Finite Group Databasebeta

← C3×#82037 #11899712

Subgroups — C3×#82037 #11899712

Order 1152.

765 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5771C211yesmaximal, characteristic, complemented
2C3×#5757C211yesmaximal, characteristic, complemented
3C32×#509C211yesmaximal, characteristic, complemented
4C3×#5194C211yesmaximal, characteristic, complemented
5C2×C3×#827C211yesmaximal, characteristic, complemented
6C3×#4915C212yesmaximal, complemented
7C3×D3×SD32C212yesmaximal, complemented
8C3×#4912C212yesmaximal, complemented
9C3×#4911C212yesmaximal, complemented
10C3×#4909C211yesmaximal, characteristic, complemented
11C3×#4907C211yesmaximal, characteristic, complemented
12C3×#3416—31nomaximal
13#82037C311yesmaximal, characteristic, direct product factor
14C3×#1010—21no—
15C3×#1004—21no—
16C32×#135C2211yescharacteristic, complemented
17C2×C32×D8C2211yescharacteristic
18C3×#951—21no—
19C3×#932—21no—
20C3×D3×Q16C2212yescomplemented
21C3×#915—22no—
22C3×#912C2212yes—
23C3×#912C2212yescomplemented
24C3×#911—22no—
25C3×D3×D8C2212yes—
26C2×C3×Dic12C2211yescharacteristic, complemented
27C3×#904C2211yescharacteristic
28C3×#901C2211yescharacteristic
29C32×SD32C2212yescomplemented
30C32×D16C2212yescomplemented
31C32×M32C2211yescharacteristic, complemented
32C3×#829C2211yescharacteristic, complemented
33C3×#828C2211yescharacteristic, complemented
34C3×#827C2212yescomplemented
35C3×#827C2211yescharacteristic, complemented
36C3×#827C2212yescomplemented
37C3×#826C2211yescharacteristic, complemented
38C2×C3×C3⋊C16C2211yescharacteristic, complemented
39C3×Dic24C2212yescomplemented
40C3×#800C2212yescomplemented
41C3×#798C2212yescomplemented
42C3×D3×C16C2212yescomplemented
43C3×#578—31no—
44C3×#576—31no—
45#5771C2×C311yescharacteristic, complemented
46#5757C2×C311yescharacteristic, complemented
47C3×#510—32no—
48C3×#510—31no—
49C3×#509—21no—
50C3×#509—32no—
51C3×#509D311yescharacteristic, complemented
52C3×#509C2×C311yescharacteristic, complemented
53C3×#508—31no—
54C3×#508—32no—
55C2×C3×SD32—31no—
56C2×C3×SD32—32no—
57C3×#504—31no—
58#5194C2×C311yescharacteristic, complemented
59C2×#827C2×C311yescharacteristic, complemented
60#4915C2×C312yescomplemented
61D3×SD32C2×C312yescomplemented
62#4912C2×C312yescomplemented
63#4911C2×C312yescomplemented
64#4909C2×C311yescharacteristic, complemented
65#4907C2×C311yescharacteristic, complemented
66C32×#41—21no—
67C2×C32×D4—21no—
68C2×C3×#67—42no—
69C3×D3×Q8—22no—
70C3×#237—22no—
71C3×#237—22no—
72C3×#237—42no—
73C3×#237—41no—
74C3×D3×D4—22no—
75C3×D3×D4—41no—
76C3×#235—21no—
77C3×#235—41no—
78C3×#235D411yescharacteristic
79C2×C3×Dic6—21no—
80C2×C3×Dic6—41no—
81C32×Q16C2311yescharacteristic
82C32×SD16—21no—
83C32×D8—21no—
84C32×D8C2312yes—
85C32×D8C2311yescharacteristic
86C2×C32×C8C2311yescharacteristic
87C3×#216—22no—
88C3×#214—22no—
89C3×#214—22no—
90C3×#213—22no—
91C3×#208D411yescharacteristic
92C3×Dic12—21no—
93C3×Dic12C2311yescharacteristic
94C3×Dic12C2312yes—
95C3×D24C2311yescharacteristic
96C3×#204—21no—
97C3×ΓL(1,𝔽25)D412yes—
