Subgroups — C₃×#2957 #84821

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 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
2	C₃×#568	C₂	1	2	yes	maximal, complemented
3	C₃×#444	C₂	1	4	yes	maximal, complemented
4	C₃×#443	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#421	C₂	1	2	yes	maximal, complemented
7	C₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
8	#2957	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#41	C₂²	1	2	yes	—
10	C₃×#135	C₄	1	2	yes	complemented
11	C₃×#135	C₄	1	2	yes	complemented
12	C₃×#135	C₂²	1	2	yes	—
13	C₃×#135	C₂²	1	2	yes	—
14	C₂×C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×SD₁₆	C₄	1	2	yes	complemented
16	C₂×C₃×D₈	C₄	1	1	yes	characteristic, complemented
17	C₃×#131	—	2	4	no	—
18	C₃×#131	C₂²	1	8	yes	complemented
19	C₂×C₃×M₁₆	—	2	4	no	—
20	C₂×C₃×M₁₆	C₂²	1	2	yes	—
21	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
23	C₃×#117	C₂²	1	1	yes	characteristic
24	C₃×#108	C₂²	1	2	yes	complemented
25	C₃×#108	C₂²	1	2	yes	complemented
26	C₃×#104	C₂²	1	8	yes	complemented
27	C₃×#97	C₂²	1	2	yes	complemented
28	C₃×C₄×C₈	C₂²	1	2	yes	complemented
29	C₂×#135	C₂×C₃	1	1	yes	characteristic
30	#568	C₂×C₃	1	2	yes	complemented
31	#444	C₂×C₃	1	4	yes	complemented
32	#443	C₂×C₃	1	4	yes	complemented
33	C₂×#108	C₂×C₃	1	1	yes	characteristic, complemented
34	#421	C₂×C₃	1	2	yes	complemented
35	#405	C₂×C₃	1	1	yes	characteristic, complemented
36	C₃×#41	—	2	2	no	—
37	C₃×#41	—	2	2	no	—
38	C₃×#41	C₂×C₄	1	4	yes	—
39	C₃×#41	C₂³	1	4	yes	—
40	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
41	C₂×C₃×D₄	—	2	2	no	—
42	C₂×C₃×D₄	C₂×C₄	1	2	yes	—
43	C₂²×C₃×C₄	—	2	2	no	—
44	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
45	C₃×Q₁₆	C₂×C₄	1	2	yes	—
46	C₃×Q₁₆	C₂×C₄	1	2	yes	—
47	C₃×SD₁₆	C₂×C₄	1	4	yes	—
48	C₃×SD₁₆	C₂×C₄	1	4	yes	—
49	C₃×D₈	C₂×C₄	1	2	yes	—
50	C₃×D₈	C₂×C₄	1	2	yes	—
51	C₃×M₁₆	—	2	2	no	—
52	C₃×M₁₆	—	2	8	no	—
53	C₃×M₁₆	D₄	1	2	yes	—
54	C₃×M₁₆	D₄	1	2	yes	—
55	C₃×M₁₆	C₂³	1	4	yes	—
56	C₂×C₃×C₈	—	2	2	no	—
57	C₂×C₃×C₈	—	2	8	no	—
58	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂³	1	2	yes	—
65	C₃×C₄⋊C₄	—	2	1	no	—
66	C₃×C₂²⋊C₄	—	2	1	no	—
67	C₃×C₄²	C₂³	1	2	yes	—
68	C₂×#41	C₂²×C₃	1	2	yes	—
69	#135	C₃×C₄	1	2	yes	complemented
70	#135	C₃×C₄	1	2	yes	complemented
71	#135	C₂²×C₃	1	2	yes	—
72	#135	C₂²×C₃	1	2	yes	—
73	C₂×Q₁₆	C₃×C₄	1	1	yes	characteristic, complemented
74	C₂×SD₁₆	C₃×C₄	1	2	yes	complemented
75	C₂×D₈	C₃×C₄	1	1	yes	characteristic, complemented
76	#131	—	2	4	no	—
77	#131	C₂²×C₃	1	8	yes	complemented
78	C₂×M₁₆	—	2	4	no	—
79	C₂×M₁₆	C₂²×C₃	1	2	yes	—
80	C₂×M₁₆	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
82	#117	C₂²×C₃	1	1	yes	characteristic
83	#108	C₂²×C₃	1	2	yes	complemented
84	#108	C₂²×C₃	1	2	yes	complemented
85	#104	C₂²×C₃	1	8	yes	complemented
86	#97	C₂²×C₃	1	2	yes	complemented
87	C₄×C₈	C₂²×C₃	1	2	yes	complemented
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	#41	1	1	yes	characteristic
90	C₃×Q₈	—	2	2	no	—
91	C₃×Q₈	C₂²×C₄	1	4	yes	—
92	C₃×D₄	—	2	4	no	—
93	C₃×D₄	—	2	4	no	—
94	C₃×D₄	—	2	2	no	—
95	C₃×D₄	C₂²×C₄	1	4	yes	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	4	no	—
98	C₂×C₃×C₄	—	2	4	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃×C₄	#41	1	1	yes	characteristic
105	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
106	C₃×C₈	—	2	4	no	—
107	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₃×C₈	C₂×D₄	1	4	yes	—
112	#41	—	2	2	no	—
113	#41	—	2	2	no	—
114	#41	C₂×C₃×C₄	1	4	yes	—
115	#41	C₂³×C₃	1	4	yes	—
116	C₂×Q₈	C₂×C₃×C₄	1	2	yes	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	C₂×C₃×C₄	1	2	yes	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
121	Q₁₆	C₂×C₃×C₄	1	2	yes	—
122	Q₁₆	C₂×C₃×C₄	1	2	yes	—
123	SD₁₆	C₂×C₃×C₄	1	4	yes	—
124	SD₁₆	C₂×C₃×C₄	1	4	yes	—
125	D₈	C₂×C₃×C₄	1	2	yes	—
