Subgroups — C₃×#3208 #85072

Order 384.

488 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₃×#492	C₂	1	2	yes	maximal, complemented
4	C₃×#486	C₂	1	2	yes	maximal, complemented
5	C₃×#460	C₂	1	4	yes	maximal, complemented
6	C₃×#450	C₂	1	4	yes	maximal, complemented
7	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	#3208	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×D₄	—	2	1	no	—
10	C₂²×C₃×D₄	—	2	1	no	—
11	C₂²×C₃×D₄	C₂²	1	2	yes	—
12	C₂×C₃×SD₁₆	C₂²	1	8	yes	complemented
13	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₃×#127	—	2	1	no	—
16	C₃×#127	—	2	1	no	—
17	C₃×#127	C₂²	1	2	yes	—
18	C₃×#127	C₂²	1	2	yes	—
19	C₃×#125	—	2	1	no	—
20	C₃×C₂²⋊Q₈	—	2	1	no	—
21	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₃×C₄×Q₈	—	2	1	no	—
23	C₃×C₄×Q₈	C₂²	1	2	yes	complemented
24	C₃×#117	—	2	1	no	—
25	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊C₈	C₂²	1	2	yes	complemented
27	C₃×#102	C₂²	1	8	yes	complemented
28	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	2	yes	complemented
30	#533	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#127	C₂×C₃	1	1	yes	characteristic
32	#492	C₂×C₃	1	2	yes	complemented
33	#486	C₂×C₃	1	2	yes	complemented
34	#460	C₂×C₃	1	4	yes	complemented
35	#450	C₂×C₃	1	4	yes	complemented
36	#431	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂⁴×C₃	—	2	2	no	—
38	C₂×C₃×Q₈	—	2	1	no	—
39	C₂×C₃×Q₈	C₂³	1	2	yes	—
40	C₂×C₃×D₄	—	2	2	no	—
41	C₂×C₃×D₄	—	2	2	no	—
42	C₂×C₃×D₄	—	2	2	no	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	—	4	4	no	—
46	C₂×C₃×D₄	—	4	4	no	—
47	C₂×C₃×D₄	—	4	4	no	—
48	C₂×C₃×D₄	C₂³	1	4	yes	—
49	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
50	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
52	C₃×SD₁₆	—	4	8	no	—
53	C₂×C₃×C₈	C₂³	1	4	yes	—
54	C₃×C₄⋊C₄	—	2	1	no	—
55	C₃×C₄⋊C₄	—	2	1	no	—
56	C₃×C₄⋊C₄	—	2	1	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	2	2	no	—
59	C₃×C₄⋊C₄	C₂³	1	2	yes	—
60	C₃×C₂²⋊C₄	—	2	2	no	—
61	C₃×C₄²	—	2	2	no	—
62	C₃×C₄²	D₄	1	1	yes	characteristic
63	C₃×C₄²	D₄	1	1	yes	characteristic
64	C₃×C₄²	C₂³	1	1	yes	characteristic
65	C₃×C₄²	C₂³	1	1	yes	characteristic
66	C₂²×D₄	—	2	1	no	—
67	C₂²×D₄	—	2	1	no	—
68	C₂²×D₄	C₂²×C₃	1	2	yes	—
69	C₂×SD₁₆	C₂²×C₃	1	8	yes	complemented
70	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
72	#127	—	2	1	no	—
73	#127	—	2	1	no	—
74	#127	C₂²×C₃	1	2	yes	—
75	#127	C₂²×C₃	1	2	yes	—
76	#125	—	2	1	no	—
77	C₂²⋊Q₈	—	2	1	no	—
78	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
79	C₄×Q₈	—	2	1	no	—
80	C₄×Q₈	C₂²×C₃	1	2	yes	complemented
81	#117	—	2	1	no	—
82	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
84	#102	C₂²×C₃	1	8	yes	complemented
85	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
86	C₄×C₈	C₂²×C₃	1	2	yes	complemented
87	C₂³×C₃	—	2	4	no	—
88	C₂³×C₃	—	4	4	no	—
89	C₂³×C₃	—	4	4	no	—
90	C₂³×C₃	—	4	4	no	—
91	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
92	C₃×Q₈	—	2	4	no	—
93	C₃×Q₈	—	4	2	no	—
94	C₃×D₄	—	2	8	no	—
95	C₃×D₄	—	4	4	no	—
96	C₃×D₄	—	4	4	no	—
97	C₃×D₄	—	4	4	no	—
98	C₃×D₄	—	4	4	no	—
99	C₃×D₄	—	4	4	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	SD₁₆	1	4	yes	complemented
105	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
111	C₃×C₈	—	2	4	no	—
112	C₂⁴	—	2	2	no	—
113	C₂×Q₈	—	2	1	no	—
114	C₂×Q₈	C₂³×C₃	1	2	yes	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	—	2	2	no	—
119	C₂×D₄	—	2	2	no	—
120	C₂×D₄	—	4	4	no	—
121	C₂×D₄	—	4	4	no	—
122	C₂×D₄	—	4	4	no	—
