Subgroups — C₃×#3285 #85149

Order 384.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#548	C₂	1	2	yes	maximal, complemented
4	C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#464	C₂	1	2	yes	maximal, complemented
8	C₃×#452	C₂	1	2	yes	maximal, complemented
9	C₃×#448	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	#3285	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂²×C₃×Q₈	C₂²	1	2	yes	—
15	C₃×#135	—	2	4	no	—
16	C₂×C₃×Q₁₆	—	2	4	no	—
17	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
21	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
22	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
25	C₃×#124	—	2	2	no	—
26	C₃×#124	C₂²	1	2	yes	—
27	C₃×C₂²⋊Q₈	—	2	2	no	—
28	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₃×#121	—	2	2	no	—
30	C₃×#121	C₂²	1	2	yes	—
31	C₃×C₄×Q₈	C₂²	1	2	yes	—
32	C₃×C₄×D₄	—	2	2	no	—
33	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₃×#107	C₂²	1	1	yes	characteristic
35	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₃×#103	C₂²	1	2	yes	—
37	C₃×#103	C₂²	1	2	yes	—
38	C₃×#102	C₂²	1	2	yes	—
39	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	C₂×#135	C₂×C₃	1	1	yes	characteristic, complemented
43	C₂²×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
44	#548	C₂×C₃	1	2	yes	complemented
45	#495	C₂×C₃	1	1	yes	characteristic, complemented
46	#467	C₂×C₃	1	1	yes	characteristic, complemented
47	#466	C₂×C₃	1	1	yes	characteristic, complemented
48	#464	C₂×C₃	1	2	yes	complemented
49	#452	C₂×C₃	1	2	yes	complemented
50	#448	C₂×C₃	1	2	yes	complemented
51	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
52	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
53	C₃×#41	—	2	4	no	—
54	C₃×#41	—	4	2	no	—
55	C₂×C₃×Q₈	—	2	2	no	—
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	—	4	2	no	—
58	C₂×C₃×Q₈	C₂³	1	2	yes	—
59	C₂×C₃×Q₈	C₂³	1	2	yes	—
60	C₂×C₃×D₄	—	2	2	no	—
61	C₂×C₃×D₄	—	2	2	no	—
62	C₂×C₃×D₄	D₄	1	1	yes	characteristic
63	C₂×C₃×D₄	C₂³	1	2	yes	—
64	C₂²×C₃×C₄	—	2	2	no	—
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
68	C₃×Q₁₆	—	2	4	no	—
69	C₃×Q₁₆	—	2	2	no	—
70	C₃×Q₁₆	D₄	1	4	yes	complemented
71	C₃×SD₁₆	—	2	4	no	—
72	C₃×SD₁₆	—	4	2	no	—
73	C₃×D₈	—	2	2	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
83	C₃×C₄⋊C₄	C₂³	1	2	yes	—
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	2	no	—
88	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
89	C₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	C₂³	1	1	yes	characteristic
92	C₂×#41	C₂²×C₃	1	2	yes	—
93	C₂²×Q₈	C₂²×C₃	1	2	yes	—
94	#135	—	2	4	no	—
95	C₂×Q₁₆	—	2	4	no	—
96	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
97	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
98	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
99	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
100	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
101	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
104	#124	—	2	2	no	—
105	#124	C₂²×C₃	1	2	yes	—
106	C₂²⋊Q₈	—	2	2	no	—
107	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
108	#121	—	2	2	no	—
109	#121	C₂²×C₃	1	2	yes	—
110	C₄×Q₈	C₂²×C₃	1	2	yes	—
111	C₄×D₄	—	2	2	no	—
112	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
113	#107	C₂²×C₃	1	1	yes	characteristic
114	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
115	#103	C₂²×C₃	1	2	yes	—
116	#103	C₂²×C₃	1	2	yes	—
117	#102	C₂²×C₃	1	2	yes	—
118	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
119	C₂²⋊C₈	C₂²×C₃	1	1	yes	characteristic
120	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
121	C₂³×C₃	—	2	2	no	—
122	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
123	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
