Subgroups — C₃×#3211 #85075

Order 384.

410 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#495	C₂	1	2	yes	maximal, complemented
4	C₃×#488	C₂	1	2	yes	maximal, complemented
5	C₃×#464	C₂	1	2	yes	maximal, complemented
6	C₃×#463	C₂	1	2	yes	maximal, complemented
7	C₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#448	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#447	C₂	1	1	yes	maximal, characteristic
11	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	#3211	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₁₆	C₂²	1	2	yes	complemented
16	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₃×SD₁₆	C₂²	1	2	yes	complemented
18	C₂×C₃×D₈	C₂²	1	2	yes	—
19	C₃×#126	—	2	1	no	—
20	C₃×#125	C₂²	1	1	yes	characteristic, complemented
21	C₃×#124	—	2	1	no	—
22	C₃×#124	—	2	1	no	—
23	C₃×#124	C₂²	1	2	yes	—
24	C₃×#124	C₂²	1	2	yes	—
25	C₃×#124	C₂²	1	1	yes	characteristic, complemented
26	C₃×#123	—	2	1	no	—
27	C₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
28	C₃×#121	C₂²	1	1	yes	characteristic
29	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
30	C₃×C₄×D₄	—	2	1	no	—
31	C₃×C₄×D₄	C₂²	1	1	yes	characteristic, complemented
32	C₃×#117	—	2	1	no	—
33	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
34	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
36	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₃×#103	C₂²	1	2	yes	complemented
38	C₃×#103	C₂²	1	2	yes	complemented
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	2	yes	—
41	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
42	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
43	C₃×C₄×C₈	C₂²	1	2	yes	complemented
44	#529	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×#124	C₂×C₃	1	1	yes	characteristic, complemented
46	#495	C₂×C₃	1	2	yes	complemented
47	#488	C₂×C₃	1	2	yes	complemented
48	#464	C₂×C₃	1	2	yes	complemented
49	#463	C₂×C₃	1	2	yes	complemented
50	#451	C₂×C₃	1	1	yes	characteristic, complemented
51	#450	C₂×C₃	1	1	yes	characteristic, complemented
52	#448	C₂×C₃	1	1	yes	characteristic, complemented
53	#447	C₂×C₃	1	1	yes	characteristic
54	#431	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂⁴×C₃	—	2	1	no	—
56	C₂×C₃×Q₈	—	2	1	no	—
57	C₂×C₃×Q₈	—	4	2	no	—
58	C₂×C₃×Q₈	C₂³	1	2	yes	—
59	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	2	1	no	—
62	C₂×C₃×D₄	—	4	2	no	—
63	C₂×C₃×D₄	C₂³	1	2	yes	—
64	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	1	no	—
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
70	C₃×Q₁₆	—	4	2	no	—
71	C₃×SD₁₆	—	4	2	no	—
72	C₃×SD₁₆	—	4	2	no	—
73	C₃×D₈	—	4	2	no	—
74	C₂×C₃×C₈	C₂³	1	2	yes	—
75	C₂×C₃×C₈	C₂³	1	2	yes	—
76	C₃×C₄⋊C₄	—	2	1	no	—
77	C₃×C₄⋊C₄	—	2	1	no	—
78	C₃×C₄⋊C₄	—	2	1	no	—
79	C₃×C₄⋊C₄	—	2	1	no	—
80	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
81	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
82	C₃×C₂²⋊C₄	—	2	1	no	—
83	C₃×C₂²⋊C₄	—	2	2	no	—
84	C₃×C₂²⋊C₄	—	2	2	no	—
85	C₃×C₂²⋊C₄	—	2	2	no	—
86	C₃×C₂²⋊C₄	—	2	2	no	—
87	C₃×C₂²⋊C₄	—	2	1	no	—
88	C₃×C₂²⋊C₄	—	2	1	no	—
89	C₃×C₂²⋊C₄	—	2	1	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₄²	—	2	1	no	—
92	C₃×C₄²	D₄	1	1	yes	characteristic
93	C₃×C₄²	D₄	1	1	yes	characteristic
94	C₃×C₄²	C₂³	1	1	yes	characteristic
95	C₃×C₄²	C₂³	1	1	yes	characteristic
96	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
97	C₂²×D₄	C₂²×C₃	1	1	yes	characteristic
98	C₂×Q₁₆	C₂²×C₃	1	2	yes	complemented
99	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
100	C₂×SD₁₆	C₂²×C₃	1	2	yes	complemented
101	C₂×D₈	C₂²×C₃	1	2	yes	—
102	#126	—	2	1	no	—
103	#125	C₂²×C₃	1	1	yes	characteristic, complemented
104	#124	—	2	1	no	—
105	#124	—	2	1	no	—
106	#124	C₂²×C₃	1	2	yes	—
107	#124	C₂²×C₃	1	2	yes	—
108	#124	C₂²×C₃	1	1	yes	characteristic, complemented
109	#123	—	2	1	no	—
110	C₂²⋊Q₈	C₂²×C₃	1	1	yes	characteristic
111	#121	C₂²×C₃	1	1	yes	characteristic
112	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
113	C₄×D₄	—	2	1	no	—
114	C₄×D₄	C₂²×C₃	1	1	yes	characteristic, complemented
115	#117	—	2	1	no	—
116	C₂×C₂²⋊C₄	—	2	1	no	—
117	C₂×C₂²⋊C₄	—	2	1	no	—
118	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
119	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
120	#103	C₂²×C₃	1	2	yes	complemented
121	#103	C₂²×C₃	1	2	yes	complemented
122	#102	C₂²×C₃	1	2	yes	—
123	#102	C₂²×C₃	1	2	yes	—
124	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