98C3×D3×C8C2312yes—
99C32×C16C2312yes—
100C3×C3⋊C16C2311yescharacteristic
101C3×C3⋊C16C2311yescharacteristic
102#3416—31noSylow
103C3×2−1+4—61no—
104C3×H(5,𝔽2)—61no—
105#1010—21no—
106#1004—21no—
107C3×#137—62no—
108C3×#137—61no—
109C3×AGL(1,ℤ/8)—62no—
110C3×AGL(1,ℤ/8)—61no—
111C3×#135—21no—
112C3×#135—32no—
113C3×#135—32no—
114C3×#135—31no—
115C3×#135C2×D311yescharacteristic, complemented
116C3×#135C22×C311yescharacteristic, complemented
117C2×C3×Q16—32no—
118C2×C3×Q16—31no—
119C2×C3×D8—21no—
120C2×C3×D8—32no—
121C2×C3×D8C2×D311yescharacteristic
122C2×C3×D8C22×C311yescharacteristic
123C3×#131—31no—
124#951—21no—
125#932—21no—
126D3×Q16C22×C312yescomplemented
127#915—22no—
128#912C22×C312yes—
129#912C22×C312yescomplemented
130#911—22no—
131D3×D8C22×C312yes—
132C2×Dic12C22×C311yescharacteristic, complemented
133#904C22×C311yescharacteristic
134#901C22×C311yescharacteristic
135C3×Q32—31no—
136C3×Q32—32no—
137C3×SD32—22no—
138C3×SD32—31no—
139C3×SD32—32no—
140C3×SD32—32no—
141C3×SD32—31no—
142C3×SD32—32no—
143C3×SD32C2×D312yescomplemented
144C3×SD32C22×C312yescomplemented
145C3×D16—22no—
146C3×D16—31no—
147C3×D16C2×D312yescomplemented
148C3×D16C22×C312yescomplemented
149C3×M32—21no—
150C3×M32—32no—
151C3×M32C2×D311yescharacteristic, complemented
152C3×M32C22×C311yescharacteristic, complemented
153C2×C3×C16—31no—
154C2×C3×C16—32no—
155#829C22×C311yescharacteristic, complemented
156#828C22×C311yescharacteristic, complemented
157#827C22×C312yescomplemented
158#827C22×C312yescomplemented
159#827C22×C311yescharacteristic, complemented
160#826C22×C311yescharacteristic, complemented
161C2×C3⋊C16C22×C311yescharacteristic, complemented
162Dic24C22×C312yescomplemented
163#800C22×C312yescomplemented
164#798C22×C312yescomplemented
165D3×C16C22×C312yescomplemented
166C23×C32—41no—
167C22×C3×D3—42no—
168C32×Q8—21no—
169C32×D4—22no—
170C32×D4—21no—
171C32×D4—41no—
172C32×D4—41no—
173C2×C32×C4—41no—
174C2×C32×C4C2×D411yescharacteristic
175C3×#67—41no—
176C3×#67—42no—
177C3×#67—42no—
178C3×#67—42no—
179C3×#67D812yes—
180C2×C3×Dic3—41no—
181C2×C3×Dic3—42no—
182C2×C3×Dic3—42no—
183C3×D12—21no—
184C3×D12D811yescharacteristic
185C3×C4×D3—42no—
186C3×C4×D3—41no—
187C3×C4×D3C2×D412yes—
188C3×Dic6—21no—
189C3×Dic6—22no—
190C3×Dic6—41no—
191C3×Dic6—42no—
192C3×Dic6D811yescharacteristic
193C32×C8C2×D411yescharacteristic
194C32×C8C2411yescharacteristic
195C3×C3⋊C8C2×D412yes—
196#578—31no—
197#576—31no—
198#510—32no—
199#510—31no—
200#509—32no—
201#509C3×D311yescharacteristic, complemented
202#508—31no—
203#508—32no—
204C2×SD32—31no—
205C2×SD32—32no—
206#504—31no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—31no—
210C3×#41—41no—
211C3×#41—62no—
212C3×#41—62no—
213C3×#41—61no—
214C3×#41—121no—
215C3×#41—122no—
216C3×#41—121no—
217C2×C3×Q8—62no—
218C2×C3×Q8—61no—
219C2×C3×Q8—121no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—41no—
223C2×C3×D4—62no—
224C2×C3×D4—121no—
225C2×C3×D4—122no—
226C2×#67—42no—
227D3×Q8—22no—
228#237—22no—
229#237—22no—
230#237—42no—
231#237—41no—
232D3×D4—22no—
