Finite Group Databasebeta

← C3×#82024 #11899700

Subgroups — C3×#82024 #11899700

Order 1152.

765 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5766C211yesmaximal, characteristic, complemented
2C3×#5757C211yesmaximal, characteristic, complemented
3C2×C32×SD32C211yesmaximal, characteristic, complemented
4C3×#5179C211yesmaximal, characteristic, complemented
5C3×#5147C211yesmaximal, characteristic, complemented
6C3×#4916C212yesmaximal, complemented
7C3×#4915C212yesmaximal, complemented
8C3×#4914C212yesmaximal, complemented
9C3×D3×SD32C212yesmaximal, complemented
10C3×#4904C211yesmaximal, characteristic, complemented
11C3×#4901C211yesmaximal, characteristic, complemented
12C3×#3416—31nomaximal
13#82024C311yesmaximal, characteristic, direct product factor
14C3×#1007—21no—
15C3×#1004—21no—
16C2×C32×Q16C2211yescharacteristic, complemented
17C2×C32×D8C2211yescharacteristic
18C3×#942—21no—
19C3×#932—21no—
20C3×#919C2212yescomplemented
21C3×#918—22no—
22C3×D3×Q16C2212yescomplemented
23C3×#912C2212yes—
24C3×#911—22no—
25C3×D3×D8C2212yes—
26C3×#904C2211yescharacteristic
27C3×#904C2211yescharacteristic, complemented
28C3×#901C2211yescharacteristic
29C32×SD32C2212yescomplemented
30C32×SD32C2212yescomplemented
31C2×C32×C16C2211yescharacteristic, complemented
32C3×#829C2212yescomplemented
33C3×#828C2212yescomplemented
34C3×#827C2212yescomplemented
35C3×#826C2212yescomplemented
36C3×#812C2211yescharacteristic, complemented
37C3×Dic24C2211yescharacteristic, complemented
38C3×#800C2211yescharacteristic, complemented
39C3×#800C2211yescharacteristic, complemented
40C3×D48C2211yescharacteristic, complemented
41C3×#798C2212yescomplemented
42C3×D3×C16C2212yescomplemented
43C3×#578—31no—
44C3×#576—31no—
45#5766C2×C311yescharacteristic, complemented
46#5757C2×C311yescharacteristic, complemented
47C3×#510—31no—
48C3×#510—32no—
49C3×#509—31no—
50C3×#509—32no—
51C3×#508—32no—
52C3×#508—31no—
53C2×C3×SD32—21no—
54C2×C3×SD32—32no—
55C2×C3×SD32D311yescharacteristic, complemented
56C2×C3×SD32C2×C311yescharacteristic, complemented
57C3×#504—31no—
58#5179C2×C311yescharacteristic, complemented
59#5147C2×C311yescharacteristic, complemented
60#4916C2×C312yescomplemented
61#4915C2×C312yescomplemented
62#4914C2×C312yescomplemented
63D3×SD32C2×C312yescomplemented
64#4904C2×C311yescharacteristic, complemented
65#4901C2×C311yescharacteristic, complemented
66C2×C32×Q8—21no—
67C2×C32×D4—21no—
68C2×C3×#67—42no—
69C3×#239—22no—
70C3×#239—41no—
71C3×D3×Q8—22no—
72C3×D3×Q8—41no—
73C3×#237—22no—
74C3×#237—41no—
75C3×D3×D4—22no—
76C3×D3×D4—41no—
77C3×#235—21no—
78C3×#235—21no—
79C3×#235—42no—
80C3×#235D411yescharacteristic
81C32×Q16—21no—
82C32×Q16C2312yes—
83C32×D8—21no—
84C32×D8C2312yes—
85C2×C32×C8C2311yescharacteristic
86C3×#216—22no—
87C3×#215—22no—
88C3×#214—22no—
89C3×#213—22no—
90C3×#208D411yescharacteristic
91C3×Dic12C2311yescharacteristic
92C3×Dic12C2311yescharacteristic
93C3×D24C2311yescharacteristic
94C3×D24C2311yescharacteristic