126	D₈	C₂×C₃×C₄	1	2	yes	—
127	M₁₆	—	2	2	no	—
128	M₁₆	—	2	8	no	—
129	M₁₆	C₃×D₄	1	2	yes	—
130	M₁₆	C₃×D₄	1	2	yes	—
131	M₁₆	C₂³×C₃	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	8	no	—
134	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
136	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
137	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
138	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
139	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
140	C₂×C₈	C₂³×C₃	1	2	yes	—
141	C₄⋊C₄	—	2	1	no	—
142	C₂²⋊C₄	—	2	1	no	—
143	C₄²	C₂³×C₃	1	2	yes	—
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	4	1	no	—
146	C₂²×C₃	—	8	2	no	—
147	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
148	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
149	C₂²×C₃	C₂×#41	1	1	yes	characteristic
150	C₃×C₄	—	2	4	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	C₄×D₄	1	1	yes	characteristic
153	C₃×C₄	C₄×D₄	1	1	yes	characteristic
154	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
155	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
156	C₂³	—	2	2	no	—
157	C₂³	C₃×#41	1	1	yes	characteristic
158	Q₈	—	2	2	no	—
159	Q₈	C₂²×C₃×C₄	1	4	yes	—
160	D₄	—	2	4	no	—
161	D₄	—	2	4	no	—
162	D₄	—	2	2	no	—
163	D₄	C₂²×C₃×C₄	1	4	yes	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
170	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂×C₄	C₃×#41	1	1	yes	characteristic
173	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
174	C₈	—	2	4	no	—
175	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₈	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₈	C₂×C₃×D₄	1	4	yes	—
180	C₂×C₃	—	2	1	no	—
181	C₂×C₃	—	2	1	no	—
182	C₂×C₃	—	2	1	no	—
183	C₂×C₃	—	4	4	no	—
184	C₂×C₃	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²	—	2	4	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	8	2	no	—
188	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
189	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
190	C₂²	C₂×C₃×#41	1	1	yes	characteristic
191	C₄	—	2	4	no	—
192	C₄	—	4	2	no	—
193	C₄	C₃×C₄×D₄	1	1	yes	characteristic
194	C₄	C₃×C₄×D₄	1	1	yes	characteristic
195	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
196	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
197	C₃	#2957	1	1	yes	characteristic, direct product factor, Sylow
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	4	no	—
202	C₂	C₂×C₃×C₄×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂²×D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂×C₃	C₂×C₄×D₄
prime radical	C₂×C₃×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#568	C₂	semidirect
3. C₃×#444	C₂	semidirect
4. C₃×#443	C₂	semidirect
5. C₂×C₃×#108	C₂	semidirect
6. C₃×#421	C₂	semidirect
7. C₃×#405	C₂	semidirect
8. #2957	C₃	direct (N × H)
10. C₃×#135	C₄	semidirect
11. C₃×#135	C₄	semidirect
14. C₂×C₃×Q₁₆	C₄	semidirect
15. C₂×C₃×SD₁₆	C₄	semidirect
16. C₂×C₃×D₈	C₄	semidirect
18. C₃×#131	C₂²	semidirect
24. C₃×#108	C₂²	semidirect
25. C₃×#108	C₂²	semidirect
26. C₃×#104	C₂²	semidirect
27. C₃×#97	C₂²	semidirect
28. C₃×C₄×C₈	C₂²	semidirect
30. #568	C₂×C₃	semidirect
31. #444	C₂×C₃	semidirect
32. #443	C₂×C₃	semidirect
33. C₂×#108	C₂×C₃	semidirect
34. #421	C₂×C₃	semidirect
35. #405	C₂×C₃	semidirect
69. #135	C₃×C₄	semidirect
70. #135	C₃×C₄	semidirect
73. C₂×Q₁₆	C₃×C₄	semidirect
74. C₂×SD₁₆	C₃×C₄	semidirect
75. C₂×D₈	C₃×C₄	semidirect
77. #131	C₂²×C₃	semidirect
83. #108	C₂²×C₃	semidirect
84. #108	C₂²×C₃	semidirect
85. #104	C₂²×C₃	semidirect
86. #97	C₂²×C₃	semidirect
87. C₄×C₈	C₂²×C₃	semidirect
197. C₃	#2957	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
C₃×#568	2
C₃×#444	2
C₃×#443	2
C₂×C₃×#108	2
C₃×#421	2
C₃×#405	2
#2957	3

Subgroups — C3×#2957 #84821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2957 #84821