Subgroups — C₃×#3352 #85216

Order 384.

336 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#552	C₂	1	2	yes	maximal
2	C₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#485	C₂	1	2	yes	maximal, complemented
4	C₃×#484	C₂	1	2	yes	maximal, complemented
5	C₃×#479	C₂	1	2	yes	maximal, complemented
6	C₃×#470	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#436	C₂	1	1	yes	maximal, characteristic
9	C₃×#428	C₂	1	2	yes	maximal, complemented
10	#3352	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×M₁₆	C₂²	1	2	yes	—
13	C₃×#126	—	2	2	no	—
14	C₃×#125	—	2	2	no	—
15	C₃×#125	C₂²	1	2	yes	—
16	C₃×#123	—	2	2	no	—
17	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₃×C₄×Q₈	C₂²	1	2	yes	—
20	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
21	C₃×#117	—	2	2	no	—
22	C₃×#117	C₂²	1	2	yes	—
23	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₃×#107	C₂²	1	1	yes	characteristic
25	C₃×#107	C₂²	1	2	yes	—
26	C₃×#107	C₂²	1	1	yes	characteristic
27	C₃×#107	C₂²	1	1	yes	characteristic
28	C₃×#106	C₂²	1	2	yes	—
29	C₃×#106	C₂²	1	2	yes	—
30	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃×#103	C₂²	1	2	yes	—
32	C₃×#103	C₂²	1	2	yes	—
33	C₃×#103	C₂²	1	2	yes	—
34	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
36	#552	C₂×C₃	1	2	yes	—
37	#499	C₂×C₃	1	1	yes	characteristic, complemented
38	#485	C₂×C₃	1	2	yes	complemented
39	#484	C₂×C₃	1	2	yes	complemented
40	#479	C₂×C₃	1	2	yes	complemented
41	#470	C₂×C₃	1	2	yes	complemented
42	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
43	#436	C₂×C₃	1	1	yes	characteristic
44	#428	C₂×C₃	1	2	yes	complemented
45	C₂×C₃×Q₈	C₂³	1	2	yes	—
46	C₂×C₃×D₄	D₄	1	1	yes	characteristic
47	C₂²×C₃×C₄	—	2	2	no	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₃×Q₁₆	—	2	2	no	—
50	C₃×M₁₆	—	2	4	no	—
51	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₈	C₂³	1	2	yes	—
53	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₄⋊C₄	—	2	2	no	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	2	2	no	—
59	C₃×C₄⋊C₄	—	2	2	no	—
60	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₃×C₄⋊C₄	C₂³	1	2	yes	—
62	C₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₃×C₂²⋊C₄	—	2	2	no	—
65	C₃×C₂²⋊C₄	—	2	2	no	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₂²⋊C₄	D₄	1	2	yes	—
68	C₃×C₄²	—	2	2	no	—
69	C₃×C₄²	—	2	2	no	—
70	C₃×C₄²	C₂³	1	1	yes	characteristic
71	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
72	C₂×M₁₆	C₂²×C₃	1	2	yes	—
73	#126	—	2	2	no	—
74	#125	—	2	2	no	—
75	#125	C₂²×C₃	1	2	yes	—
76	#123	—	2	2	no	—
77	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
78	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
79	C₄×Q₈	C₂²×C₃	1	2	yes	—
80	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
81	#117	—	2	2	no	—
82	#117	C₂²×C₃	1	2	yes	—
83	C₂×C₄⋊C₄	C₂²×C₃	1	2	yes	—
84	#107	C₂²×C₃	1	1	yes	characteristic
85	#107	C₂²×C₃	1	2	yes	—
86	#107	C₂²×C₃	1	1	yes	characteristic
87	#107	C₂²×C₃	1	1	yes	characteristic
88	#106	C₂²×C₃	1	2	yes	—
89	#106	C₂²×C₃	1	2	yes	—
90	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
91	#103	C₂²×C₃	1	2	yes	—
92	#103	C₂²×C₃	1	2	yes	—
93	#103	C₂²×C₃	1	2	yes	—
94	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
95	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
96	C₂³×C₃	C₂×D₄	1	2	yes	—
97	C₃×Q₈	—	2	2	no	—
98	C₃×Q₈	—	2	2	no	—
99	C₃×D₄	—	4	1	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	2	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	4	4	no	—
107	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
109	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
111	C₃×C₈	—	2	2	no	—
112	C₃×C₈	—	2	1	no	—
113	C₃×C₈	#41	1	2	yes	—
114	C₂×Q₈	C₂³×C₃	1	2	yes	—
115	C₂×D₄	C₃×D₄	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂³×C₃	1	2	yes	—
118	Q₁₆	—	2	2	no	—
119	M₁₆	—	2	4	no	—
120	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
121	C₂×C₈	C₂³×C₃	1	2	yes	—
122	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	2	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
130	C₄⋊C₄	C₂³×C₃	1	2	yes	—
131	C₄⋊C₄	C₂³×C₃	1	2	yes	—
132	C₄⋊C₄	C₂³×C₃	1	2	yes	—
133	C₂²⋊C₄	—	2	2	no	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₂²⋊C₄	C₃×D₄	1	2	yes	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	2	no	—
139	C₄²	C₂³×C₃	1	1	yes	characteristic
140	C₂²×C₃	—	2	2	no	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	—	2	2	no	—
143	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	2	1	no	—
146	C₃×C₄	—	2	1	no	—
147	C₃×C₄	—	4	2	no	—
148	C₃×C₄	—	4	2	no	—
149	C₃×C₄	—	4	2	no	—
150	C₃×C₄	—	4	2	no	—
151	C₃×C₄	C₂×#41	1	1	yes	characteristic
152	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
153	C₂³	C₂×C₃×D₄	1	2	yes	—
154	Q₈	—	2	2	no	—
155	Q₈	—	2	2	no	—
156	D₄	—	4	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
166	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	—	2	1	no	—
170	C₈	C₃×#41	1	2	yes	—
171	C₂×C₃	—	4	2	no	—
172	C₂×C₃	#547	1	1	yes	characteristic
173	C₂×C₃	#575	1	1	yes	characteristic
174	C₂×C₃	#578	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	—	4	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	2	no	—
186	C₄	C₂×C₃×#41	1	1	yes	characteristic
187	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
188	C₃	#3352	1	1	yes	characteristic, direct product factor, Sylow
189	C₂	—	4	2	no	—
190	C₂	C₃×#547	1	1	yes	characteristic
191	C₂	C₃×#575	1	1	yes	characteristic
192	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	C₂×C₃×D₄	D₄
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
2. C₃×#499	C₂	semidirect
3. C₃×#485	C₂	semidirect
4. C₃×#484	C₂	semidirect
5. C₃×#479	C₂	semidirect
6. C₃×#470	C₂	semidirect
7. C₃×C₄×Q₁₆	C₂	semidirect
9. C₃×#428	C₂	semidirect
10. #3352	C₃	direct (N × H)
37. #499	C₂×C₃	semidirect
38. #485	C₂×C₃	semidirect
39. #484	C₂×C₃	semidirect
40. #479	C₂×C₃	semidirect
41. #470	C₂×C₃	semidirect
42. C₄×Q₁₆	C₂×C₃	semidirect
44. #428	C₂×C₃	semidirect
188. C₃	#3352	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×#552	2
C₃×#499	2
C₃×#485	2
C₃×#484	2
C₃×#479	2
C₃×#470	2
C₃×C₄×Q₁₆	2
C₃×#436	2
C₃×#428	2
#3352	3

Subgroups — C3×#3352 #85216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3352 #85216