Finite Group Databasebeta

← C3×#82927 #11863921

Subgroups — C3×#82927 #11863921

Order 1152.

2153 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×D12C211yesmaximal, characteristic, complemented
2C32×#525C211yesmaximal, characteristic, complemented
3C2×C3×#949C212yesmaximal, complemented
4C3×#5207C211yesmaximal, characteristic, complemented
5C3×#5046C214yesmaximal, complemented
6C3×#5036C214yesmaximal, complemented
7C3×#5001C212yesmaximal, complemented
8C3×#3003—31nomaximal
9#82927C311yesmaximal, characteristic, direct product factor
10C24×C3×D3—21no—
11C2×C32×#41C2211yescharacteristic, complemented
12C23×C32×C4C2211yescharacteristic
13C22×C3×D12—22no—
14C22×C3×D12—21no—
15C22×C3×D12—28no—
16C22×C3×D12C2214yes—
17C22×C3×D12C2212yes—
18C32×#123—21no—
19C32×C22⋊Q8C2212yescomplemented
20C32×#121C2212yescomplemented
21C32×#120—21no—
22C32×C4×D4—22no—
23C32×#117C2211yescharacteristic, complemented
24C3×#949—28no—
25C2×C3×#215C2214yescomplemented
26C2×C3×#213C2214yescomplemented
27C2×C3×#208C2212yescomplemented
28C3×#830C2214yescomplemented
29C3×#809C2218yescomplemented
30C23×C3×D4—31no—
31C23×D12C2×C311yescharacteristic, complemented
32C2×C3×AGL(1,ℤ/8)—32no—
33C3×#525—21no—
34C3×#525D311yescharacteristic, complemented
35C3×#525C2×C311yescharacteristic, complemented
36C2×#949C2×C312yescomplemented
37C3×#450—34no—
38C3×#447—34no—
39C3×#418—32no—
40C3×#407—31no—
41#5207C2×C311yescharacteristic, complemented
42#5046C2×C314yescomplemented
43#5036C2×C314yescomplemented
44#5001C2×C312yescomplemented
45C24×C32D411yescharacteristic
46C23×C3×D3—28no—
47C23×C3×D3—24no—
48C23×C3×D3—22no—
49C23×C3×D3—48no—
50C32×#41—24no—
51C2×C32×Q8C2311yescharacteristic, complemented
52C2×C32×D4—21no—
53C2×C32×D4—22no—
54C2×C32×D4C2311yescharacteristic, complemented
55C22×C32×C4—24no—
56C22×C32×C4—21no—
57C22×C32×C4D412yes—
58C22×C32×C4D411yescharacteristic
59C22×C32×C4C2312yes—
60C22×C32×C4C2311yescharacteristic
61C2×C3×D12—28no—
62C2×C3×D12—216no—
63C2×C3×D12—24no—
64C2×C3×D12—22no—
65C2×C3×D12—24no—
66C2×C3×D12—216no—
67C2×C3×D12D418yescomplemented
68C2×C3×D12C2314yes—
69C32×C4⋊C4—22no—
70C32×C4⋊C4C2312yescomplemented
71C32×C22⋊C4—21no—
72C32×C22⋊C4—22no—
73C32×C22⋊C4—22no—
74C32×C42—21no—
75C3×#215—28no—
76C3×#213—28no—
77C3×#208—24no—
78C2×C3×C3⋊C8C2314yescomplemented
79#3003—31noSylow
80C25×C3—61no—
81C24×D3—21no—
82C2×C3×#41—21no—
83C2×C3×#41C2×D311yescharacteristic, complemented
84C2×C3×#41C22×C311yescharacteristic, complemented
85C22×C3×D4—32no—
86C22×C3×D4—34no—
87C22×C3×D4—68no—
88C22×C3×D4—62no—
89C22×C3×D4—61no—
90C23×C3×C4—21no—
91C23×C3×C4C2×D311yescharacteristic
92C23×C3×C4C22×C311yescharacteristic
93C22×D12—22no—
94C22×D12—21no—
95C22×D12—28no—
96C22×D12C22×C314yes—
97C22×D12C22×C312yes—
98C3×AGL(1,ℤ/8)—68no—
99C2×C3×SD16—34no—
100C2×C3×D8—34no—
101C2×C3×M16—32no—
102C3×#123—22no—
103C3×#123—21no—
