Subgroups — C₃×#3304 #85168

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#550	C₂	1	1	yes	maximal, characteristic
3	C₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#482	C₂	1	2	yes	maximal, complemented
5	C₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#465	C₂	1	2	yes	maximal, complemented
7	C₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#452	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#416	C₂	1	2	yes	maximal, complemented
12	#3304	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#41	C₂²	1	2	yes	—
14	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
16	C₂²×C₃×C₈	C₂²	1	2	yes	—
17	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₃×#127	—	2	1	no	—
19	C₃×#127	C₂²	1	1	yes	characteristic
20	C₃×#126	—	2	2	no	—
21	C₃×#124	—	2	1	no	—
22	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₃×#121	—	2	2	no	—
24	C₃×#121	C₂²	1	2	yes	—
25	C₃×C₄×Q₈	—	2	1	no	—
26	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
27	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
28	C₃×C₄×D₄	—	2	1	no	—
29	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
30	C₃×#117	C₂²	1	2	yes	—
31	C₃×#107	C₂²	1	1	yes	characteristic
32	C₃×#106	C₂²	1	2	yes	—
33	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₃×#103	C₂²	1	1	yes	characteristic
35	C₃×#103	C₂²	1	2	yes	—
36	C₃×#103	C₂²	1	1	yes	characteristic
37	C₃×#102	C₂²	1	2	yes	—
38	C₃×#102	C₂²	1	2	yes	—
39	C₃×#102	C₂²	1	2	yes	—
40	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
42	#556	C₂×C₃	1	1	yes	characteristic, complemented
43	#550	C₂×C₃	1	1	yes	characteristic
44	#486	C₂×C₃	1	1	yes	characteristic, complemented
45	#482	C₂×C₃	1	2	yes	complemented
46	#475	C₂×C₃	1	1	yes	characteristic, complemented
47	#465	C₂×C₃	1	2	yes	complemented
48	#460	C₂×C₃	1	1	yes	characteristic, complemented
49	#452	C₂×C₃	1	2	yes	complemented
50	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
51	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
52	#416	C₂×C₃	1	2	yes	complemented
53	C₃×#41	—	2	4	no	—
54	C₃×#41	—	4	2	no	—
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
57	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	1	no	—
61	C₂×C₃×D₄	—	2	1	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	C₂³	1	2	yes	—
66	C₃×Q₁₆	—	2	2	no	—
67	C₃×SD₁₆	—	4	2	no	—
68	C₂×C₃×C₈	—	4	2	no	—
69	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂³	1	2	yes	—
72	C₃×C₄⋊C₄	—	2	2	no	—
73	C₃×C₄⋊C₄	—	2	1	no	—
74	C₃×C₄⋊C₄	—	2	1	no	—
75	C₃×C₄⋊C₄	—	2	1	no	—
76	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
77	C₃×C₄⋊C₄	C₂³	1	2	yes	—
78	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
79	C₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₂²⋊C₄	—	2	2	no	—
82	C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×C₂²⋊C₄	D₄	1	2	yes	—
84	C₃×C₄²	—	2	2	no	—
85	C₃×C₄²	—	2	1	no	—
86	C₃×C₄²	—	2	1	no	—
87	C₃×C₄²	C₂³	1	1	yes	characteristic
88	C₂×#41	C₂²×C₃	1	2	yes	—
89	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
90	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
91	C₂²×C₈	C₂²×C₃	1	2	yes	—
92	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
93	#127	—	2	1	no	—
94	#127	C₂²×C₃	1	1	yes	characteristic
95	#126	—	2	2	no	—
96	#124	—	2	1	no	—
97	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
98	#121	—	2	2	no	—
99	#121	C₂²×C₃	1	2	yes	—
100	C₄×Q₈	—	2	1	no	—
101	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
102	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
103	C₄×D₄	—	2	1	no	—
104	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
105	#117	C₂²×C₃	1	2	yes	—
106	#107	C₂²×C₃	1	1	yes	characteristic
107	#106	C₂²×C₃	1	2	yes	—
108	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
109	#103	C₂²×C₃	1	1	yes	characteristic
110	#103	C₂²×C₃	1	2	yes	—
111	#103	C₂²×C₃	1	1	yes	characteristic
112	#102	C₂²×C₃	1	2	yes	—