123	C₂×D₄	C₂³×C₃	1	4	yes	—
124	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
125	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
126	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
127	SD₁₆	—	4	8	no	—
128	C₂×C₈	C₂³×C₃	1	4	yes	—
129	C₄⋊C₄	—	2	1	no	—
130	C₄⋊C₄	—	2	1	no	—
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	C₂³×C₃	1	2	yes	—
135	C₂²⋊C₄	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	C₃×D₄	1	1	yes	characteristic
138	C₄²	C₃×D₄	1	1	yes	characteristic
139	C₄²	C₂³×C₃	1	1	yes	characteristic
140	C₄²	C₂³×C₃	1	1	yes	characteristic
141	C₂²×C₃	—	2	1	no	—
142	C₂²×C₃	—	2	1	no	—
143	C₂²×C₃	—	4	4	no	—
144	C₂²×C₃	—	4	4	no	—
145	C₂²×C₃	—	4	4	no	—
146	C₂²×C₃	—	8	8	no	—
147	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
148	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
149	C₃×C₄	—	2	1	no	—
150	C₃×C₄	—	2	1	no	—
151	C₃×C₄	—	4	2	no	—
152	C₃×C₄	—	4	2	no	—
153	C₃×C₄	C₂×SD₁₆	1	2	yes	—
154	C₃×C₄	C₂×SD₁₆	1	2	yes	—
155	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
156	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
157	C₂³	—	2	4	no	—
158	C₂³	—	4	4	no	—
159	C₂³	—	4	4	no	—
160	C₂³	—	4	4	no	—
161	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
162	Q₈	—	2	4	no	—
163	Q₈	—	4	2	no	—
164	D₄	—	2	8	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	4	no	—
169	D₄	—	4	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	C₃×SD₁₆	1	4	yes	complemented
175	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
179	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
180	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
181	C₈	—	2	4	no	—
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	—	8	4	no	—
184	C₂×C₃	#535	1	1	yes	characteristic
185	C₂×C₃	C₂²×SD₁₆	1	1	yes	characteristic
186	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	4	4	no	—
190	C₂²	—	4	4	no	—
191	C₂²	—	4	4	no	—
192	C₂²	—	8	8	no	—
193	C₂²	C₂×C₃×SD₁₆	1	2	yes	—
194	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	C₂×C₃×SD₁₆	1	2	yes	—
200	C₄	C₂×C₃×SD₁₆	1	2	yes	—
201	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
202	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
203	C₃	#3208	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	—	2	2	no	—
205	C₂	—	8	4	no	—
206	C₂	C₃×#535	1	1	yes	characteristic
207	C₂	C₂²×C₃×SD₁₆	1	1	yes	characteristic
208	C₂	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#533	C₂	semidirect
3. C₃×#492	C₂	semidirect
4. C₃×#486	C₂	semidirect
5. C₃×#460	C₂	semidirect
6. C₃×#450	C₂	semidirect
7. C₃×#431	C₂	semidirect
8. #3208	C₃	direct (N × H)
12. C₂×C₃×SD₁₆	C₂²	semidirect
13. C₃×C₄⋊Q₈	C₂²	semidirect
14. C₃×C₄⋊Q₈	C₂²	semidirect
23. C₃×C₄×Q₈	C₂²	semidirect
26. C₃×C₄⋊C₈	C₂²	semidirect
27. C₃×#102	C₂²	semidirect
29. C₃×C₄×C₈	C₂²	semidirect
30. #533	C₂×C₃	semidirect
32. #492	C₂×C₃	semidirect
33. #486	C₂×C₃	semidirect
34. #460	C₂×C₃	semidirect
35. #450	C₂×C₃	semidirect
36. #431	C₂×C₃	semidirect
69. C₂×SD₁₆	C₂²×C₃	semidirect
70. C₄⋊Q₈	C₂²×C₃	semidirect
71. C₄⋊Q₈	C₂²×C₃	semidirect
80. C₄×Q₈	C₂²×C₃	semidirect
83. C₄⋊C₈	C₂²×C₃	semidirect
84. #102	C₂²×C₃	semidirect
86. C₄×C₈	C₂²×C₃	semidirect
104. C₂×C₃×C₄	SD₁₆	semidirect
174. C₂×C₄	C₃×SD₁₆	semidirect
203. C₃	#3208	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×#533	2
C₂×C₃×#127	2
C₃×#492	2
C₃×#486	2
C₃×#460	2
C₃×#450	2
C₃×#431	2
#3208	3

Subgroups — C3×#3208 #85072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3208 #85072