Subgroups — C₃×#3325 #85189

Order 384.

460 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×D₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#482	C₂	1	2	yes	maximal, complemented
5	C₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#469	C₂	1	2	yes	maximal, complemented
7	C₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#447	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#418	C₂	1	2	yes	maximal, complemented
12	#3325	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×D₄	—	2	1	no	—
14	C₂²×C₃×D₄	C₂²	1	2	yes	—
15	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₈	C₂²	1	2	yes	complemented
17	C₂×C₃×M₁₆	C₂²	1	2	yes	complemented
18	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₃×#127	C₂²	1	1	yes	characteristic
20	C₃×#124	—	2	2	no	—
21	C₃×#121	—	2	1	no	—
22	C₃×#121	—	2	1	no	—
23	C₃×#121	C₂²	1	2	yes	—
24	C₃×#121	C₂²	1	2	yes	complemented
25	C₃×#120	—	2	2	no	—
26	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
27	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
28	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
29	C₃×#117	—	2	1	no	—
30	C₃×#117	C₂²	1	2	yes	complemented
31	C₃×#107	C₂²	1	1	yes	characteristic, complemented
32	C₃×#106	C₂²	1	2	yes	complemented
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
34	C₃×#102	C₂²	1	1	yes	characteristic
35	C₃×#102	C₂²	1	2	yes	—
36	C₃×#102	C₂²	1	2	yes	complemented
37	C₃×#102	C₂²	1	2	yes	complemented
38	C₃×#102	C₂²	1	1	yes	characteristic, complemented
39	C₃×#102	C₂²	1	2	yes	complemented
40	C₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic, complemented
42	#555	C₂×C₃	1	1	yes	characteristic, complemented
43	D₄²	C₂×C₃	1	1	yes	characteristic, complemented
44	#486	C₂×C₃	1	1	yes	characteristic, complemented
45	#482	C₂×C₃	1	2	yes	complemented
46	#476	C₂×C₃	1	1	yes	characteristic, complemented
47	#469	C₂×C₃	1	2	yes	complemented
48	#459	C₂×C₃	1	1	yes	characteristic, complemented
49	#447	C₂×C₃	1	2	yes	complemented
50	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
51	#436	C₂×C₃	1	1	yes	characteristic, complemented
52	#418	C₂×C₃	1	2	yes	complemented
53	C₂⁴×C₃	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	1	no	—
55	C₂×C₃×D₄	—	2	4	no	—
56	C₂×C₃×D₄	—	2	2	no	—
57	C₂×C₃×D₄	—	2	2	no	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	1	no	—
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂×C₃×D₄	—	4	4	no	—
63	C₂×C₃×D₄	D₄	1	1	yes	characteristic
64	C₂×C₃×D₄	C₂³	1	2	yes	—
65	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
66	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	—	2	1	no	—
68	C₂²×C₃×C₄	—	2	1	no	—
69	C₂²×C₃×C₄	C₂³	1	2	yes	—
70	C₃×D₈	—	2	2	no	—
71	C₃×D₈	—	4	2	no	—
72	C₃×M₁₆	—	4	2	no	—
73	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂³	1	2	yes	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
78	C₃×C₄⋊C₄	C₂³	1	2	yes	—
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₂²⋊C₄	—	2	1	no	—
84	C₃×C₂²⋊C₄	—	2	1	no	—
85	C₃×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₂²⋊C₄	D₄	1	2	yes	—
87	C₃×C₄²	—	2	2	no	—
88	C₃×C₄²	C₂³	1	1	yes	characteristic
89	C₂²×D₄	—	2	1	no	—
90	C₂²×D₄	C₂²×C₃	1	2	yes	—
91	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
92	C₂×D₈	C₂²×C₃	1	2	yes	complemented
93	C₂×M₁₆	C₂²×C₃	1	2	yes	complemented
94	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
95	#127	C₂²×C₃	1	1	yes	characteristic
96	#124	—	2	2	no	—
97	#121	—	2	1	no	—
98	#121	—	2	1	no	—
99	#121	C₂²×C₃	1	2	yes	—
100	#121	C₂²×C₃	1	2	yes	complemented
101	#120	—	2	2	no	—
102	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
103	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
104	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
105	#117	—	2	1	no	—
106	#117	C₂²×C₃	1	2	yes	complemented
107	#107	C₂²×C₃	1	1	yes	characteristic, complemented
108	#106	C₂²×C₃	1	2	yes	complemented
109	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic, complemented
110	#102	C₂²×C₃	1	1	yes	characteristic
111	#102	C₂²×C₃	1	2	yes	—
112	#102	C₂²×C₃	1	2	yes	complemented
113	#102	C₂²×C₃	1	2	yes	complemented
114	#102	C₂²×C₃	1	1	yes	characteristic, complemented
115	#102	C₂²×C₃	1	2	yes	complemented
116	C₂²⋊C₈	C₂²×C₃	1	2	yes	complemented
117	C₄×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	—	2	2	no	—
121	C₂³×C₃	—	4	4	no	—
122	C₂³×C₃	—	4	4	no	—
123	C₂³×C₃	—	8	2	no	—
124	C₂³×C₃	C₂×D₄	1	2	yes	—
125	C₃×Q₈	—	8	1	no	—
126	C₃×D₄	—	2	2	no	—
127	C₃×D₄	—	2	1	no	—
128	C₃×D₄	—	2	2	no	—
129	C₃×D₄	—	2	1	no	—
130	C₃×D₄	—	2	1	no	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	4	4	no	—
