Subgroups — C₃×#3403 #85267

Order 384.

380 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#559	C₂	1	2	yes	maximal, complemented
2	C₃×#496	C₂	1	2	yes	maximal, complemented
3	C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#464	C₂	1	2	yes	maximal, complemented
5	C₃×#463	C₂	1	2	yes	maximal, complemented
6	C₃×#459	C₂	1	2	yes	maximal, complemented
7	C₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#440	C₂	1	2	yes	maximal, complemented
9	C₃×C₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#3403	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₃×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₈	C₂²	1	2	yes	—
17	C₃×#127	C₂²	1	2	yes	—
18	C₃×#124	—	2	2	no	—
19	C₃×#124	C₂²	1	2	yes	—
20	C₃×#124	C₂²	1	2	yes	—
21	C₃×#123	—	2	2	no	—
22	C₃×#123	—	2	2	no	—
23	C₃×C₄×Q₈	C₂²	1	2	yes	—
24	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
25	C₃×C₄×D₄	—	2	2	no	—
26	C₃×C₄×D₄	C₂²	1	2	yes	—
27	C₃×C₄×D₄	C₂²	1	2	yes	—
28	C₃×#107	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃×#103	C₂²	1	2	yes	—
32	C₃×#102	C₂²	1	2	yes	—
33	C₃×#102	C₂²	1	2	yes	—
34	C₃×#97	C₂²	1	2	yes	—
35	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
36	#559	C₂×C₃	1	2	yes	complemented
37	#496	C₂×C₃	1	2	yes	complemented
38	#495	C₂×C₃	1	1	yes	characteristic, complemented
39	#464	C₂×C₃	1	2	yes	complemented
40	#463	C₂×C₃	1	2	yes	complemented
41	#459	C₂×C₃	1	2	yes	complemented
42	#446	C₂×C₃	1	1	yes	characteristic, complemented
43	#440	C₂×C₃	1	2	yes	complemented
44	C₄×D₈	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
46	C₂×C₃×Q₈	C₂³	1	2	yes	—
47	C₂×C₃×D₄	—	2	2	no	—
48	C₂×C₃×D₄	—	2	2	no	—
49	C₂×C₃×D₄	C₂³	1	2	yes	—
50	C₂×C₃×D₄	C₂³	1	2	yes	—
51	C₂²×C₃×C₄	—	2	2	no	—
52	C₂²×C₃×C₄	—	2	2	no	—
53	C₃×Q₁₆	—	4	1	no	—
54	C₃×SD₁₆	—	2	4	no	—
55	C₃×SD₁₆	—	4	2	no	—
56	C₃×D₈	—	2	2	no	—
57	C₃×D₈	—	4	2	no	—
58	C₃×D₈	—	4	1	no	—
59	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂³	1	2	yes	—
61	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	D₄	1	2	yes	—
64	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₂²⋊C₄	—	2	2	no	—
69	C₃×C₂²⋊C₄	—	2	2	no	—
70	C₃×C₂²⋊C₄	—	2	2	no	—
71	C₃×C₂²⋊C₄	—	2	2	no	—
72	C₃×C₄²	—	2	2	no	—
73	C₃×C₄²	C₂³	1	1	yes	characteristic
74	C₃×C₄²	C₂³	1	2	yes	—
75	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
77	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
78	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
79	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
80	C₂×D₈	C₂²×C₃	1	2	yes	—
81	#127	C₂²×C₃	1	2	yes	—
82	#124	—	2	2	no	—
83	#124	C₂²×C₃	1	2	yes	—
84	#124	C₂²×C₃	1	2	yes	—
85	#123	—	2	2	no	—
86	#123	—	2	2	no	—
87	C₄×Q₈	C₂²×C₃	1	2	yes	—
88	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
89	C₄×D₄	—	2	2	no	—
90	C₄×D₄	C₂²×C₃	1	2	yes	—
91	C₄×D₄	C₂²×C₃	1	2	yes	—
92	#107	C₂²×C₃	1	1	yes	characteristic
93	C₄⋊C₈	C₂²×C₃	1	2	yes	—
94	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
95	#103	C₂²×C₃	1	2	yes	—
96	#102	C₂²×C₃	1	2	yes	—
97	#102	C₂²×C₃	1	2	yes	—
98	#97	C₂²×C₃	1	2	yes	—
99	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
100	C₂³×C₃	—	2	2	no	—
101	C₂³×C₃	—	2	2	no	—
102	C₃×Q₈	—	2	2	no	—
103	C₃×Q₈	—	2	2	no	—
104	C₃×Q₈	—	4	1	no	—
105	C₃×D₄	—	2	2	no	—
106	C₃×D₄	—	2	2	no	—