95C3×#204—21no—
96C3×#204—21no—
97C3×ΓL(1,𝔽25)D412yes—
98C3×D3×C8C2312yes—
99C32×C16C2311yescharacteristic
100C32×C16C2311yescharacteristic
101C3×C3⋊C16C2312yes—
102#3416—31noSylow
103C3×2−1+4—61no—
104C3×H(5,𝔽2)—61no—
105#1007—21no—
106#1004—21no—
107C3×#137—62no—
108C3×#137—61no—
109C3×AGL(1,ℤ/8)—62no—
110C3×AGL(1,ℤ/8)—61no—
111C3×#135—32no—
112C3×#135—32no—
113C3×#135—31no—
114C3×#135—31no—
115C2×C3×Q16—21no—
116C2×C3×Q16—32no—
117C2×C3×Q16C2×D311yescharacteristic, complemented
118C2×C3×Q16C22×C311yescharacteristic, complemented
119C2×C3×D8—21no—
120C2×C3×D8—32no—
121C2×C3×D8C2×D311yescharacteristic
122C2×C3×D8C22×C311yescharacteristic
123C3×#131—31no—
124#942—21no—
125#932—21no—
126#919C22×C312yescomplemented
127#918—22no—
128D3×Q16C22×C312yescomplemented
129#912C22×C312yes—
130#911—22no—
131D3×D8C22×C312yes—
132#904C22×C311yescharacteristic
133#904C22×C311yescharacteristic, complemented
134#901C22×C311yescharacteristic
135C3×Q32—32no—
136C3×Q32—31no—
137C3×SD32—22no—
138C3×SD32—22no—
139C3×SD32—32no—
140C3×SD32—32no—
141C3×SD32—31no—
142C3×SD32—31no—
143C3×SD32C2×D312yescomplemented
144C3×SD32C2×D312yescomplemented
145C3×SD32C22×C312yescomplemented
146C3×SD32C22×C312yescomplemented
147C3×D16—32no—
148C3×D16—31no—
149C3×M32—31no—
150C3×M32—32no—
151C2×C3×C16—21no—
152C2×C3×C16—32no—
153C2×C3×C16C2×D311yescharacteristic, complemented
154C2×C3×C16C22×C311yescharacteristic, complemented
155#829C22×C312yescomplemented
156#828C22×C312yescomplemented
157#827C22×C312yescomplemented
158#826C22×C312yescomplemented
159#812C22×C311yescharacteristic, complemented
160Dic24C22×C311yescharacteristic, complemented
161#800C22×C311yescharacteristic, complemented
162#800C22×C311yescharacteristic, complemented
163D48C22×C311yescharacteristic, complemented
164#798C22×C312yescomplemented
165D3×C16C22×C312yescomplemented
166C23×C32—41no—
167C22×C3×D3—42no—
168C32×Q8—22no—
169C32×Q8—41no—
170C32×D4—22no—
171C32×D4—41no—
172C2×C32×C4—41no—
173C2×C32×C4C2×D411yescharacteristic
174C3×#67—41no—
175C3×#67—41no—
176C3×#67—42no—
177C3×#67—42no—
178C3×#67D812yes—
179C2×C3×Dic3—42no—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12—42no—
183C3×D12D811yescharacteristic
184C3×C4×D3—42no—
185C3×C4×D3—41no—
186C3×C4×D3—41no—
187C3×C4×D3—42no—
188C3×C4×D3C2×D412yes—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—42no—
192C3×Dic6D811yescharacteristic
193C32×C8C2×D411yescharacteristic
194C32×C8C2411yescharacteristic
195C3×C3⋊C8C2×D412yes—
196#578—31no—
197#576—31no—
198#510—31no—
199#510—32no—
200#509—31no—
201#509—32no—
202#508—32no—
203#508—31no—
204C2×SD32—32no—
205C2×SD32C3×D311yescharacteristic, complemented
206#504—31no—
207C3×#41—31no—
208C3×#41—62no—
209C3×#41—62no—
210C3×#41—61no—
211C3×#41—61no—
212C3×#41—122no—
213C3×#41—121no—
214C3×#41—121no—
215C2×C3×Q8—21no—
216C2×C3×Q8—21no—
217C2×C3×Q8—41no—
218C2×C3×Q8—62no—
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—
227#239—22no—
228#239—41no—
229D3×Q8—22no—
230D3×Q8—41no—