104C3×#123—21no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8C2×D312yescomplemented
107C3×C22⋊Q8C22×C312yescomplemented
108C3×#121—22no—
109C3×#121C2×D312yescomplemented
110C3×#121C22×C312yescomplemented
111C3×#120—22no—
112C3×#120—21no—
113C3×#120—21no—
114C3×C4×D4—24no—
115C3×C4×D4—22no—
116C3×C4×D4—22no—
117C3×#117—21no—
118C3×#117C2×D311yescharacteristic, complemented
119C3×#117C22×C311yescharacteristic, complemented
120#949—28no—
121C2×#215C22×C314yescomplemented
122C2×#213C22×C314yescomplemented
123C2×#208C22×C312yescomplemented
124C3×#102—38no—
125C3×C22⋊C8—34no—
126#830C22×C314yescomplemented
127#809C22×C318yescomplemented
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—24no—
131C23×C32C2×D411yescharacteristic
132C23×C32C2×D412yes—
133C22×C3×D3—216no—
134C22×C3×D3—28no—
135C22×C3×D3—24no—
136C22×C3×D3—432no—
137C22×C3×D3—48no—
138C22×C3×D3—48no—
139C22×C3×D3—48no—
140C32×Q8—22no—
141C32×D4—22no—
142C32×D4—42no—
143C32×D4—44no—
144C2×C32×C4—22no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—24no—
148C2×C32×C4—24no—
149C2×C32×C4—42no—
150C2×C32×C4C2×D414yes—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4C2411yescharacteristic
154C3×D12—216no—
155C3×D12—28no—
156C3×D12—24no—
157C3×D12C2×D418yes—
158C3×C3⋊C8—24no—
159C23×D4—31no—
160C2×AGL(1,ℤ/8)—32no—
161#525C3×D311yescharacteristic, complemented
162#450—34no—
163#447—34no—
164#418—32no—
165#407—31no—
166C24×C3—21no—
167C24×C3—68no—
168C24×C3—64no—
169C24×C3—62no—
170C24×C3—128no—
171C24×C3#6711yescharacteristic
172C24×C3C3×D411yescharacteristic
173C23×D3—28no—
174C23×D3—24no—
175C23×D3—22no—
176C23×D3—48no—
177C3×#41—24no—
178C3×#41—24no—
179C3×#41—44no—
180C2×C3×Q8—21no—
181C2×C3×Q8C22×D311yescharacteristic, complemented
182C2×C3×Q8C23×C311yescharacteristic, complemented
183C2×C3×D4—24no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—21no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—38no—
191C2×C3×D4—34no—
192C2×C3×D4—616no—
193C2×C3×D4—68no—
194C2×C3×D4—616no—
195C2×C3×D4—64no—
196C2×C3×D4—64no—
197C2×C3×D4—62no—
198C2×C3×D4C22×D311yescharacteristic, complemented
199C2×C3×D4C23×C311yescharacteristic, complemented
200C22×C3×C4—24no—
201C22×C3×C4—24no—
202C22×C3×C4—22no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—22no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—22no—
209C22×C3×C4—44no—
210C22×C3×C4#6712yes—
211C22×C3×C4#6711yescharacteristic
212C22×C3×C4C3×D412yes—
213C22×C3×C4C3×D411yescharacteristic
214C22×C3×C4C22×D312yes—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C23×C312yes—
217C22×C3×C4C23×C311yescharacteristic
218C2×D12—28no—
219C2×D12—216no—
220C2×D12—24no—
221C2×D12—24no—
222C2×D12—22no—
223C2×D12—216no—
224C2×D12C3×D418yescomplemented
225C2×D12C23×C314yes—
226C3×SD16—68no—
227C3×D8—68no—
228C3×M16—64no—
229C2×C3×C8—34no—
