Finite Group Databasebeta

← C3×#3264 #85128

Subgroups — C3×#3264 #85128

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2C22×C3×D8C211yesmaximal, characteristic, complemented
3C3×#546C211yesmaximal, characteristic, complemented
4C3×D42C211yesmaximal, characteristic, complemented
5C3×#496C211yesmaximal, characteristic, complemented
6C3×#469C211yesmaximal, characteristic, complemented
7C3×#465C211yesmaximal, characteristic, complemented
8C3×#463C211yesmaximal, characteristic, complemented
9C3×#459C211yesmaximal, characteristic, complemented
10C3×#450C211yesmaximal, characteristic, complemented
11C3×#449C211yesmaximal, characteristic, complemented
12C3×#447C211yesmaximal, characteristic, complemented
13C3×#447C211yesmaximal, characteristic, complemented
14C3×#442C211yesmaximal, characteristic, complemented
15C3×#435C211yesmaximal, characteristic, complemented
16#3264C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×#41C2211yescharacteristic, complemented
18C22×C3×D4—21no—
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic, complemented
21C22×C3×D4C2211yescharacteristic, complemented
22C3×AGL(1,ℤ/8)—24no—
23C2×C3×SD16C2211yescharacteristic, complemented
24C2×C3×SD16C2211yescharacteristic, complemented
25C2×C3×SD16C2211yescharacteristic, complemented
26C2×C3×D8—24no—
27C2×C3×D8C2211yescharacteristic
28C2×C3×D8C2211yescharacteristic
29C2×C3×D8C2211yescharacteristic, complemented
30C2×C3×D8C2211yescharacteristic, complemented
31C2×C3×D8C2211yescharacteristic, complemented
32C2×C3×D8C2211yescharacteristic, complemented
33C2×C3×M16C2211yescharacteristic, complemented
34C22×C3×C8C2211yescharacteristic, complemented
35C3×#127C2211yescharacteristic, complemented
36C3×#124C2211yescharacteristic, complemented
37C3×#123—21no—
38C3×C22⋊Q8C2211yescharacteristic, complemented
39C3×#121—21no—
40C3×#121—21no—
41C3×#121C2211yescharacteristic
42C3×#121C2211yescharacteristic, complemented
43C3×#121C2211yescharacteristic, complemented
44C3×#120—21no—
45C3×#120—21no—
46C3×#120—21no—
47C3×C4×D4C2211yescharacteristic
48C3×C4×D4C2211yescharacteristic, complemented
49C3×C4×D4C2211yescharacteristic, complemented
50C2×C3×C22⋊C4—21no—
51C3×#106C2211yescharacteristic, complemented
52C3×C4⋊C8C2211yescharacteristic, complemented
53C3×#103C2211yescharacteristic, complemented
54C3×#102C2211yescharacteristic, complemented
55C3×#102C2211yescharacteristic, complemented
56C3×#102C2211yescharacteristic, complemented
57C3×#102C2211yescharacteristic, complemented
58C3×#102C2211yescharacteristic, complemented
59C3×C22⋊C8C2211yescharacteristic, complemented
60C3×C22⋊C8C2211yescharacteristic, complemented
61C3×#97C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)C2×C311yescharacteristic, complemented
63C22×D8C2×C311yescharacteristic, complemented
64#546C2×C311yescharacteristic, complemented
65D42C2×C311yescharacteristic, complemented
66#496C2×C311yescharacteristic, complemented
67#469C2×C311yescharacteristic, complemented
68#465C2×C311yescharacteristic, complemented
69#463C2×C311yescharacteristic, complemented
70#459C2×C311yescharacteristic, complemented
71#450C2×C311yescharacteristic, complemented
72#449C2×C311yescharacteristic, complemented
73#447C2×C311yescharacteristic, complemented