231#237—22no—
232#237—41no—
233D3×D4—22no—
234D3×D4—41no—
235#235—21no—
236#235—21no—
237#235—42no—
238#235C3×D411yescharacteristic
239C3×Q16—21no—
240C3×Q16—21no—
241C3×Q16—22no—
242C3×Q16—31no—
243C3×Q16—31no—
244C3×Q16—41no—
245C3×Q16—62no—
246C3×Q16C22×D312yes—
247C3×Q16C23×C312yes—
248C3×SD16—62no—
249C3×SD16—62no—
250C3×SD16—61no—
251C3×SD16—61no—
252C3×D8—21no—
253C3×D8—21no—
254C3×D8—22no—
255C3×D8—31no—
256C3×D8—31no—
257C3×D8—41no—
258C3×D8—62no—
259C3×D8C22×D312yes—
260C3×D8C23×C312yes—
261C3×M16—31no—
262C3×M16—32no—
263C2×C3×C8—21no—
264C2×C3×C8—32no—
265C2×C3×C8C22×D311yescharacteristic
266C2×C3×C8C23×C311yescharacteristic
267#216—22no—
268#215—22no—
269#214—22no—
270#213—22no—
271#208C3×D411yescharacteristic
272Dic12C23×C311yescharacteristic
273Dic12C23×C311yescharacteristic
274D24C23×C311yescharacteristic
275D24C23×C311yescharacteristic
276#204—21no—
277#204—21no—
278ΓL(1,𝔽25)C3×D412yes—
279D3×C8C23×C312yes—
280C3×C16—21no—
281C3×C16—21no—
282C3×C16—32no—
283C3×C16C22×D311yescharacteristic
284C3×C16C22×D311yescharacteristic
285C3×C16C23×C311yescharacteristic
286C3×C16C23×C311yescharacteristic
287C3⋊C16C23×C312yes—
288C22×C32—42no—
289C22×C32—161no—
290C22×C32C2×D811yescharacteristic
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C2×C3×D3—162no—
294C2×C3×D3C2×D812yes—
295C32×C4—42no—
296C32×C4C2×D811yescharacteristic
297C32×C4C22×D411yescharacteristic
298C3×Dic3—41no—
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#135—31no—
311C2×Q16—32no—
312C2×Q16C2×C3×D311yescharacteristic, complemented
313C2×D8—32no—
314C2×D8C2×C3×D311yescharacteristic
315#131—31no—
316Q32—32no—
317Q32—31no—
318SD32—32no—
319SD32—32no—
320SD32—31no—
321SD32—31no—
322SD32C2×C3×D312yescomplemented
323SD32C2×C3×D312yescomplemented
324D16—32no—
325D16—31no—
326M32—31no—
327M32—32no—
328C2×C16—32no—
329C2×C16C2×C3×D311yescharacteristic, complemented
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—81no—
333C23×C3—122no—
334C22×D3—42no—
335C3×Q8—22no—
336C3×Q8—22no—
337C3×Q8—31no—
338C3×Q8—42no—
339C3×Q8—41no—
340C3×Q8—41no—
341C3×Q8—61no—
342C3×Q8—61no—
343C3×Q8—81no—
344C3×Q8—122no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—32no—
348C3×D4—31no—
349C3×D4—42no—
350C3×D4—41no—
351C3×D4—41no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—81no—
355C3×D4—122no—
356C3×D4—122no—
357C3×D4—122no—
358C3×D4—121no—
359C3×D4—121no—
360C2×C3×C4—21no—
361C2×C3×C4—32no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—81no—
365C2×C3×C4—122no—
366C2×C3×C4—122no—
367C2×C3×C4—122no—
368C2×C3×C4—121no—
369C2×C3×C4—121no—
370C2×C3×C4D3×D411yescharacteristic
371C2×C3×C4C2×C3×D411yescharacteristic
372#67—41no—
373#67—41no—
374#67—42no—
375#67—42no—
376#67C3×D812yes—
377C2×Dic3—42no—
378D12—21no—
379D12—21no—
380D12—42no—
381D12C3×D811yescharacteristic
382C4×D3—42no—
383C4×D3—41no—
384C4×D3—41no—
385C4×D3—42no—
386C4×D3C2×C3×D412yes—
387Dic6—21no—
388Dic6—21no—
389Dic6—42no—
390Dic6C3×D811yescharacteristic
391C3×C8—21no—
