Subgroups — C₃×#3126 #84990

Order 384.

382 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	C₃×#540	C₂	1	1	yes	maximal, characteristic
2	C₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#482	C₂	1	2	yes	maximal, complemented
4	C₃×#479	C₂	1	2	yes	maximal, complemented
5	C₃×#460	C₂	1	2	yes	maximal, complemented
6	C₃×#448	C₂	1	2	yes	maximal, complemented
7	C₃×#440	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×SD₁₆	C₂	1	2	yes	maximal, complemented
9	C₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
10	#3126	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
14	C₃×#127	C₂²	1	1	yes	characteristic
15	C₃×#125	C₂²	1	1	yes	characteristic
16	C₃×#123	—	2	2	no	—
17	C₃×#121	—	2	2	no	—
18	C₃×#121	C₂²	1	2	yes	—
19	C₃×C₄×Q₈	—	2	2	no	—
20	C₃×C₄×Q₈	C₂²	1	2	yes	—
21	C₃×C₄×Q₈	C₂²	1	2	yes	—
22	C₃×C₄×D₄	C₂²	1	2	yes	—
23	C₃×#117	—	2	2	no	—
24	C₃×#117	C₂²	1	1	yes	characteristic
25	C₃×#117	C₂²	1	1	yes	characteristic
26	C₃×#107	C₂²	1	2	yes	—
27	C₃×#106	C₂²	1	2	yes	—
28	C₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₃×#103	C₂²	1	2	yes	—
30	C₃×#103	C₂²	1	2	yes	—
31	C₃×#102	C₂²	1	2	yes	—
32	C₃×#102	C₂²	1	2	yes	—
33	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₃×#97	C₂²	1	1	yes	characteristic
35	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
36	#540	C₂×C₃	1	1	yes	characteristic
37	#519	C₂×C₃	1	1	yes	characteristic, complemented
38	#482	C₂×C₃	1	2	yes	complemented
39	#479	C₂×C₃	1	2	yes	complemented
40	#460	C₂×C₃	1	2	yes	complemented
41	#448	C₂×C₃	1	2	yes	complemented
42	#440	C₂×C₃	1	2	yes	complemented
43	C₄×SD₁₆	C₂×C₃	1	2	yes	complemented
44	#433	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×Q₈	—	2	4	no	—
46	C₂×C₃×Q₈	—	2	1	no	—
47	C₂×C₃×Q₈	C₂³	1	2	yes	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	—	2	2	no	—
50	C₂×C₃×D₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	—	2	1	no	—
52	C₂²×C₃×C₄	—	2	2	no	—
53	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
54	C₃×SD₁₆	—	2	4	no	—
55	C₃×SD₁₆	—	2	4	no	—
56	C₂×C₃×C₈	C₂³	1	2	yes	—
57	C₂×C₃×C₈	C₂³	1	2	yes	—
58	C₃×C₄⋊C₄	—	2	2	no	—
59	C₃×C₄⋊C₄	—	2	2	no	—
60	C₃×C₄⋊C₄	—	2	2	no	—
61	C₃×C₄⋊C₄	D₄	1	2	yes	—
62	C₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₃×C₂²⋊C₄	—	2	1	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	—	2	1	no	—
71	C₃×C₄²	C₂³	1	1	yes	characteristic
72	C₃×C₄²	C₂³	1	1	yes	characteristic
73	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
75	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
76	#127	C₂²×C₃	1	1	yes	characteristic
77	#125	C₂²×C₃	1	1	yes	characteristic
78	#123	—	2	2	no	—
79	#121	—	2	2	no	—
80	#121	C₂²×C₃	1	2	yes	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	C₂²×C₃	1	2	yes	—
83	C₄×Q₈	C₂²×C₃	1	2	yes	—
84	C₄×D₄	C₂²×C₃	1	2	yes	—
85	#117	—	2	2	no	—
86	#117	C₂²×C₃	1	1	yes	characteristic
87	#117	C₂²×C₃	1	1	yes	characteristic
88	#107	C₂²×C₃	1	2	yes	—
89	#106	C₂²×C₃	1	2	yes	—
90	C₄⋊C₈	C₂²×C₃	1	2	yes	—
91	#103	C₂²×C₃	1	2	yes	—
92	#103	C₂²×C₃	1	2	yes	—
93	#102	C₂²×C₃	1	2	yes	—
94	#102	C₂²×C₃	1	2	yes	—
95	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
96	#97	C₂²×C₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
