C2×D311yescharacteristic
128C2×#124C2×D312yescomplemented
129C2×#121—34no—
130C2×#121—32no—
131C2×#121C2×D314yescomplemented
132C2×#120—34no—
133C2×#120—34no—
134C2×C4×D4—34no—
135C2×C4×D4—34no—
136C2×C4×D4—32no—
137C2×C4×D4C2×D314yescomplemented
138C22×C22⋊C4—34no—
139C22×C42C2×D311yescharacteristic, complemented
140#397—38no—
141#394—38no—
142#393—38no—
143#392—38no—
144#390—34no—
145#390—34no—
146#390—32no—
147#389—32no—
148#388—34no—
149#386—34no—
150#386—34no—
151#386—38no—
152#386—34no—
153#386—38no—
154#382—34no—
155#381—34no—
156#380—32no—
157#380—34no—
158#380—34no—
159#379—38no—
160C4×C4⋊C4—32no—
161C4×C22⋊C4—34no—
162C24×C3C2414yes—
163C24×C3C2411yescharacteristic
164C23×D3—28no—
165C23×D3—216no—
166C23×D3—44no—
167C23×D3—48no—
168C23×D3—44no—
169C23×D3C2414yes—
170C3×#41—432no—
171C2×C3×Q8—24no—
172C2×C3×Q8—42no—
173C2×C3×Q8—42no—
174C2×C3×D4—28no—
175C2×C3×D4—28no—
176C2×C3×D4—28no—
177C2×C3×D4—28no—
178C2×C3×D4—42no—
179C2×C3×D4—42no—
180C2×C3×D4—44no—
181C2×C3×D4C2×D4116yes—
182C2×C3×D4C2×D418yes—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—28no—
187C22×C3×C4—24no—
188C22×C3×C4—24no—
189C22×C3×C4—28no—
190C22×C3×C4—24no—
191C22×C3×C4—44no—
192C22×C3×C4C2414yes—
193C22×C3×C4C2412yes—
194C22×C3×C4C2411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C22×C3×C4C2412yes—
197C2×#67—216no—
198C2×#67—28no—
199C2×#67—28no—
200C2×#67—216no—
201C2×#67—44no—
202C2×#67—48no—
203C22×Dic3—44no—
204C22×Dic3—48no—
205C22×Dic3C2414yes—
206C22×Dic3C2414yes—
207C22×Dic3C2414yes—
208D3×D4—464no—
209C2×D12—24no—
210C2×D12—24no—
211C2×D12#4118yescomplemented
212C2×C4×D3—28no—
213C2×C4×D3—28no—
214C2×C4×D3—28no—
215C3×C4⋊C4—28no—
216C3×C22⋊C4—216no—
217C3×C42—24no—
218C3×C42—22no—
219C3×C42C2×D414yes—
220#217—28no—
221#217—44no—
222#217—48no—
223#217—44no—
224(C2×D3)⋊C4—28no—
225(C2×D3)⋊C4—28no—
226(C2×D3)⋊C4—28no—
227(C2×D3)⋊C4—48no—
228(C2×D3)⋊C4—44no—
229(C2×D3)⋊C4—48no—
230C4⋊Dic3—24no—
231C4⋊Dic3—24no—
232C4⋊Dic3—24no—
233Dic3⋊C4—44no—
234C4×Dic3—28no—
235C25—34no—
236C2×#41—28no—
237C2×#41—44no—
238C2×#41—44no—
239C22×Q8C22×D312yes—
240C22×D4—34no—
241C22×D4—38no—
242C22×D4—32no—
243C22×D4—616no—
244C22×D4—616no—
245C22×D4—128no—
246C22×D4C22×D314yes—
247C22×D4C22×D314yes—
248C22×D4C22×D312yes—
249C23×C4—34no—
250C23×C4—34no—
251C23×C4C22×D312yes—
252C23×C4C22×D311yescharacteristic
253C23×C4C22×D314yes—
254C4⋊Q8—24no—
255#127#6718yescomplemented
256#124—28no—
257#121—216no—
258#121—68no—
259#121—616no—
260#120—616no—
261#120—128no—
262C4×D4—216no—
263C4×D4—68no—
264C4×D4—128no—
265C4×D4—128no—
266C2×C4⋊C4—32no—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—34no—
270C2×C4⋊C4C22×D314yes—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—34no—
273C2×C22⋊C4—34no—
274C2×C22⋊C4—34no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—34no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—34no—
280C2×C22⋊C4—38no—
281C2×C22⋊C4—128no—
282C2×C22⋊C4C22×D318yes—
283C2×C42—24no—
284C2×C42—34no—
285C2×C42C22×D312yes—
286C2×C42C22×D311yescharacteristic