392C3×C8—21no—
393C3×C8—32no—
394C3×C8D3×D411yescharacteristic
395C3×C8C2×C3×D411yescharacteristic
396C3×C8C23×D311yescharacteristic
397C3×C8C24×C311yescharacteristic
398C3⋊C8C2×C3×D412yes—
399C2×C32—21no—
400C2×C32—82no—
401C2×C32C22×D811yescharacteristic
402C3×D3—22no—
403C3×D3—81no—
404C3×D3—81no—
405#41—31no—
406#41—62no—
407#41—62no—
408#41—61no—
409#41—61no—
410#41—122no—
411#41—121no—
412#41—121no—
413C2×Q8—21no—
414C2×Q8—62no—
415C2×Q8—121no—
416C2×D4—21no—
417C2×D4—62no—
418C2×D4—121no—
419C2×D4—122no—
420Q16—21no—
421Q16—31no—
422Q16—31no—
423Q16—62no—
424Q16C22×C3×D312yes—
425SD16—62no—
426SD16—62no—
427SD16—61no—
428SD16—61no—
429D8—21no—
430D8—31no—
431D8—31no—
432D8—62no—
433D8C22×C3×D312yes—
434M16—31no—
435M16—32no—
436C2×C8—32no—
437C2×C8C22×C3×D311yescharacteristic
438C16—32no—
439C16C22×C3×D311yescharacteristic
440C16C22×C3×D311yescharacteristic
441C22×C3—21no—
442C22×C3—32no—
443C22×C3—42no—
444C22×C3—42no—
445C22×C3—82no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—161no—
449C22×C3—161no—
450C22×C3—162no—
451C22×C3—482no—
452C22×C3D3×D811yescharacteristic
453C22×C3C2×C3×D811yescharacteristic
454C2×D3—41no—
455C2×D3—41no—
456C2×D3—162no—
457C2×D3C2×C3×D812yes—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—32no—
461C3×C4—42no—
462C3×C4—42no—
463C3×C4—82no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4D3×D811yescharacteristic
467C3×C4C2×C3×D811yescharacteristic
468C3×C4C2×D3×D411yescharacteristic
469C3×C4C22×C3×D411yescharacteristic
470Dic3—41no—
471Dic3—41no—
472Dic3C2×C3×D812yes—
473C32#341611yescharacteristic, complemented, Sylow
474C23—41no—
475C23—122no—
476Q8—22no—
477Q8—31no—
478Q8—41no—
479Q8—61no—
480Q8—61no—
481Q8—122no—
482D4—22no—
483D4—32no—
484D4—31no—
485D4—41no—
486D4—61no—
487D4—61no—
488D4—122no—
489D4—122no—
490D4—122no—
491D4—121no—
492D4—121no—
493C2×C4—32no—
494C2×C4—41no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4C3×D3×D411yescharacteristic
501C8—32no—
502C8C3×D3×D411yescharacteristic
503C8C23×C3×D311yescharacteristic
504C2×C3—22no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—62no—
509C2×C3—82no—
510C2×C3—82no—
511C2×C3—162no—
512C2×C3—241no—
513C2×C3—241no—
514C2×C3C2×D3×D811yescharacteristic
515C2×C3C22×C3×D811yescharacteristic
516D3—22no—
517D3—81no—
518D3—81no—
519C22—32no—
520C22—42no—
521C22—121no—
522C22—121no—
523C22—161no—
524C22—482no—
525C22C3×D3×D811yescharacteristic
526C4—32no—
527C4—42no—
528C4—121no—
529C4—121no—
530C4C3×D3×D811yescharacteristic
531C4C2×C3×D3×D411yescharacteristic
532C3—21no—
533C3#8202411yescharacteristic, direct product factor
534C3C3×#341611yescharacteristic, complemented
535C2—21no—
536C2—62no—
537C2—82no—
538C2—241no—
539C2—241no—
540C2C2×C3×D3×D811yescharacteristic
subgroupG/H
centerC2×C3C2×D3×D8
commutator, cosocleC3×C8C24×C3
FrattiniC8C23×C3×D3
FittingC2×C32×SD32C2
socleC2×C32C22×D8
hypercenterC2×C3×SD32D3
hypocenterC3C3×#3416
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×SD32

