Subgroups — C₃×#3205 #85069

Order 384.

532 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#493	C₂	1	2	yes	maximal, complemented
4	C₃×#486	C₂	1	2	yes	maximal, complemented
5	C₃×#459	C₂	1	4	yes	maximal, complemented
6	C₃×#447	C₂	1	4	yes	maximal, complemented
7	C₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	#3205	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#41	—	2	1	no	—
10	C₂²×C₃×D₄	—	2	1	no	—
11	C₂²×C₃×D₄	—	2	1	no	—
12	C₂²×C₃×D₄	C₂²	1	2	yes	—
13	C₂×C₃×D₈	C₂²	1	8	yes	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×#127	C₂²	1	1	yes	characteristic, complemented
20	C₃×#124	—	2	1	no	—
21	C₃×#121	—	2	1	no	—
22	C₃×#121	C₂²	1	2	yes	—
23	C₃×C₄×D₄	—	2	1	no	—
24	C₃×C₄×D₄	C₂²	1	2	yes	complemented
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	#532	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×#127	C₂×C₃	1	1	yes	characteristic, complemented
32	#493	C₂×C₃	1	2	yes	complemented
33	#486	C₂×C₃	1	2	yes	complemented
34	#459	C₂×C₃	1	4	yes	complemented
35	#447	C₂×C₃	1	4	yes	complemented
36	#431	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂⁴×C₃	—	2	2	no	—
38	C₃×#41	—	4	4	no	—
39	C₂×C₃×Q₈	—	2	1	no	—
40	C₂×C₃×D₄	—	2	2	no	—
41	C₂×C₃×D₄	—	2	1	no	—
42	C₂×C₃×D₄	—	2	1	no	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	—	2	2	no	—
46	C₂×C₃×D₄	—	2	2	no	—
47	C₂×C₃×D₄	—	2	2	no	—
48	C₂×C₃×D₄	—	4	4	no	—
49	C₂×C₃×D₄	—	4	4	no	—
50	C₂×C₃×D₄	—	4	4	no	—
51	C₂×C₃×D₄	C₂³	1	4	yes	—
52	C₂×C₃×D₄	C₂³	1	2	yes	—
53	C₂²×C₃×C₄	—	2	2	no	—
54	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
56	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
57	C₃×D₈	—	4	8	no	—
58	C₂×C₃×C₈	C₂³	1	4	yes	—
59	C₃×C₄⋊C₄	—	2	1	no	—
60	C₃×C₄⋊C₄	C₂³	1	2	yes	—
61	C₃×C₂²⋊C₄	—	2	2	no	—
62	C₃×C₂²⋊C₄	—	2	2	no	—
63	C₃×C₄²	D₄	1	1	yes	characteristic
64	C₃×C₄²	D₄	1	1	yes	characteristic
65	C₃×C₄²	C₂³	1	1	yes	characteristic
66	C₃×C₄²	C₂³	1	1	yes	characteristic
67	C₂×#41	—	2	1	no	—
68	C₂²×D₄	—	2	1	no	—
69	C₂²×D₄	—	2	1	no	—
70	C₂²×D₄	C₂²×C₃	1	2	yes	—
71	C₂×D₈	C₂²×C₃	1	8	yes	complemented
72	C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
73	#127	—	2	1	no	—
74	#127	—	2	1	no	—
75	#127	C₂²×C₃	1	2	yes	—
76	#127	C₂²×C₃	1	2	yes	—
77	#127	C₂²×C₃	1	1	yes	characteristic, complemented
78	#124	—	2	1	no	—
79	#121	—	2	1	no	—
80	#121	C₂²×C₃	1	2	yes	—
81	C₄×D₄	—	2	1	no	—
82	C₄×D₄	C₂²×C₃	1	2	yes	complemented
83	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₄⋊C₈	C₂²×C₃	1	2	yes	complemented
85	#102	C₂²×C₃	1	8	yes	complemented
86	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
87	C₄×C₈	C₂²×C₃	1	2	yes	complemented
88	C₂³×C₃	—	2	2	no	—
89	C₂³×C₃	—	2	4	no	—
90	C₂³×C₃	—	4	4	no	—
91	C₂³×C₃	—	4	4	no	—
92	C₂³×C₃	—	4	4	no	—
93	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
94	C₃×Q₈	—	4	2	no	—
95	C₃×D₄	—	2	4	no	—
96	C₃×D₄	—	2	8	no	—
97	C₃×D₄	—	4	4	no	—
98	C₃×D₄	—	4	4	no	—
99	C₃×D₄	—	4	4	no	—
100	C₃×D₄	—	4	4	no	—
101	C₃×D₄	—	4	4	no	—
102	C₃×D₄	—	4	2	no	—
103	C₃×D₄	—	4	2	no	—
104	C₃×D₄	—	4	4	no	—
105	C₂×C₃×C₄	—	2	2	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	4	4	no	—
109	C₂×C₃×C₄	D₈	1	4	yes	complemented
110	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
116	C₃×C₈	—	2	4	no	—
117	C₂⁴	—	2	2	no	—
118	#41	—	4	4	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×D₄	—	2	2	no	—
121	C₂×D₄	—	2	1	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	—	2	2	no	—
124	C₂×D₄	—	2	2	no	—