133	C₃×D₄	—	4	4	no	—
134	C₃×D₄	—	4	2	no	—
135	C₃×D₄	—	8	1	no	—
136	C₃×D₄	—	8	2	no	—
137	C₃×D₄	#41	1	2	yes	—
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
149	C₃×C₈	—	2	1	no	—
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	2	2	no	—
152	C₂⁴	—	2	2	no	—
153	C₂×Q₈	—	2	1	no	—
154	C₂×D₄	—	2	4	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	1	no	—
159	C₂×D₄	—	2	1	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	4	no	—
162	C₂×D₄	C₃×D₄	1	1	yes	characteristic
163	C₂×D₄	C₂³×C₃	1	2	yes	—
164	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
165	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	C₂³×C₃	1	2	yes	—
169	D₈	—	2	2	no	—
170	D₈	—	4	2	no	—
171	M₁₆	—	4	2	no	—
172	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
174	C₂×C₈	C₂³×C₃	1	2	yes	—
175	C₄⋊C₄	—	2	1	no	—
176	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
177	C₄⋊C₄	C₂³×C₃	1	2	yes	—
178	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	1	no	—
183	C₂²⋊C₄	—	2	1	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	C₃×D₄	1	2	yes	—
186	C₄²	—	2	2	no	—
187	C₄²	C₂³×C₃	1	1	yes	characteristic
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	4	1	no	—
193	C₂²×C₃	—	4	1	no	—
194	C₂²×C₃	—	4	2	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	2	no	—
200	C₂²×C₃	—	8	8	no	—
201	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	4	1	no	—
205	C₃×C₄	—	4	1	no	—
206	C₃×C₄	—	4	2	no	—
207	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
208	C₃×C₄	C₂×#41	1	1	yes	characteristic
209	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	2	no	—
216	C₂³	C₂×C₃×D₄	1	2	yes	—
217	Q₈	—	8	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	2	1	no	—
223	D₄	—	2	2	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	D₄	—	8	1	no	—
228	D₄	—	8	2	no	—
229	D₄	C₃×#41	1	2	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
238	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	2	no	—
244	C₂×C₃	—	4	2	no	—
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	—	8	1	no	—
247	C₂×C₃	—	8	2	no	—
248	C₂×C₃	#546	1	1	yes	characteristic
249	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₂×C₃	#576	1	1	yes	characteristic
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	8	8	no	—
264	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	1	no	—
268	C₄	—	4	1	no	—
269	C₄	—	4	2	no	—
270	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
271	C₄	C₂×C₃×#41	1	1	yes	characteristic
272	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
273	C₃	#3325	1	1	yes	characteristic, direct product factor, Sylow
274	C₂	—	4	2	no	—
275	C₂	—	4	2	no	—
276	C₂	—	8	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₃×#546	1	1	yes	characteristic
279	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
280	C₂	C₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#555	C₂	semidirect
2. C₃×D₄²	C₂	semidirect
3. C₃×#486	C₂	semidirect
4. C₃×#482	C₂	semidirect
5. C₃×#476	C₂	semidirect
6. C₃×#469	C₂	semidirect
7. C₃×#459	C₂	semidirect
8. C₃×#447	C₂	semidirect
9. C₃×C₄×D₈	C₂	semidirect
10. C₃×#436	C₂	semidirect
11. C₃×#418	C₂	semidirect
12. #3325	C₃	direct (N × H)
16. C₂×C₃×D₈	C₂²	semidirect
17. C₂×C₃×M₁₆	C₂²	semidirect
18. C₃×C₄⋊Q₈	C₂²	semidirect
24. C₃×#121	C₂²	semidirect
28. C₃×C₄×D₄	C₂²	semidirect
30. C₃×#117	C₂²	semidirect
31. C₃×#107	C₂²	semidirect
32. C₃×#106	C₂²	semidirect
33. C₃×C₄⋊C₈	C₂²	semidirect
36. C₃×#102	C₂²	semidirect
37. C₃×#102	C₂²	semidirect
38. C₃×#102	C₂²	semidirect
39. C₃×#102	C₂²	semidirect
40. C₃×C₂²⋊C₈	C₂²	semidirect
41. C₃×C₄×C₈	C₂²	semidirect
42. #555	C₂×C₃	semidirect
43. D₄²	C₂×C₃	semidirect
44. #486	C₂×C₃	semidirect
45. #482	C₂×C₃	semidirect
46. #476	C₂×C₃	semidirect
47. #469	C₂×C₃	semidirect
48. #459	C₂×C₃	semidirect
49. #447	C₂×C₃	semidirect
50. C₄×D₈	C₂×C₃	semidirect
51. #436	C₂×C₃	semidirect
52. #418	C₂×C₃	semidirect
92. C₂×D₈	C₂²×C₃	semidirect
93. C₂×M₁₆	C₂²×C₃	semidirect
94. C₄⋊Q₈	C₂²×C₃	semidirect
100. #121	C₂²×C₃	semidirect
104. C₄×D₄	C₂²×C₃	semidirect
106. #117	C₂²×C₃	semidirect
107. #107	C₂²×C₃	semidirect
108. #106	C₂²×C₃	semidirect
109. C₄⋊C₈	C₂²×C₃	semidirect
112. #102	C₂²×C₃	semidirect
113. #102	C₂²×C₃	semidirect
114. #102	C₂²×C₃	semidirect
115. #102	C₂²×C₃	semidirect
116. C₂²⋊C₈	C₂²×C₃	semidirect
117. C₄×C₈	C₂²×C₃	semidirect
273. C₃	#3325	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#555	2
C₃×D₄²	2
C₃×#486	2
C₃×#482	2
C₃×#476	2
C₃×#469	2
C₃×#459	2
C₃×#447	2
C₃×C₄×D₈	2
C₃×#436	2
C₃×#418	2
#3325	3

Subgroups — C3×#3325 #85189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3325 #85189