Complements

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

NH ≅ G/Ntype
1. C3×#5766C2semidirect
2. C3×#5757C2semidirect
3. C2×C32×SD32C2semidirect
4. C3×#5179C2semidirect
5. C3×#5147C2semidirect
6. C3×#4916C2semidirect
7. C3×#4915C2semidirect
8. C3×#4914C2semidirect
9. C3×D3×SD32C2semidirect
10. C3×#4904C2semidirect
11. C3×#4901C2semidirect
13. #82024C3direct (N × H)
16. C2×C32×Q16C22semidirect
20. C3×#919C22semidirect
22. C3×D3×Q16C22semidirect
27. C3×#904C22semidirect
29. C32×SD32C22semidirect
30. C32×SD32C22semidirect
31. C2×C32×C16C22semidirect
32. C3×#829C22semidirect
33. C3×#828C22semidirect
34. C3×#827C22semidirect
35. C3×#826C22semidirect
36. C3×#812C22semidirect
37. C3×Dic24C22semidirect
38. C3×#800C22semidirect
39. C3×#800C22semidirect
40. C3×D48C22semidirect
41. C3×#798C22semidirect
42. C3×D3×C16C22semidirect
45. #5766C2×C3semidirect
46. #5757C2×C3semidirect
55. C2×C3×SD32D3semidirect
56. C2×C3×SD32C2×C3semidirect
58. #5179C2×C3semidirect
59. #5147C2×C3semidirect
60. #4916C2×C3semidirect
61. #4915C2×C3semidirect
62. #4914C2×C3semidirect
63. D3×SD32C2×C3semidirect
64. #4904C2×C3semidirect
65. #4901C2×C3semidirect
117. C2×C3×Q16C2×D3semidirect
118. C2×C3×Q16C22×C3semidirect
126. #919C22×C3semidirect
128. D3×Q16C22×C3semidirect
133. #904C22×C3semidirect
143. C3×SD32C2×D3semidirect
144. C3×SD32C2×D3semidirect
145. C3×SD32C22×C3semidirect
146. C3×SD32C22×C3semidirect
153. C2×C3×C16C2×D3semidirect
154. C2×C3×C16C22×C3semidirect
155. #829C22×C3semidirect
156. #828C22×C3semidirect
157. #827C22×C3semidirect
158. #826C22×C3semidirect
159. #812C22×C3semidirect
160. Dic24C22×C3semidirect
161. #800C22×C3semidirect
162. #800C22×C3semidirect
163. D48C22×C3semidirect
164. #798C22×C3semidirect
165. D3×C16C22×C3semidirect
205. C2×SD32C3×D3semidirect
312. C2×Q16C2×C3×D3semidirect
322. SD32C2×C3×D3semidirect
323. SD32C2×C3×D3semidirect
329. C2×C16C2×C3×D3semidirect
473. C32#3416semidirect
533. C3#82024direct (N × H)
534. C3C3×#3416semidirect

Maximal subgroups

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

subgroupindex
C3×#57662
C3×#57572
C2×C32×SD322
C3×#51792
C3×#51472
C3×#49162
C3×#49152
C3×#49142
C3×D3×SD322
C3×#49042
C3×#49012
C3×#34163
#820243