Subgroups — C₃×#3276 #85140

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#137	C₂	1	2	yes	maximal, complemented
2	C₃×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#547	C₂	1	1	yes	maximal, characteristic
4	C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#470	C₂	1	2	yes	maximal, complemented
6	C₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#452	C₂	1	2	yes	maximal, complemented
9	C₃×#451	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
12	#3276	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×#137	—	2	8	no	—
16	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
18	C₂×C₃×Q₁₆	C₂²	1	2	yes	—
19	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
20	C₂×C₃×SD₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×M₁₆	C₂²	1	2	yes	—
22	C₃×C₄⋊Q₈	—	2	1	no	—
23	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃×#123	—	2	2	no	—
26	C₃×C₂²⋊Q₈	—	2	2	no	—
27	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
29	C₃×#121	—	2	1	no	—
30	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
31	C₃×C₄×D₄	—	2	1	no	—
32	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
33	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
34	C₂×C₃×C₄⋊C₄	—	2	1	no	—
35	C₃×#106	C₂²	1	1	yes	characteristic
36	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃×#103	C₂²	1	1	yes	characteristic
38	C₃×#103	C₂²	1	2	yes	—
39	C₃×#103	C₂²	1	2	yes	—
40	C₃×#102	C₂²	1	1	yes	characteristic
41	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
43	C₂×#137	C₂×C₃	1	2	yes	complemented
44	D₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
45	#547	C₂×C₃	1	1	yes	characteristic
46	#494	C₂×C₃	1	1	yes	characteristic, complemented
47	#470	C₂×C₃	1	2	yes	complemented
48	#462	C₂×C₃	1	1	yes	characteristic, complemented
49	#461	C₂×C₃	1	1	yes	characteristic, complemented
50	#452	C₂×C₃	1	2	yes	complemented
51	#451	C₂×C₃	1	2	yes	complemented
52	C₄×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
53	#436	C₂×C₃	1	1	yes	characteristic, complemented
54	C₃×#41	—	2	4	no	—
55	C₃×#41	—	4	2	no	—
56	C₂×C₃×Q₈	—	2	2	no	—
57	C₂×C₃×Q₈	—	2	4	no	—
58	C₂×C₃×Q₈	—	4	2	no	—
59	C₂×C₃×Q₈	C₂³	1	2	yes	—
60	C₂×C₃×Q₈	C₂³	1	2	yes	—
61	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	D₄	1	1	yes	characteristic
64	C₂×C₃×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₃×C₄	—	2	2	no	—
66	C₂²×C₃×C₄	—	2	1	no	—
67	C₂²×C₃×C₄	—	2	2	no	—
68	C₂²×C₃×C₄	C₂³	1	2	yes	—
69	C₃×Q₁₆	—	2	4	no	—
70	C₃×Q₁₆	—	2	4	no	—
71	C₃×Q₁₆	—	4	2	no	—
72	C₃×SD₁₆	—	2	4	no	—
73	C₃×SD₁₆	D₄	1	4	yes	complemented
74	C₃×M₁₆	—	2	4	no	—
75	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂³	1	2	yes	—
77	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
78	C₃×C₄⋊C₄	—	2	1	no	—
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₄⋊C₄	—	2	1	no	—
81	C₃×C₄⋊C₄	—	2	1	no	—
82	C₃×C₄⋊C₄	—	2	1	no	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
86	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
87	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
88	C₃×C₂²⋊C₄	—	2	2	no	—
89	C₃×C₂²⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	—	2	1	no	—
91	C₃×C₂²⋊C₄	D₄	1	2	yes	—
92	C₃×C₄²	—	2	1	no	—
93	C₃×C₄²	C₂³	1	1	yes	characteristic
94	C₂×#41	C₂²×C₃	1	2	yes	—
95	C₂²×Q₈	C₂²×C₃	1	2	yes	—
96	#137	—	2	8	no	—
97	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
98	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
99	C₂×Q₁₆	C₂²×C₃	1	2	yes	—
100	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
101	C₂×SD₁₆	C₂²×C₃	1	1	yes	characteristic
102	C₂×M₁₆	C₂²×C₃	1	2	yes	—
103	C₄⋊Q₈	—	2	1	no	—
104	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
105	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
106	#123	—	2	2	no	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
109	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
110	#121	—	2	1	no	—
111	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
112	C₄×D₄	—	2	1	no	—
113	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
114	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	1	no	—
116	#106	C₂²×C₃	1	1	yes	characteristic
117	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