74#447C2×C311yescharacteristic, complemented
75#442C2×C311yescharacteristic, complemented
76#435C2×C311yescharacteristic, complemented
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—21no—
80C3×#41—22no—
81C3×#41—41no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—21no—
87C2×C3×D4—21no—
88C2×C3×D4—21no—
89C2×C3×D4—21no—
90C2×C3×D4—21no—
91C2×C3×D4—21no—
92C2×C3×D4—21no—
93C2×C3×D4—21no—
94C2×C3×D4—21no—
95C2×C3×D4—41no—
96C2×C3×D4—41no—
97C2×C3×D4—41no—
98C2×C3×D4—44no—
99C2×C3×D4D411yescharacteristic
100C2×C3×D4C2311yescharacteristic
101C2×C3×D4C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2311yescharacteristic
104C2×C3×D4C2311yescharacteristic
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2311yescharacteristic
110C3×SD16—22no—
111C3×SD16—22no—
112C3×SD16—41no—
113C3×D8—22no—
114C3×D8—22no—
115C3×D8—22no—
116C3×D8—22no—
117C3×D8—41no—
118C3×D8D414yescomplemented
119C3×M16—22no—
120C2×C3×C8—22no—
121C2×C3×C8C2311yescharacteristic
122C2×C3×C8C2311yescharacteristic
123C2×C3×C8C2311yescharacteristic
124C2×C3×C8C2311yescharacteristic
125C3×C4⋊C4—21no—
126C3×C4⋊C4D411yescharacteristic
127C3×C4⋊C4C2311yescharacteristic
128C3×C4⋊C4C2311yescharacteristic
129C3×C22⋊C4—21no—
130C3×C22⋊C4—21no—
131C3×C22⋊C4—21no—
132C3×C22⋊C4—21no—
133C3×C22⋊C4—21no—
134C3×C22⋊C4—21no—
135C3×C22⋊C4—21no—
136C3×C22⋊C4—21no—
137C3×C22⋊C4D411yescharacteristic
138C3×C22⋊C4D411yescharacteristic
139C3×C42C2311yescharacteristic
140C2×#41C22×C311yescharacteristic, complemented
141C22×D4—21no—
142C22×D4C22×C311yescharacteristic
143C22×D4C22×C311yescharacteristic, complemented
144C22×D4C22×C311yescharacteristic, complemented
145AGL(1,ℤ/8)—24no—
146C2×SD16C22×C311yescharacteristic, complemented
147C2×SD16C22×C311yescharacteristic, complemented
148C2×SD16C22×C311yescharacteristic, complemented
149C2×D8—24no—
150C2×D8C22×C311yescharacteristic
151C2×D8C22×C311yescharacteristic
152C2×D8C22×C311yescharacteristic, complemented
153C2×D8C22×C311yescharacteristic, complemented
154C2×D8C22×C311yescharacteristic, complemented
155C2×D8C22×C311yescharacteristic, complemented
156C2×M16C22×C311yescharacteristic, complemented
157C22×C8C22×C311yescharacteristic, complemented
158#127C22×C311yescharacteristic, complemented
159#124C22×C311yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C22×C311yescharacteristic, complemented
162#121—21no—
163#121—21no—
164#121C22×C311yescharacteristic
165#121C22×C311yescharacteristic, complemented
166#121C22×C311yescharacteristic, complemented
167#120—21no—
168#120—21no—
169#120—21no—
170C4×D4C22×C311yescharacteristic
171C4×D4C22×C311yescharacteristic, complemented
172C4×D4C22×C311yescharacteristic, complemented
173C2×C22⋊C4—21no—
174#106C22×C311yescharacteristic, complemented
175C4⋊C8C22×C311yescharacteristic, complemented
176#103C22×C311yescharacteristic, complemented
177#102C22×C311yescharacteristic, complemented
178#102C22×C311yescharacteristic, complemented
179#102C22×C311yescharacteristic, complemented
180#102C22×C311yescharacteristic, complemented
181#102C22×C311yescharacteristic, complemented
182C22⋊C8C22×C311yescharacteristic, complemented
183C22⋊C8C22×C311yescharacteristic, complemented
184#97C22×C311yescharacteristic, complemented
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—42no—