230C3×C4⋊C4—24no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4C22×D312yescomplemented
235C3×C4⋊C4C23×C312yescomplemented
236C3×C22⋊C4—22no—
237C3×C22⋊C4—24no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C42—22no—
246C3×C42—21no—
247C3×C42—21no—
248#215—28no—
249#213—28no—
250#208—24no—
251C2×C3⋊C8C23×C314yescomplemented
252C22×C32—28no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32#12012yes—
259C22×C32AGL(1,ℤ/8)14yescomplemented
260C22×C32C22×D411yescharacteristic
261C2×C3×D3—28no—
262C2×C3×D3—416no—
263C2×C3×D3—432no—
264C2×C3×D3—48no—
265C32×C4—22no—
266C32×C4—41no—
267C32×C4—42no—
268C32×C4#12012yes—
269C32×C4C22×D412yes—
270C25—61no—
271C2×#41C2×C3×D311yescharacteristic, complemented
272C22×D4—32no—
273C22×D4—34no—
274C22×D4—68no—
275C22×D4—62no—
276C22×D4—61no—
277C23×C4C2×C3×D311yescharacteristic
278AGL(1,ℤ/8)—68no—
279C2×SD16—34no—
280C2×D8—34no—
281C2×M16—32no—
282#123—21no—
283C22⋊Q8C2×C3×D312yescomplemented
284#121C2×C3×D312yescomplemented
285#120—21no—
286C4×D4—22no—
287#117C2×C3×D311yescharacteristic, complemented
288#102—38no—
289C22⋊C8—34no—
290C23×C3—22no—
291C23×C3—24no—
292C23×C3—22no—
293C23×C3—24no—
294C23×C3—22no—
295C23×C3—21no—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—44no—
301C23×C3—616no—
302C23×C3—68no—
303C23×C3—64no—
304C23×C3—1232no—
305C23×C3—128no—
306C23×C3—128no—
307C23×C3—128no—
308C23×C3C2×#6711yescharacteristic
309C23×C3C2×#6712yes—
310C23×C3C2×C3×D411yescharacteristic
311C23×C3C2×C3×D412yes—
312C22×D3—216no—
313C22×D3—28no—
314C22×D3—24no—
315C22×D3—432no—
316C22×D3—48no—
317C22×D3—48no—
318C22×D3—48no—
319C3×Q8—22no—
320C3×Q8—22no—
321C3×Q8—42no—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—38no—
325C3×D4—42no—
326C3×D4—48no—
327C3×D4—44no—
328C3×D4—42no—
329C3×D4—44no—
330C3×D4—42no—
331C3×D4—44no—
332C3×D4—616no—
333C3×D4—68no—
334C3×D4—64no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—24no—
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—24no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—44no—
352C2×C3×C4—44no—
353C2×C3×C4—42no—
354C2×C3×C4—44no—
355C2×C3×C4—42no—
356C2×C3×C4—42no—
357C2×C3×C4D3×D414yes—
358C2×C3×C4C2×#6712yes—
359C2×C3×C4C2×#6711yescharacteristic
360C2×C3×C4C2×C3×D414yes—
361C2×C3×C4C2×C3×D412yes—
362C2×C3×C4C2×C3×D411yescharacteristic
363C2×C3×C4C23×D311yescharacteristic
364C2×C3×C4C24×C311yescharacteristic
365D12—216no—
366D12—28no—
367D12—24no—
368D12C2×C3×D418yes—
369C3×C8—64no—
370C3⋊C8—24no—
371C2×C32—22no—
372C2×C32—24no—
373C2×C32—81no—
374C2×C32C2×#12011yescharacteristic
375C2×C32C2×AGL(1,ℤ/8)12yes—
376C3×D3—48no—
377C24—68no—
378C24—64no—
379C24—62no—
380C24—128no—
381C24C3×#6711yescharacteristic
382#41—24no—
383C2×Q8C22×C3×D311yescharacteristic, complemented
384C2×D4—22no—
385C2×D4—21no—
386C2×D4—34no—
387C2×D4—38no—
388C2×D4—616no—
389C2×D4—68no—
390C2×D4—616no—
391C2×D4—64no—
392C2×D4—64no—
393C2×D4—62no—