113	#102	C₂²×C₃	1	2	yes	—
114	#102	C₂²×C₃	1	2	yes	—
115	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
116	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
117	C₂³×C₃	—	2	2	no	—
118	C₂³×C₃	C₂×D₄	1	2	yes	—
119	C₃×Q₈	—	2	1	no	—
120	C₃×Q₈	—	2	1	no	—
121	C₃×Q₈	—	2	1	no	—
122	C₃×Q₈	—	4	2	no	—
123	C₃×Q₈	#41	1	2	yes	—
124	C₃×D₄	—	2	2	no	—
125	C₃×D₄	—	2	2	no	—
126	C₃×D₄	—	2	2	no	—
127	C₃×D₄	—	4	4	no	—
128	C₃×D₄	—	4	2	no	—
129	C₃×D₄	—	8	1	no	—
130	C₃×D₄	—	8	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	4	4	no	—
139	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
140	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	2	no	—
146	#41	—	2	4	no	—
147	#41	—	4	2	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
150	C₂×Q₈	C₂³×C₃	1	1	yes	characteristic
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	—	2	1	no	—
155	C₂×D₄	C₃×D₄	1	1	yes	characteristic
156	C₂×D₄	C₂³×C₃	1	2	yes	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂³×C₃	1	2	yes	—
159	Q₁₆	—	2	2	no	—
160	SD₁₆	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
163	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
164	C₂×C₈	C₂³×C₃	1	2	yes	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
170	C₄⋊C₄	C₂³×C₃	1	2	yes	—
171	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
172	C₄⋊C₄	C₂³×C₃	1	1	yes	characteristic
173	C₂²⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	2	2	no	—
176	C₂²⋊C₄	C₃×D₄	1	2	yes	—
177	C₄²	—	2	2	no	—
178	C₄²	—	2	1	no	—
179	C₄²	—	2	1	no	—
180	C₄²	C₂³×C₃	1	1	yes	characteristic
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	—	4	2	no	—
185	C₂²×C₃	—	4	2	no	—
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	4	1	no	—
192	C₃×C₄	—	4	2	no	—
193	C₃×C₄	—	4	1	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	#135	1	2	yes	—
196	C₃×C₄	C₂×#41	1	1	yes	characteristic
197	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂³	—	2	2	no	—
199	C₂³	C₂×C₃×D₄	1	2	yes	—
200	Q₈	—	2	1	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	4	2	no	—
204	Q₈	C₃×#41	1	2	yes	—
205	D₄	—	2	2	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	4	no	—
209	D₄	—	4	2	no	—
210	D₄	—	8	1	no	—
211	D₄	—	8	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
221	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	2	2	no	—
227	C₂×C₃	—	4	2	no	—
228	C₂×C₃	—	8	2	no	—
229	C₂×C₃	#546	1	1	yes	characteristic
230	C₂×C₃	C₂×#135	1	1	yes	characteristic
231	C₂×C₃	#577	1	1	yes	characteristic
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	—	4	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	4	1	no	—
246	C₄	C₃×#135	1	2	yes	—
247	C₄	C₂×C₃×#41	1	1	yes	characteristic
248	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
249	C₃	#3304	1	1	yes	characteristic, direct product factor, Sylow
250	C₂	—	4	2	no	—
251	C₂	—	8	2	no	—
252	C₂	C₃×#546	1	1	yes	characteristic
253	C₂	C₂×C₃×#135	1	1	yes	characteristic
254	C₂	C₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#550	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. C₃×#556	C₂	semidirect
3. C₃×#486	C₂	semidirect
4. C₃×#482	C₂	semidirect
5. C₃×#475	C₂	semidirect
6. C₃×#465	C₂	semidirect
7. C₃×#460	C₂	semidirect
8. C₃×#452	C₂	semidirect
9. C₃×C₄×Q₁₆	C₂	semidirect
10. C₃×C₈×D₄	C₂	semidirect
11. C₃×#416	C₂	semidirect
12. #3304	C₃	direct (N × H)
42. #556	C₂×C₃	semidirect
44. #486	C₂×C₃	semidirect
45. #482	C₂×C₃	semidirect
46. #475	C₂×C₃	semidirect
47. #465	C₂×C₃	semidirect
48. #460	C₂×C₃	semidirect
49. #452	C₂×C₃	semidirect
50. C₄×Q₁₆	C₂×C₃	semidirect
51. C₈×D₄	C₂×C₃	semidirect
52. #416	C₂×C₃	semidirect
249. C₃	#3304	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×#556	2
C₃×#550	2
C₃×#486	2
C₃×#482	2
C₃×#475	2
C₃×#465	2
C₃×#460	2
C₃×#452	2
C₃×C₄×Q₁₆	2
C₃×C₈×D₄	2
C₃×#416	2
#3304	3

Subgroups — C3×#3304 #85168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3304 #85168