191C23×C3—42no—
192C23×C3—42no—
193C23×C3—42no—
194C23×C3—42no—
195C23×C3—42no—
196C23×C3—81no—
197C23×C3—81no—
198C23×C3—81no—
199C23×C3C2×D411yescharacteristic
200C23×C3C2×D411yescharacteristic
201C3×Q8—21no—
202C3×Q8—21no—
203C3×D4—22no—
204C3×D4—21no—
205C3×D4—21no—
206C3×D4—21no—
207C3×D4—21no—
208C3×D4—21no—
209C3×D4—21no—
210C3×D4—21no—
211C3×D4—21no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—42no—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—42no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4C2×D412yes—
222C3×D4C2×D412yescomplemented
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—81no—
233C2×C3×C4C2×D411yescharacteristic
234C2×C3×C4C2×D411yescharacteristic
235C2×C3×C4C2×D411yescharacteristic
236C2×C3×C4C2×D411yescharacteristic
237C2×C3×C4C2411yescharacteristic
238C3×C8—21no—
239C3×C8—21no—
240C3×C8—21no—
241C3×C8C2×D412yescomplemented
242C24—21no—
243C24—21no—
244C24—21no—
245#41—22no—
246#41—41no—
247C2×Q8C23×C311yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—44no—
264C2×D4C3×D411yescharacteristic
265C2×D4C23×C311yescharacteristic
266C2×D4C23×C311yescharacteristic
267C2×D4C23×C311yescharacteristic
268C2×D4C23×C311yescharacteristic
269C2×D4C23×C311yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4C23×C311yescharacteristic
274C22×C4C23×C311yescharacteristic
275SD16—22no—
276SD16—22no—
277SD16—41no—
278D8—22no—
279D8—22no—
280D8—22no—
281D8—22no—
282D8—41no—
283D8C3×D414yescomplemented
284M16—22no—
285C2×C8—22no—
286C2×C8C23×C311yescharacteristic
287C2×C8C23×C311yescharacteristic
288C2×C8C23×C311yescharacteristic
289C2×C8C23×C311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4C3×D411yescharacteristic
292C4⋊C4C23×C311yescharacteristic
293C4⋊C4C23×C311yescharacteristic
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4C3×D411yescharacteristic
303C22⋊C4C3×D411yescharacteristic
304C42C23×C311yescharacteristic
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—84no—
326C22×C3—84no—
327C22×C3—84no—
328C22×C3AGL(1,ℤ/8)12yescomplemented
329C22×C3C22×D411yescharacteristic
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337C3×C4C22×D411yescharacteristic
338C3×C4H(5,𝔽2)11yescharacteristic
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23C2×C3×D411yescharacteristic
354C23C2×C3×D411yescharacteristic
355Q8—21no—
356Q8—21no—
357D4—22no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—81no—
374D4—81no—
375D4C2×C3×D412yes—
376D4C2×C3×D412yescomplemented
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—81no—
387C2×C4C2×C3×D411yescharacteristic
388C2×C4C2×C3×D411yescharacteristic
389C2×C4C2×C3×D411yescharacteristic
390C2×C4C2×C3×D411yescharacteristic
391C2×C4C24×C311yescharacteristic
392C8—21no—
393C8—21no—
394C8—21no—
395C8C2×C3×D412yescomplemented
396C2×C3—22no—
397C2×C3—41no—
398C2×C3—42no—
399C2×C3—42no—
400C2×C3—81no—
401C2×C3—81no—
402C2×C3—81no—
403C2×C3D4211yescharacteristic
404C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
405C2×C3#57611yescharacteristic
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—22no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—84no—
427C22—84no—
428C22—84no—
429C22C3×AGL(1,ℤ/8)12yescomplemented