124	C₃×Q₈	—	2	2	no	—
125	C₃×Q₈	—	2	2	no	—
126	C₃×Q₈	—	2	2	no	—
127	C₃×Q₈	—	4	4	no	—
128	C₃×Q₈	C₂×D₄	1	4	yes	—
129	C₃×D₄	—	2	2	no	—
130	C₃×D₄	—	2	2	no	—
131	C₃×D₄	—	2	2	no	—
132	C₃×D₄	—	4	4	no	—
133	C₃×D₄	—	4	4	no	—
134	C₂×C₃×C₄	—	2	2	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	4	4	no	—
142	C₂×C₃×C₄	—	4	4	no	—
143	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
148	C₃×C₈	—	2	1	no	—
149	C₃×C₈	—	2	1	no	—
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	C₂×D₄	1	2	yes	—
152	#41	—	2	4	no	—
153	#41	—	4	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	C₂³×C₃	1	2	yes	—
158	C₂×Q₈	C₂³×C₃	1	2	yes	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	—	2	2	no	—
161	C₂×D₄	C₃×D₄	1	1	yes	characteristic
162	C₂×D₄	C₂³×C₃	1	2	yes	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
166	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
167	Q₁₆	—	2	4	no	—
168	Q₁₆	—	2	2	no	—
169	Q₁₆	C₃×D₄	1	4	yes	complemented
170	SD₁₆	—	2	4	no	—
171	SD₁₆	—	4	2	no	—
172	D₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
182	C₄⋊C₄	C₂³×C₃	1	2	yes	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₂²⋊C₄	—	2	2	no	—
185	C₂²⋊C₄	—	2	2	no	—
186	C₂²⋊C₄	—	2	2	no	—
187	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
188	C₂²⋊C₄	C₃×D₄	1	1	yes	characteristic
189	C₄²	—	2	2	no	—
190	C₄²	C₂³×C₃	1	1	yes	characteristic
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	—	4	2	no	—
198	C₂²×C₃	—	4	2	no	—
199	C₂²×C₃	#135	1	2	yes	complemented
200	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	4	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	4	2	no	—
207	C₃×C₄	—	4	2	no	—
208	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
209	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂³	—	2	2	no	—
211	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	4	no	—
217	Q₈	C₂×C₃×D₄	1	4	yes	—
218	D₄	—	2	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	2	2	no	—
221	D₄	—	4	4	no	—
222	D₄	—	4	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	C₂×C₃×D₄	1	2	yes	—
241	C₂×C₃	—	2	2	no	—
242	C₂×C₃	—	4	1	no	—
243	C₂×C₃	—	8	2	no	—
244	C₂×C₃	D₄²	1	1	yes	characteristic
245	C₂×C₃	C₂×#135	1	1	yes	characteristic
246	C₂×C₃	#578	1	1	yes	characteristic
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	C₃×#135	1	2	yes	complemented
256	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
265	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
266	C₃	#3285	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	8	2	no	—
270	C₂	C₃×D₄²	1	1	yes	characteristic
271	C₂	C₂×C₃×#135	1	1	yes	characteristic
272	C₂	C₃×#578	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. C₂×C₃×#135	C₂	semidirect
2. C₂²×C₃×Q₁₆	C₂	semidirect
3. C₃×#548	C₂	semidirect
4. C₃×#495	C₂	semidirect
5. C₃×#467	C₂	semidirect
6. C₃×#466	C₂	semidirect
7. C₃×#464	C₂	semidirect
8. C₃×#452	C₂	semidirect
9. C₃×#448	C₂	semidirect
10. C₃×C₄×Q₁₆	C₂	semidirect
11. C₃×C₈×D₄	C₂	semidirect
12. #3285	C₃	direct (N × H)
42. C₂×#135	C₂×C₃	semidirect
43. C₂²×Q₁₆	C₂×C₃	semidirect
44. #548	C₂×C₃	semidirect
45. #495	C₂×C₃	semidirect
46. #467	C₂×C₃	semidirect
47. #466	C₂×C₃	semidirect
48. #464	C₂×C₃	semidirect
49. #452	C₂×C₃	semidirect
50. #448	C₂×C₃	semidirect
51. C₄×Q₁₆	C₂×C₃	semidirect
52. C₈×D₄	C₂×C₃	semidirect
70. C₃×Q₁₆	D₄	semidirect
169. Q₁₆	C₃×D₄	semidirect
199. C₂²×C₃	#135	semidirect
255. C₂²	C₃×#135	semidirect
266. C₃	#3285	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#135	2
C₂²×C₃×Q₁₆	2
C₃×#548	2
C₃×#495	2
C₃×#467	2
C₃×#466	2
C₃×#464	2
C₃×#452	2
C₃×#448	2
C₃×C₄×Q₁₆	2
C₃×C₈×D₄	2
#3285	3

Subgroups — C3×#3285 #85149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3285 #85149