394C2×D4C22×C3×D311yescharacteristic, complemented
395C22×C4—24no—
396C22×C4—21no—
397C22×C4C3×#6712yes—
398C22×C4C3×#6711yescharacteristic
399C22×C4C22×C3×D312yes—
400C22×C4C22×C3×D311yescharacteristic
401SD16—68no—
402D8—68no—
403M16—64no—
404C2×C8—34no—
405C4⋊C4—22no—
406C4⋊C4C22×C3×D312yescomplemented
407C22⋊C4—21no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C42—21no—
411C22×C3—28no—
412C22×C3—28no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—22no—
420C22×C3—24no—
421C22×C3—21no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—48no—
426C22×C3—42no—
427C22×C3—44no—
428C22×C3—42no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3—41no—
432C22×C3—68no—
433C22×C3—1216no—
434C22×C3—1232no—
435C22×C3—128no—
436C22×C3#93712yes—
437C22×C3#94914yescomplemented
438C22×C3C3×#12012yes—
439C22×C3C3×AGL(1,ℤ/8)14yescomplemented
440C22×C3C22×#6711yescharacteristic
441C22×C3C22×C3×D411yescharacteristic
442C2×D3—28no—
443C2×D3—416no—
444C2×D3—432no—
445C2×D3—48no—
446C3×C4—22no—
447C3×C4—22no—
448C3×C4—22no—
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—42no—
452C3×C4—44no—
453C3×C4—42no—
454C3×C4—41no—
455C3×C4—42no—
456C3×C4—41no—
457C3×C4#93712yes—
458C3×C4C3×#12012yes—
459C3×C4C2×D3×D412yes—
460C3×C4C22×C3×D412yes—
461C32#300311yescharacteristic, complemented, Sylow
462C23—22no—
463C23—24no—
464C23—21no—
465C23—616no—
466C23—68no—
467C23—64no—
468C23—1232no—
469C23—128no—
470C23—128no—
471C23—128no—
472C23C2×C3×#6711yescharacteristic
473C23C2×C3×#6712yes—
474Q8—22no—
475D4—22no—
476D4—38no—
477D4—42no—
478D4—44no—
479D4—616no—
480D4—68no—
481D4—64no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—42no—
488C2×C4C3×D3×D414yes—
489C2×C4C2×C3×#6712yes—
490C2×C4C2×C3×#6711yescharacteristic
491C2×C4C23×C3×D311yescharacteristic
492C8—64no—
493C2×C3—22no—
494C2×C3—24no—
495C2×C3—22no—
496C2×C3—24no—
497C2×C3—21no—
498C2×C3—22no—
499C2×C3—42no—
500C2×C3—44no—
501C2×C3—82no—
502C2×C3—81no—
503C2×C3—81no—
504C2×C3—128no—
505C2×C3C2×#93711yescharacteristic
506C2×C3C2×#94912yes—
507C2×C3C2×C3×#12011yescharacteristic
508C2×C3C2×C3×AGL(1,ℤ/8)12yes—
509D3—48no—
510C22—28no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—41no—
515C22—42no—
516C22—68no—
517C22—1216no—
518C22—1232no—
519C22—128no—
520C22C3×#93712yes—
521C22C3×#94914yescomplemented
522C22C22×C3×#6711yescharacteristic
523C4—22no—
524C4—41no—
525C4—42no—
526C4C3×#93712yes—
527C4C2×C3×D3×D412yes—
528C3—21no—
529C3#8292711yescharacteristic, direct product factor
530C3C3×#300311yescharacteristic, complemented
531C2—22no—
532C2—24no—
533C2—81no—
534C2—128no—
535C2C2×C3×#93711yescharacteristic
536C2C2×C3×#94912yes—
subgroupG/H
centerC22×C3C22×#67
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#525C2
socleC22×C32C22×D4
hypercenterC3×#525D3
hypocenterC3C3×#3003
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C3×#525