233D3×D4—41no—
234#235—21no—
235#235—41no—
236#235C3×D411yescharacteristic
237C2×Dic6—21no—
238C2×Dic6—41no—
239C3×Q16—21no—
240C3×Q16—32no—
241C3×Q16—31no—
242C3×Q16—62no—
243C3×Q16—61no—
244C3×Q16C22×D311yescharacteristic
245C3×Q16C23×C311yescharacteristic
246C3×SD16—21no—
247C3×SD16—21no—
248C3×SD16—41no—
249C3×SD16—62no—
250C3×SD16—62no—
251C3×SD16—61no—
252C3×D8—21no—
253C3×D8—21no—
254C3×D8—21no—
255C3×D8—22no—
256C3×D8—31no—
257C3×D8—41no—
258C3×D8—62no—
259C3×D8C22×D311yescharacteristic
260C3×D8C22×D312yes—
261C3×D8C23×C311yescharacteristic
262C3×D8C23×C312yes—
263C3×M16—31no—
264C3×M16—32no—
265C2×C3×C8—21no—
266C2×C3×C8—32no—
267C2×C3×C8C22×D311yescharacteristic
268C2×C3×C8C23×C311yescharacteristic
269#216—22no—
270#214—22no—
271#214—22no—
272#213—22no—
273#208C3×D411yescharacteristic
274Dic12—21no—
275Dic12C23×C312yes—
276Dic12C23×C311yescharacteristic
277D24C23×C311yescharacteristic
278#204—21no—
279ΓL(1,𝔽25)C3×D412yes—
280D3×C8C23×C312yes—
281C3×C16—22no—
282C3×C16—31no—
283C3×C16—31no—
284C3×C16C22×D312yes—
285C3×C16C23×C312yes—
286C3⋊C16C23×C311yescharacteristic
287C3⋊C16C23×C311yescharacteristic
288C22×C32—41no—
289C22×C32—42no—
290C22×C32—161no—
291C22×C32C2×D811yescharacteristic
292C2×C3×D3—41no—
293C2×C3×D3—162no—
294C2×C3×D3C2×D812yes—
295C32×C4—41no—
296C32×C4C2×D811yescharacteristic
297C32×C4C22×D411yescharacteristic
298C3×Dic3—42no—
299C3×Dic3—41no—
300C3×Dic3C2×D812yes—
3012−1+4—61no—
302H(5,𝔽2)—61no—
303#137—62no—
304#137—61no—
305AGL(1,ℤ/8)—62no—
306AGL(1,ℤ/8)—61no—
307#135—32no—
308#135—32no—
309#135—31no—
310#135C2×C3×D311yescharacteristic, complemented
311C2×Q16—32no—
312C2×Q16—31no—
313C2×D8—32no—
314C2×D8C2×C3×D311yescharacteristic
315#131—31no—
316Q32—31no—
317Q32—32no—
318SD32—31no—
319SD32—32no—
320SD32—32no—
321SD32—31no—
322SD32—32no—
323SD32C2×C3×D312yescomplemented
324D16—31no—
325D16C2×C3×D312yescomplemented
326M32—32no—
327M32C2×C3×D311yescharacteristic, complemented
328C2×C16—31no—
329C2×C16—32no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—81no—
333C23×C3—122no—
334C22×D3—42no—
335C3×Q8—21no—
336C3×Q8—21no—
337C3×Q8—31no—
338C3×Q8—41no—
339C3×Q8—61no—
340C3×Q8—62no—
341C3×Q8—122no—
342C3×Q8—121no—
343C3×D4—22no—
344C3×D4—21no—
345C3×D4—22no—
346C3×D4—21no—
347C3×D4—32no—
348C3×D4—31no—
349C3×D4—42no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—61no—
356C3×D4—81no—
357C3×D4—81no—
358C3×D4—122no—
359C3×D4—122no—
360C3×D4—122no—
361C3×D4—121no—
362C2×C3×C4—21no—
363C2×C3×C4—32no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—81no—
367C2×C3×C4—122no—
368C2×C3×C4—122no—
369C2×C3×C4—122no—
370C2×C3×C4—121no—
371C2×C3×C4—121no—
372C2×C3×C4D3×D411yescharacteristic
373C2×C3×C4C2×C3×D411yescharacteristic
374#67—41no—
375#67—42no—
376#67—42no—
377#67—42no—
378#67C3×D812yes—
379C2×Dic3—41no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382D12—21no—
383D12C3×D811yescharacteristic
384C4×D3—42no—
385C4×D3—41no—
386C4×D3C2×C3×D412yes—
387Dic6—21no—
388Dic6—22no—
389Dic6—41no—
390Dic6—42no—
391Dic6C3×D811yescharacteristic