125	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
126	C₄×C₈	C₂²×C₃	1	2	yes	complemented
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	—	2	2	no	—
129	C₂³×C₃	—	4	2	no	—
130	C₂³×C₃	—	4	2	no	—
131	C₂³×C₃	—	4	2	no	—
132	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
133	C₃×Q₈	—	2	2	no	—
134	C₃×Q₈	—	2	2	no	—
135	C₃×Q₈	—	2	2	no	—
136	C₃×Q₈	—	4	2	no	—
137	C₃×D₄	—	2	2	no	—
138	C₃×D₄	—	2	2	no	—
139	C₃×D₄	—	2	2	no	—
140	C₃×D₄	—	4	2	no	—
141	C₃×D₄	—	4	2	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	4	1	no	—
149	C₂×C₃×C₄	—	4	2	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₂⁴	—	2	1	no	—
160	C₂×Q₈	—	2	1	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	C₂³×C₃	1	2	yes	—
163	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
164	C₂×D₄	—	2	1	no	—
165	C₂×D₄	—	2	1	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂×D₄	C₂³×C₃	1	2	yes	—
168	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
172	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
174	Q₁₆	—	4	2	no	—
175	SD₁₆	—	4	2	no	—
176	SD₁₆	—	4	2	no	—
177	D₈	—	4	2	no	—
178	C₂×C₈	C₂³×C₃	1	2	yes	—
179	C₂×C₈	C₂³×C₃	1	2	yes	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
186	C₂²⋊C₄	—	2	1	no	—
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	1	no	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	1	no	—
194	C₂²⋊C₄	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	C₃×D₄	1	1	yes	characteristic
197	C₄²	C₃×D₄	1	1	yes	characteristic
198	C₄²	C₂³×C₃	1	1	yes	characteristic
199	C₄²	C₂³×C₃	1	1	yes	characteristic
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	—	4	1	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	4	2	no	—
206	C₂²×C₃	—	4	1	no	—
207	C₂²×C₃	—	4	2	no	—
208	C₂²×C₃	—	8	4	no	—
209	C₂²×C₃	#135	1	2	yes	—
210	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	—	4	1	no	—
218	C₃×C₄	—	4	1	no	—
219	C₃×C₄	—	4	1	no	—
220	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	2	2	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	4	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
246	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
247	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
248	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₂×C₃	—	2	2	no	—
254	C₂×C₃	—	8	1	no	—
255	C₂×C₃	—	8	2	no	—
256	C₂×C₃	#535	1	1	yes	characteristic
257	C₂×C₃	C₂×#135	1	1	yes	characteristic
258	C₂×C₃	#575	1	1	yes	characteristic
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	4	no	—
268	C₂²	C₃×#135	1	2	yes	—
269	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	2	no	—
278	C₄	—	4	1	no	—
279	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
280	C₃	#3211	1	1	yes	characteristic, direct product factor, Sylow
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₃×#535	1	1	yes	characteristic
285	C₂	C₂×C₃×#135	1	1	yes	characteristic
286	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#529	C₂	semidirect
2. C₂×C₃×#124	C₂	semidirect
3. C₃×#495	C₂	semidirect
4. C₃×#488	C₂	semidirect
5. C₃×#464	C₂	semidirect
6. C₃×#463	C₂	semidirect
7. C₃×#451	C₂	semidirect
8. C₃×#450	C₂	semidirect
9. C₃×#448	C₂	semidirect
11. C₃×#431	C₂	semidirect
12. #3211	C₃	direct (N × H)
15. C₂×C₃×Q₁₆	C₂²	semidirect
17. C₂×C₃×SD₁₆	C₂²	semidirect
20. C₃×#125	C₂²	semidirect
25. C₃×#124	C₂²	semidirect
29. C₃×C₄×Q₈	C₂²	semidirect
31. C₃×C₄×D₄	C₂²	semidirect
36. C₃×C₄⋊C₈	C₂²	semidirect
37. C₃×#103	C₂²	semidirect
38. C₃×#103	C₂²	semidirect
43. C₃×C₄×C₈	C₂²	semidirect
44. #529	C₂×C₃	semidirect
45. C₂×#124	C₂×C₃	semidirect
46. #495	C₂×C₃	semidirect
47. #488	C₂×C₃	semidirect
48. #464	C₂×C₃	semidirect
49. #463	C₂×C₃	semidirect
50. #451	C₂×C₃	semidirect
51. #450	C₂×C₃	semidirect
52. #448	C₂×C₃	semidirect
54. #431	C₂×C₃	semidirect
98. C₂×Q₁₆	C₂²×C₃	semidirect
100. C₂×SD₁₆	C₂²×C₃	semidirect
103. #125	C₂²×C₃	semidirect
108. #124	C₂²×C₃	semidirect
112. C₄×Q₈	C₂²×C₃	semidirect
114. C₄×D₄	C₂²×C₃	semidirect
119. C₄⋊C₈	C₂²×C₃	semidirect
120. #103	C₂²×C₃	semidirect
121. #103	C₂²×C₃	semidirect
126. C₄×C₈	C₂²×C₃	semidirect
280. C₃	#3211	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×#529	2
C₂×C₃×#124	2
C₃×#495	2
C₃×#488	2
C₃×#464	2
C₃×#463	2
C₃×#451	2
C₃×#450	2
C₃×#448	2
C₃×#447	2
C₃×#431	2
#3211	3

Subgroups — C3×#3211 #85075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3211 #85075