287C2×C42C22×D311yescharacteristic
288#95—34no—
289#95—34no—
290#95—34no—
291#95—38no—
292#95—38no—
293C23×C3—21no—
294C23×C3—22no—
295C23×C3—28no—
296C23×C3—22no—
297C23×C3—28no—
298C23×C3—24no—
299C23×C3—24no—
300C23×C3—21no—
301C23×C3C22×D418yes—
302C23×C3H(5,𝔽2)12yes—
303C23×C3C2511yescharacteristic
304C22×D3—24no—
305C22×D3—24no—
306C22×D3—24no—
307C22×D3—24no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3—416no—
311C22×D3—432no—
312C22×D3—816no—
313C22×D3—88no—
314C22×D3—88no—
315C22×D3C2×#4118yes—
316C3×Q8—48no—
317C3×D4—232no—
318C3×D4—48no—
319C3×D4—416no—
320C2×C3×C4—28no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—416no—
332C2×C3×C4—48no—
333C2×C3×C4C22×D412yes—
334C2×C3×C4C22×D414yes—
335C2×C3×C4C2×#4114yes—
336C2×C3×C4H(5,𝔽2)12yes—
337C2×C3×C4H(5,𝔽2)14yes—
338#67—432no—
339#67—416no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—24no—
345C2×Dic3—24no—
346C2×Dic3—24no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—88no—
350D12—216no—
351C4×D3—416no—
352C24—34no—
353C24—616no—
354C24—68no—
355C24—124no—
356C24—128no—
357C24—124no—
358C24C23×D311yescharacteristic
359C24C23×D314yes—
360#41—432no—
361C2×Q8—24no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—38no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—44no—
372C2×D4—68no—
373C2×D4—616no—
374C2×D4—68no—
375C2×D4—616no—
376C2×D4—64no—
377C2×D4—64no—
378C2×D4—124no—
379C2×D4—128no—
380C2×D4—1264no—
381C2×D4C2×#67116yes—
382C2×D4C2×#6718yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—34no—
392C22×C4—34no—
393C22×C4—34no—
394C22×C4—44no—
395C22×C4—68no—
396C22×C4—68no—
397C22×C4—68no—
398C22×C4—124no—
399C22×C4—128no—
400C22×C4C23×D314yes—
401C22×C4C23×D312yes—
402C22×C4C23×D311yescharacteristic
403C22×C4C23×D311yescharacteristic
404C22×C4C23×D312yes—
405C4⋊C4—28no—
406C4⋊C4—64no—
407C4⋊C4—64no—
408C4⋊C4—64no—
409C4⋊C4—124no—
410C22⋊C4—216no—
411C22⋊C4—68no—
412C22⋊C4—68no—
413C22⋊C4—68no—
414C22⋊C4—68no—
415C22⋊C4—128no—
416C22⋊C4—124no—
417C22⋊C4—128no—
418C22⋊C4—128no—
419C22⋊C4—124no—
420C22⋊C4—124no—
421C42—24no—
422C42—22no—
423C42—68no—
424C42C2×#6714yes—
425C22×C3—216no—
426C22×C3—22no—
427C22×C3—28no—
428C22×C3—24no—
429C22×C3—42no—
430C22×C3—44no—
431C22×C3—41no—
432C22×C3—48no—
433C22×C3—44no—
434C22×C3—41no—
435C22×C3C23×D411yescharacteristic
436C22×C3C22×#4112yes—
437C22×C3C2×H(5,𝔽2)11yescharacteristic
438C22×C3C2×H(5,𝔽2)11yescharacteristic
439C22×C3C2×H(5,𝔽2)12yes—
440C2×D3—28no—
441C2×D3—28no—
442C2×D3—28no—
443C2×D3—44no—
444C2×D3—44no—
445C2×D3—44no—
446C2×D3—44no—
447C2×D3—832no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—44no—
452C3×C4#54618yes—
453Dic3—44no—
454Dic3—44no—
455Dic3—44no—
456C23—24no—
457C23—24no—
458C23—21no—
459C23—22no—
460C23—28no—
461C23—22no—
462C23—28no—
463C23—21no—
464C23—38no—
465C23—64no—
466C23—64no—
467C23—64no—
468C23—64no—
469C23—64no—
470C23—64no—
471C23—1232no—
472C23—1216no—
473C23—248no—
474C23—2416no—
475C23—248no—
476C23#100412yes—
477C23C22×#6718yes—
478C23C24×D311yescharacteristic
479Q8—48no—