Complements

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

NH ≅ G/Ntype
1. C23×C3×D12C2semidirect
2. C32×#525C2semidirect
3. C2×C3×#949C2semidirect
4. C3×#5207C2semidirect
5. C3×#5046C2semidirect
6. C3×#5036C2semidirect
7. C3×#5001C2semidirect
9. #82927C3direct (N × H)
11. C2×C32×#41C22semidirect
19. C32×C22⋊Q8C22semidirect
20. C32×#121C22semidirect
23. C32×#117C22semidirect
25. C2×C3×#215C22semidirect
26. C2×C3×#213C22semidirect
27. C2×C3×#208C22semidirect
28. C3×#830C22semidirect
29. C3×#809C22semidirect
31. C23×D12C2×C3semidirect
34. C3×#525D3semidirect
35. C3×#525C2×C3semidirect
36. C2×#949C2×C3semidirect
41. #5207C2×C3semidirect
42. #5046C2×C3semidirect
43. #5036C2×C3semidirect
44. #5001C2×C3semidirect
51. C2×C32×Q8C23semidirect
54. C2×C32×D4C23semidirect
67. C2×C3×D12D4semidirect
70. C32×C4⋊C4C23semidirect
78. C2×C3×C3⋊C8C23semidirect
83. C2×C3×#41C2×D3semidirect
84. C2×C3×#41C22×C3semidirect
106. C3×C22⋊Q8C2×D3semidirect
107. C3×C22⋊Q8C22×C3semidirect
109. C3×#121C2×D3semidirect
110. C3×#121C22×C3semidirect
118. C3×#117C2×D3semidirect
119. C3×#117C22×C3semidirect
121. C2×#215C22×C3semidirect
122. C2×#213C22×C3semidirect
123. C2×#208C22×C3semidirect
126. #830C22×C3semidirect
127. #809C22×C3semidirect
161. #525C3×D3semidirect
181. C2×C3×Q8C22×D3semidirect
182. C2×C3×Q8C23×C3semidirect
198. C2×C3×D4C22×D3semidirect
199. C2×C3×D4C23×C3semidirect
224. C2×D12C3×D4semidirect
234. C3×C4⋊C4C22×D3semidirect
235. C3×C4⋊C4C23×C3semidirect
251. C2×C3⋊C8C23×C3semidirect
259. C22×C32AGL(1,ℤ/8)semidirect
271. C2×#41C2×C3×D3semidirect
283. C22⋊Q8C2×C3×D3semidirect
284. #121C2×C3×D3semidirect
287. #117C2×C3×D3semidirect
383. C2×Q8C22×C3×D3semidirect
394. C2×D4C22×C3×D3semidirect
406. C4⋊C4C22×C3×D3semidirect
437. C22×C3#949semidirect
439. C22×C3C3×AGL(1,ℤ/8)semidirect
461. C32#3003semidirect
521. C22C3×#949semidirect
529. C3#82927direct (N × H)
530. C3C3×#3003semidirect

Maximal subgroups

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

subgroupindex
C23×C3×D122
C32×#5252
C2×C3×#9492
C3×#52072
C3×#50462
C3×#50362
C3×#50012
C3×#30033
#829273