392C3×C8—21no—
393C3×C8—21no—
394C3×C8—32no—
395C3×C8D3×D411yescharacteristic
396C3×C8C2×C3×D411yescharacteristic
397C3×C8C23×D311yescharacteristic
398C3×C8C24×C311yescharacteristic
399C3⋊C8C2×C3×D412yes—
400C2×C32—21no—
401C2×C32—82no—
402C2×C32—81no—
403C2×C32C22×D811yescharacteristic
404C3×D3—22no—
405C3×D3—81no—
406#41—21no—
407#41—31no—
408#41—62no—
409#41—62no—
410#41—61no—
411#41—121no—
412#41—122no—
413#41—121no—
414C2×Q8—62no—
415C2×Q8—61no—
416C2×Q8—121no—
417C2×D4—21no—
418C2×D4—62no—
419C2×D4—121no—
420C2×D4—122no—
421Q16—32no—
422Q16—31no—
423Q16—62no—
424Q16—61no—
425Q16C22×C3×D311yescharacteristic
426SD16—21no—
427SD16—62no—
428SD16—62no—
429SD16—61no—
430D8—21no—
431D8—31no—
432D8—62no—
433D8C22×C3×D311yescharacteristic
434D8C22×C3×D312yes—
435M16—31no—
436M16—32no—
437C2×C8—32no—
438C2×C8C22×C3×D311yescharacteristic
439C16—31no—
440C16—31no—
441C16C22×C3×D312yes—
442C22×C3—21no—
443C22×C3—32no—
444C22×C3—41no—
445C22×C3—42no—
446C22×C3—41no—
447C22×C3—42no—
448C22×C3—81no—
449C22×C3—82no—
450C22×C3—121no—
451C22×C3—161no—
452C22×C3—161no—
453C22×C3—162no—
454C22×C3—482no—
455C22×C3D3×D811yescharacteristic
456C22×C3C2×C3×D811yescharacteristic
457C2×D3—41no—
458C2×D3—162no—
459C2×D3C2×C3×D812yes—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—32no—
463C3×C4—41no—
464C3×C4—41no—
465C3×C4—81no—
466C3×C4—121no—
467C3×C4—122no—
468C3×C4D3×D811yescharacteristic
469C3×C4C2×C3×D811yescharacteristic
470C3×C4C2×D3×D411yescharacteristic
471C3×C4C22×C3×D411yescharacteristic
472Dic3—42no—
473Dic3—41no—
474Dic3C2×C3×D812yes—
475C32#341611yescharacteristic, complemented, Sylow
476C23—41no—
477C23—122no—
478Q8—21no—
479Q8—31no—
480Q8—61no—
481Q8—62no—
482Q8—122no—
483Q8—121no—
484D4—22no—
485D4—21no—
486D4—32no—
487D4—31no—
488D4—41no—
489D4—41no—
490D4—61no—
491D4—122no—
492D4—122no—
493D4—122no—
494D4—121no—
495C2×C4—32no—
496C2×C4—41no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—121no—
501C2×C4—121no—
502C2×C4C3×D3×D411yescharacteristic
503C8—32no—
504C8C3×D3×D411yescharacteristic
505C8C23×C3×D311yescharacteristic
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—41no—
510C2×C3—62no—
511C2×C3—82no—
512C2×C3—81no—
513C2×C3—82no—
514C2×C3—81no—
515C2×C3—162no—
516C2×C3—161no—
517C2×C3—241no—
518C2×C3C2×D3×D811yescharacteristic
519C2×C3C22×C3×D811yescharacteristic
520D3—22no—
521D3—81no—
522C22—32no—
523C22—41no—
524C22—42no—
525C22—121no—
526C22—161no—
527C22—482no—
528C22C3×D3×D811yescharacteristic
529C4—32no—
530C4—41no—
531C4—121no—
532C4—122no—
533C4C3×D3×D811yescharacteristic
534C4C2×C3×D3×D411yescharacteristic
535C3—21no—
536C3#8203711yescharacteristic, direct product factor
537C3C3×#341611yescharacteristic, complemented
538C2—21no—
539C2—62no—
540C2—82no—
541C2—81no—
542C2—241no—
543C2C2×C3×D3×D811yescharacteristic
subgroupG/H
centerC2×C3C2×D3×D8
commutator, cosocleC3×C8C24×C3
FrattiniC8C23×C3×D3
FittingC32×#509C2
socleC2×C32C22×D8
hypercenterC3×#509D3
hypocenterC3C3×#3416
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C3×#509