480D4—232no—
481D4—48no—
482D4—416no—
483D4—616no—
484D4—1232no—
485D4—1216no—
486C2×C4—28no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—416no—
498C2×C4—48no—
499C2×C4—64no—
500C2×C4—64no—
501C2×C4—64no—
502C2×C4—64no—
503C2×C4—64no—
504C2×C4—64no—
505C2×C4—64no—
506C2×C4—64no—
507C2×C4—64no—
508C2×C4—1216no—
509C2×C4—248no—
510C2×C4#100412yes—
511C2×C4D3×#4114yes—
512C2×C4#100914yes—
513C2×C4C22×#6712yes—
514C2×C4C22×#6714yes—
515C2×C3—48no—
516C2×C3—42no—
517C2×C3C2×#54612yes—
518C2×C3#348111yescharacteristic
519C2×C3#348411yescharacteristic
520C2×C3#351912yes—
521C2×C3#354911yescharacteristic
522D3—48no—
523C22—22no—
524C22—28no—
525C22—216no—
526C22—24no—
527C22—44no—
528C22—41no—
529C22—42no—
530C22—48no—
531C22—44no—
532C22—41no—
533C22—68no—
534C22—68no—
535C22—68no—
536C22—124no—
537C22—124no—
538C22—124no—
539C22—124no—
540C22—2432no—
541C22C2×#100411yescharacteristic
542C22C2×#100411yescharacteristic
543C22C2×D3×#4112yes—
544C22C2×#100912yes—
545C22C23×#6711yescharacteristic
546C4—22no—
547C4—22no—
548C4—44no—
549C4—42no—
550C4—124no—
551C4—124no—
552C4—124no—
553C4#582918yes—
554C3#5904111yescharacteristic, complemented, Sylow
555C2—48no—
556C2—42no—
557C2—128no—
558C2#8623911yescharacteristic
559C2#8624712yes—
560C2#8625211yescharacteristic
561C2#8723911yescharacteristic
562C2C2×#582912yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#532C2
hypercenterC2×#532D3
hypocenterC3#59041
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#532

Complements

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

NH ≅ G/Ntype
1. C2×C3×#532C2semidirect
2. C2×#5717C2semidirect
3. #84096C2semidirect
4. #83260C2semidirect
5. #83173C2semidirect
6. #83005C2semidirect
7. #82666C2semidirect
8. #82640C2semidirect
9. #82405C2semidirect
10. D4×(C2×D3)⋊C4C2semidirect
11. C4×#930C2semidirect
13. C22×C3×#41C22semidirect
16. C2×C3×C4⋊Q8C22semidirect
18. C2×C3×#124C22semidirect
19. C2×C3×#121C22semidirect
20. C2×C3×C4×D4C22semidirect
21. C22×C3×C42C22semidirect
24. C2×#937C22semidirect
25. C2×D4×Dic3C22semidirect
27. C2×C4×#67C22semidirect
28. C22×(C2×D3)⋊C4C22semidirect
31. #5231C22semidirect
32. #5223C22semidirect
33. #5221C22semidirect
34. #5220C22semidirect
35. #5220C22semidirect
36. #5213C22semidirect
37. #5213C22semidirect
38. #5212C22semidirect
39. #5211C22semidirect
41. #4987C22semidirect
42. #4987C22semidirect
43. #4977C22semidirect
44. #4959C22semidirect
45. #4958C22semidirect
46. #4955C22semidirect
47. #4952C22semidirect
48. #4940C22semidirect
50. C4×(C2×D3)⋊C4C22semidirect
51. C4×C4⋊Dic3C22semidirect
52. #4928C22semidirect
54. C2×#532D3semidirect
84. C3×#127D4semidirect
122. C22×#41C2×D3semidirect
126. C2×C4⋊Q8C2×D3semidirect
128. C2×#124C2×D3semidirect
131. C2×#121C2×D3semidirect
137. C2×C4×D4C2×D3semidirect
139. C22×C42C2×D3semidirect
211. C2×D12#41semidirect
255. #127#67semidirect
554. C3#59041semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5322
C2×#57172
#840962
#832602
#831732
#830052
#826662
#826402
#824052
D4×(C2×D3)⋊C42
C4×#9302
#590413