430C22C22×C3×D411yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4C22×C3×D411yescharacteristic
439C4C3×H(5,𝔽2)11yescharacteristic
440C3#326411yescharacteristic, direct product factor, Sylow
441C2—22no—
442C2—41no—
443C2—42no—
444C2—42no—
445C2—81no—
446C2—81no—
447C2—81no—
448C2C3×D4211yescharacteristic
449C2C2×C3×AGL(1,ℤ/8)11yescharacteristic
450C2C3×#57611yescharacteristic
subgroupG/H
center, socleC22×C3C22×D4
commutator, Frattini, cosocleC2×C4C24×C3
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×AGL(1,ℤ/8)C2semidirect
2. C22×C3×D8C2semidirect
3. C3×#546C2semidirect
4. C3×D42C2semidirect
5. C3×#496C2semidirect
6. C3×#469C2semidirect
7. C3×#465C2semidirect
8. C3×#463C2semidirect
9. C3×#459C2semidirect
10. C3×#450C2semidirect
11. C3×#449C2semidirect
12. C3×#447C2semidirect
13. C3×#447C2semidirect
14. C3×#442C2semidirect
15. C3×#435C2semidirect
16. #3264C3direct (N × H)
17. C2×C3×#41C22semidirect
20. C22×C3×D4C22semidirect
21. C22×C3×D4C22semidirect
23. C2×C3×SD16C22semidirect
24. C2×C3×SD16C22semidirect
25. C2×C3×SD16C22semidirect
29. C2×C3×D8C22semidirect
30. C2×C3×D8C22semidirect
31. C2×C3×D8C22semidirect
32. C2×C3×D8C22semidirect
33. C2×C3×M16C22semidirect
34. C22×C3×C8C22semidirect
35. C3×#127C22semidirect
36. C3×#124C22semidirect
38. C3×C22⋊Q8C22semidirect
42. C3×#121C22semidirect
43. C3×#121C22semidirect
48. C3×C4×D4C22semidirect
49. C3×C4×D4C22semidirect
51. C3×#106C22semidirect
52. C3×C4⋊C8C22semidirect
53. C3×#103C22semidirect
54. C3×#102C22semidirect
55. C3×#102C22semidirect
56. C3×#102C22semidirect
57. C3×#102C22semidirect
58. C3×#102C22semidirect
59. C3×C22⋊C8C22semidirect
60. C3×C22⋊C8C22semidirect
61. C3×#97C22semidirect
62. C2×AGL(1,ℤ/8)C2×C3semidirect
63. C22×D8C2×C3semidirect
64. #546C2×C3semidirect
65. D42C2×C3semidirect
66. #496C2×C3semidirect
67. #469C2×C3semidirect
68. #465C2×C3semidirect
69. #463C2×C3semidirect
70. #459C2×C3semidirect
71. #450C2×C3semidirect
72. #449C2×C3semidirect
73. #447C2×C3semidirect
74. #447C2×C3semidirect
75. #442C2×C3semidirect
76. #435C2×C3semidirect
118. C3×D8D4semidirect
140. C2×#41C22×C3semidirect
143. C22×D4C22×C3semidirect
144. C22×D4C22×C3semidirect
146. C2×SD16C22×C3semidirect
147. C2×SD16C22×C3semidirect
148. C2×SD16C22×C3semidirect
152. C2×D8C22×C3semidirect
153. C2×D8C22×C3semidirect
154. C2×D8C22×C3semidirect
155. C2×D8C22×C3semidirect
156. C2×M16C22×C3semidirect
157. C22×C8C22×C3semidirect
158. #127C22×C3semidirect
159. #124C22×C3semidirect
161. C22⋊Q8C22×C3semidirect
165. #121C22×C3semidirect
166. #121C22×C3semidirect
171. C4×D4C22×C3semidirect
172. C4×D4C22×C3semidirect
174. #106C22×C3semidirect
175. C4⋊C8C22×C3semidirect
176. #103C22×C3semidirect
177. #102C22×C3semidirect
178. #102C22×C3semidirect
179. #102C22×C3semidirect
180. #102C22×C3semidirect
181. #102C22×C3semidirect
182. C22⋊C8C22×C3semidirect
183. C22⋊C8C22×C3semidirect
184. #97C22×C3semidirect
222. C3×D4C2×D4semidirect
241. C3×C8C2×D4semidirect
283. D8C3×D4semidirect
328. C22×C3AGL(1,ℤ/8)semidirect
376. D4C2×C3×D4semidirect
395. C8C2×C3×D4semidirect
429. C22C3×AGL(1,ℤ/8)semidirect
440. C3#3264direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×AGL(1,ℤ/8)2
C22×C3×D82
C3×#5462
C3×D422
C3×#4962
C3×#4692
C3×#4652
C3×#4632
C3×#4592
C3×#4502
C3×#4492
C3×#4472
C3×#4422
C3×#4352
#32643