118	#103	C₂²×C₃	1	1	yes	characteristic
119	#103	C₂²×C₃	1	2	yes	—
120	#103	C₂²×C₃	1	2	yes	—
121	#102	C₂²×C₃	1	1	yes	characteristic
122	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
123	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	C₂×D₄	1	2	yes	—
126	C₃×Q₈	—	2	1	no	—
127	C₃×Q₈	—	2	2	no	—
128	C₃×Q₈	—	2	2	no	—
129	C₃×Q₈	—	2	2	no	—
130	C₃×Q₈	—	2	2	no	—
131	C₃×Q₈	—	4	4	no	—
132	C₃×Q₈	C₂×D₄	1	2	yes	—
133	C₃×D₄	—	2	2	no	—
134	C₃×D₄	—	2	1	no	—
135	C₃×D₄	—	4	4	no	—
136	C₃×D₄	C₂×D₄	1	2	yes	—
137	C₂×C₃×C₄	—	2	1	no	—
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	1	no	—
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	4	4	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	C₂×D₄	1	2	yes	—
154	#41	—	2	4	no	—
155	#41	—	4	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂³×C₃	1	2	yes	—
160	C₂×Q₈	C₂³×C₃	1	2	yes	—
161	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	C₃×D₄	1	1	yes	characteristic
164	C₂×D₄	C₂³×C₃	1	1	yes	characteristic
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	C₂³×C₃	1	2	yes	—
169	Q₁₆	—	2	4	no	—
170	Q₁₆	—	2	4	no	—
171	Q₁₆	—	4	2	no	—
172	SD₁₆	—	2	4	no	—
173	SD₁₆	C₃×D₄	1	4	yes	complemented
174	M₁₆	—	2	4	no	—
175	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₃	1	2	yes	—
177	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	1	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
187	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	C₃×D₄	1	2	yes	—
192	C₄²	—	2	1	no	—
193	C₄²	C₂³×C₃	1	1	yes	characteristic
194	C₂²×C₃	—	2	2	no	—
195	C₂²×C₃	—	2	2	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	1	no	—
200	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	2	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	4	2	no	—
205	C₃×C₄	—	4	2	no	—
206	C₃×C₄	—	4	1	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	#137	1	2	yes	complemented
209	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
210	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂³	—	2	1	no	—
212	C₂³	C₂×C₃×D₄	1	2	yes	—
213	Q₈	—	2	1	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	2	2	no	—
217	Q₈	—	2	2	no	—
218	Q₈	—	4	4	no	—
219	Q₈	C₂×C₃×D₄	1	2	yes	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	4	4	no	—
223	D₄	C₂×C₃×D₄	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
236	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	1	no	—
240	C₈	C₂×C₃×D₄	1	2	yes	—
241	C₂×C₃	—	4	2	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	D₄²	1	1	yes	characteristic
244	C₂×C₃	C₂×#137	1	1	yes	characteristic
245	C₂×C₃	#578	1	1	yes	characteristic
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
253	C₄	—	2	1	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	4	2	no	—
259	C₄	—	4	2	no	—
260	C₄	C₃×#137	1	2	yes	complemented
261	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
262	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
263	C₃	#3276	1	1	yes	characteristic, direct product factor, Sylow
264	C₂	—	4	2	no	—
265	C₂	—	4	2	no	—
266	C₂	C₃×D₄²	1	1	yes	characteristic
267	C₂	C₂×C₃×#137	1	1	yes	characteristic
268	C₂	C₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#547	C₂
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×#137	C₂	semidirect
2. C₃×D₄×Q₈	C₂	semidirect
4. C₃×#494	C₂	semidirect
5. C₃×#470	C₂	semidirect
6. C₃×#462	C₂	semidirect
7. C₃×#461	C₂	semidirect
8. C₃×#452	C₂	semidirect
9. C₃×#451	C₂	semidirect
10. C₃×C₄×SD₁₆	C₂	semidirect
11. C₃×#436	C₂	semidirect
12. #3276	C₃	direct (N × H)
43. C₂×#137	C₂×C₃	semidirect
44. D₄×Q₈	C₂×C₃	semidirect
46. #494	C₂×C₃	semidirect
47. #470	C₂×C₃	semidirect
48. #462	C₂×C₃	semidirect
49. #461	C₂×C₃	semidirect
50. #452	C₂×C₃	semidirect
51. #451	C₂×C₃	semidirect
52. C₄×SD₁₆	C₂×C₃	semidirect
53. #436	C₂×C₃	semidirect
73. C₃×SD₁₆	D₄	semidirect
173. SD₁₆	C₃×D₄	semidirect
208. C₃×C₄	#137	semidirect
260. C₄	C₃×#137	semidirect
263. C₃	#3276	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×#137	2
C₃×D₄×Q₈	2
C₃×#547	2
C₃×#494	2
C₃×#470	2
C₃×#462	2
C₃×#461	2
C₃×#452	2
C₃×#451	2
C₃×C₄×SD₁₆	2
C₃×#436	2
#3276	3

Subgroups — C3×#3276 #85140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3276 #85140