98	C₂³×C₃	—	2	2	no	—
99	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
100	C₃×Q₈	—	2	4	no	—
101	C₃×Q₈	—	2	2	no	—
102	C₃×Q₈	#41	1	4	yes	—
103	C₃×D₄	—	2	2	no	—
104	C₃×D₄	—	2	2	no	—
105	C₃×D₄	—	8	2	no	—
106	C₃×D₄	—	8	2	no	—
107	C₂×C₃×C₄	—	2	4	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	8	2	no	—
113	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
114	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
115	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
117	C₃×C₈	—	2	2	no	—
118	C₃×C₈	—	2	2	no	—
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	2	1	no	—
121	C₂×Q₈	C₂³×C₃	1	2	yes	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	C₂³×C₃	1	2	yes	—
125	C₂²×C₄	—	2	1	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
128	SD₁₆	—	2	4	no	—
129	SD₁₆	—	2	4	no	—
130	C₂×C₈	C₂³×C₃	1	2	yes	—
131	C₂×C₈	C₂³×C₃	1	2	yes	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄⋊C₄	—	2	2	no	—
135	C₄⋊C₄	C₃×D₄	1	2	yes	—
136	C₄⋊C₄	C₂³×C₃	1	2	yes	—
137	C₄⋊C₄	C₂³×C₃	1	2	yes	—
138	C₂²⋊C₄	—	2	1	no	—
139	C₂²⋊C₄	—	2	2	no	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	C₃×D₄	1	2	yes	—
142	C₄²	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	—	2	1	no	—
145	C₄²	C₂³×C₃	1	1	yes	characteristic
146	C₄²	C₂³×C₃	1	1	yes	characteristic
147	C₂²×C₃	—	2	1	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	4	2	no	—
150	C₂²×C₃	—	4	2	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
153	C₃×C₄	—	2	4	no	—
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	4	2	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	C₂×#41	1	2	yes	—
160	C₂³	—	2	2	no	—
161	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
162	Q₈	—	2	4	no	—
163	Q₈	—	2	2	no	—
164	Q₈	C₃×#41	1	4	yes	—
165	D₄	—	2	2	no	—
166	D₄	—	2	2	no	—
167	D₄	—	8	2	no	—
168	D₄	—	8	2	no	—
169	C₂×C₄	—	2	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	—	8	2	no	—
175	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
177	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₂×C₃	—	4	1	no	—
182	C₂×C₃	—	8	2	no	—
183	C₂×C₃	#525	1	1	yes	characteristic
184	C₂×C₃	#577	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	C₂×C₃×#41	1	2	yes	—
198	C₃	#3126	1	1	yes	characteristic, direct product factor, Sylow
199	C₂	—	4	1	no	—
200	C₂	—	8	2	no	—
201	C₂	C₃×#525	1	1	yes	characteristic
202	C₂	C₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#540	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#519	C₂	semidirect
3. C₃×#482	C₂	semidirect
4. C₃×#479	C₂	semidirect
5. C₃×#460	C₂	semidirect
6. C₃×#448	C₂	semidirect
7. C₃×#440	C₂	semidirect
8. C₃×C₄×SD₁₆	C₂	semidirect
9. C₃×#433	C₂	semidirect
10. #3126	C₃	direct (N × H)
37. #519	C₂×C₃	semidirect
38. #482	C₂×C₃	semidirect
39. #479	C₂×C₃	semidirect
40. #460	C₂×C₃	semidirect
41. #448	C₂×C₃	semidirect
42. #440	C₂×C₃	semidirect
43. C₄×SD₁₆	C₂×C₃	semidirect
44. #433	C₂×C₃	semidirect
198. C₃	#3126	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×#540	2
C₃×#519	2
C₃×#482	2
C₃×#479	2
C₃×#460	2
C₃×#448	2
C₃×#440	2
C₃×C₄×SD₁₆	2
C₃×#433	2
#3126	3

Subgroups — C3×#3126 #84990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3126 #84990