107	C₃×D₄	—	2	2	no	—
108	C₃×D₄	—	2	2	no	—
109	C₃×D₄	—	4	4	no	—
110	C₃×D₄	—	8	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	8	2	no	—
114	C₂×C₃×C₄	—	8	2	no	—
115	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
116	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
119	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
120	C₃×C₈	—	2	2	no	—
121	C₃×C₈	—	2	1	no	—
122	C₃×C₈	#41	1	2	yes	—
123	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
124	C₂×Q₈	C₂³×C₃	1	2	yes	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	C₂³×C₃	1	2	yes	—
128	C₂×D₄	C₂³×C₃	1	2	yes	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	Q₁₆	—	4	1	no	—
132	SD₁₆	—	2	4	no	—
133	SD₁₆	—	4	2	no	—
134	D₈	—	2	2	no	—
135	D₈	—	4	2	no	—
136	D₈	—	4	1	no	—
137	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
138	C₂×C₈	C₂³×C₃	1	2	yes	—
139	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	C₃×D₄	1	2	yes	—
142	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
143	C₄⋊C₄	C₂³×C₃	1	2	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₂²⋊C₄	—	2	2	no	—
148	C₂²⋊C₄	—	2	2	no	—
149	C₂²⋊C₄	—	2	2	no	—
150	C₄²	—	2	2	no	—
151	C₄²	C₂³×C₃	1	1	yes	characteristic
152	C₄²	C₂³×C₃	1	2	yes	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	—	4	2	no	—
156	C₂²×C₃	—	4	2	no	—
157	C₂²×C₃	—	4	2	no	—
158	C₂²×C₃	—	4	2	no	—
159	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	4	2	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	C₂×#41	1	1	yes	characteristic
167	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	Q₈	—	4	1	no	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	4	4	no	—
178	D₄	—	8	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	8	2	no	—
182	C₂×C₄	—	8	2	no	—
183	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
184	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
187	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	2	1	no	—
190	C₈	C₃×#41	1	2	yes	—
191	C₂×C₃	—	8	2	no	—
192	C₂×C₃	—	8	2	no	—
193	C₂×C₃	#550	1	1	yes	characteristic
194	C₂×C₃	#575	1	1	yes	characteristic
195	C₂×C₃	#576	1	1	yes	characteristic
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	C₂×C₃×#41	1	1	yes	characteristic
210	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
211	C₃	#3403	1	1	yes	characteristic, direct product factor, Sylow
212	C₂	—	8	2	no	—
213	C₂	—	8	2	no	—
214	C₂	C₃×#550	1	1	yes	characteristic
215	C₂	C₃×#575	1	1	yes	characteristic
216	C₂	C₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#559	C₂	semidirect
2. C₃×#496	C₂	semidirect
3. C₃×#495	C₂	semidirect
4. C₃×#464	C₂	semidirect
5. C₃×#463	C₂	semidirect
6. C₃×#459	C₂	semidirect
7. C₃×#446	C₂	semidirect
8. C₃×#440	C₂	semidirect
9. C₃×C₄×D₈	C₂	semidirect
10. #3403	C₃	direct (N × H)
36. #559	C₂×C₃	semidirect
37. #496	C₂×C₃	semidirect
38. #495	C₂×C₃	semidirect
39. #464	C₂×C₃	semidirect
40. #463	C₂×C₃	semidirect
41. #459	C₂×C₃	semidirect
42. #446	C₂×C₃	semidirect
43. #440	C₂×C₃	semidirect
44. C₄×D₈	C₂×C₃	semidirect
211. C₃	#3403	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×#559	2
C₃×#496	2
C₃×#495	2
C₃×#464	2
C₃×#463	2
C₃×#459	2
C₃×#446	2
C₃×#440	2
C₃×C₄×D₈	2
#3403	3

Subgroups — C3×#3403 #85267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3403 #85267