Complements

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

NH ≅ G/Ntype
1. C3×#5771C2semidirect
2. C3×#5757C2semidirect
3. C32×#509C2semidirect
4. C3×#5194C2semidirect
5. C2×C3×#827C2semidirect
6. C3×#4915C2semidirect
7. C3×D3×SD32C2semidirect
8. C3×#4912C2semidirect
9. C3×#4911C2semidirect
10. C3×#4909C2semidirect
11. C3×#4907C2semidirect
13. #82037C3direct (N × H)
16. C32×#135C22semidirect
20. C3×D3×Q16C22semidirect
23. C3×#912C22semidirect
26. C2×C3×Dic12C22semidirect
29. C32×SD32C22semidirect
30. C32×D16C22semidirect
31. C32×M32C22semidirect
32. C3×#829C22semidirect
33. C3×#828C22semidirect
34. C3×#827C22semidirect
35. C3×#827C22semidirect
36. C3×#827C22semidirect
37. C3×#826C22semidirect
38. C2×C3×C3⋊C16C22semidirect
39. C3×Dic24C22semidirect
40. C3×#800C22semidirect
41. C3×#798C22semidirect
42. C3×D3×C16C22semidirect
45. #5771C2×C3semidirect
46. #5757C2×C3semidirect
51. C3×#509D3semidirect
52. C3×#509C2×C3semidirect
58. #5194C2×C3semidirect
59. C2×#827C2×C3semidirect
60. #4915C2×C3semidirect
61. D3×SD32C2×C3semidirect
62. #4912C2×C3semidirect
63. #4911C2×C3semidirect
64. #4909C2×C3semidirect
65. #4907C2×C3semidirect
115. C3×#135C2×D3semidirect
116. C3×#135C22×C3semidirect
126. D3×Q16C22×C3semidirect
129. #912C22×C3semidirect
132. C2×Dic12C22×C3semidirect
143. C3×SD32C2×D3semidirect
144. C3×SD32C22×C3semidirect
147. C3×D16C2×D3semidirect
148. C3×D16C22×C3semidirect
151. C3×M32C2×D3semidirect
152. C3×M32C22×C3semidirect
155. #829C22×C3semidirect
156. #828C22×C3semidirect
157. #827C22×C3semidirect
158. #827C22×C3semidirect
159. #827C22×C3semidirect
160. #826C22×C3semidirect
161. C2×C3⋊C16C22×C3semidirect
162. Dic24C22×C3semidirect
163. #800C22×C3semidirect
164. #798C22×C3semidirect
165. D3×C16C22×C3semidirect
201. #509C3×D3semidirect
310. #135C2×C3×D3semidirect
323. SD32C2×C3×D3semidirect
325. D16C2×C3×D3semidirect
327. M32C2×C3×D3semidirect
475. C32#3416semidirect
536. C3#82037direct (N × H)
537. C3C3×#3416semidirect

Maximal subgroups

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

subgroupindex
C3×#57712
C3×#57572
C32×#5092
C3×#51942
C2×C3×#8272
C3×#49152
C3×D3×SD322
C3×#49122
C3×#49112
C3×#49092
C3×#49072
C3×#34163
#820373