125	C₂×D₄	—	2	2	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	—	2	2	no	—
128	C₂×D₄	—	4	4	no	—
129	C₂×D₄	—	4	4	no	—
130	C₂×D₄	—	4	4	no	—
131	C₂×D₄	C₂³×C₃	1	4	yes	—
132	C₂×D₄	C₂³×C₃	1	2	yes	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
136	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
137	D₈	—	4	8	no	—
138	C₂×C₈	C₂³×C₃	1	4	yes	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	C₂³×C₃	1	2	yes	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₂²⋊C₄	—	2	2	no	—
143	C₄²	C₃×D₄	1	1	yes	characteristic
144	C₄²	C₃×D₄	1	1	yes	characteristic
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	1	no	—
149	C₂²×C₃	—	4	4	no	—
150	C₂²×C₃	—	4	2	no	—
151	C₂²×C₃	—	4	2	no	—
152	C₂²×C₃	—	4	4	no	—
153	C₂²×C₃	—	4	4	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	—	8	8	no	—
156	C₂²×C₃	C₂×D₈	1	2	yes	—
157	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	4	2	no	—
161	C₃×C₄	C₂×D₈	1	2	yes	—
162	C₃×C₄	C₂×D₈	1	2	yes	—
163	C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
164	C₃×C₄	H(5,𝔽₂)	1	2	yes	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	4	no	—
167	C₂³	—	4	4	no	—
168	C₂³	—	4	4	no	—
169	C₂³	—	4	4	no	—
170	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
171	Q₈	—	4	2	no	—
172	D₄	—	2	4	no	—
173	D₄	—	2	8	no	—
174	D₄	—	4	4	no	—
175	D₄	—	4	4	no	—
176	D₄	—	4	4	no	—
177	D₄	—	4	4	no	—
178	D₄	—	4	4	no	—
179	D₄	—	4	2	no	—
180	D₄	—	4	4	no	—
181	D₄	—	4	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₃×D₈	1	4	yes	complemented
187	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
190	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	—	8	2	no	—
196	C₂×C₃	—	8	4	no	—
197	C₂×C₃	#535	1	1	yes	characteristic
198	C₂×C₃	C₂²×D₈	1	1	yes	characteristic
199	C₂×C₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	—	4	4	no	—
205	C₂²	—	4	2	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	8	8	no	—
209	C₂²	C₂×C₃×D₈	1	2	yes	—
210	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	4	2	no	—
214	C₄	C₂×C₃×D₈	1	2	yes	—
215	C₄	C₂×C₃×D₈	1	2	yes	—
216	C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
217	C₄	C₃×H(5,𝔽₂)	1	2	yes	—
218	C₃	#3205	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	2	2	no	—
220	C₂	—	8	4	no	—
221	C₂	—	8	2	no	—
222	C₂	C₃×#535	1	1	yes	characteristic
223	C₂	C₂²×C₃×D₈	1	1	yes	characteristic
224	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
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×#532	C₂	semidirect
2. C₂×C₃×#127	C₂	semidirect
3. C₃×#493	C₂	semidirect
4. C₃×#486	C₂	semidirect
5. C₃×#459	C₂	semidirect
6. C₃×#447	C₂	semidirect
7. C₃×#431	C₂	semidirect
8. #3205	C₃	direct (N × H)
13. C₂×C₃×D₈	C₂²	semidirect
14. C₃×C₄⋊Q₈	C₂²	semidirect
19. C₃×#127	C₂²	semidirect
24. C₃×C₄×D₄	C₂²	semidirect
26. C₃×C₄⋊C₈	C₂²	semidirect
27. C₃×#102	C₂²	semidirect
29. C₃×C₄×C₈	C₂²	semidirect
30. #532	C₂×C₃	semidirect
31. C₂×#127	C₂×C₃	semidirect
32. #493	C₂×C₃	semidirect
33. #486	C₂×C₃	semidirect
34. #459	C₂×C₃	semidirect
35. #447	C₂×C₃	semidirect
36. #431	C₂×C₃	semidirect
71. C₂×D₈	C₂²×C₃	semidirect
72. C₄⋊Q₈	C₂²×C₃	semidirect
77. #127	C₂²×C₃	semidirect
82. C₄×D₄	C₂²×C₃	semidirect
84. C₄⋊C₈	C₂²×C₃	semidirect
85. #102	C₂²×C₃	semidirect
87. C₄×C₈	C₂²×C₃	semidirect
109. C₂×C₃×C₄	D₈	semidirect
186. C₂×C₄	C₃×D₈	semidirect
218. C₃	#3205	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×#532	2
C₂×C₃×#127	2
C₃×#493	2
C₃×#486	2
C₃×#459	2
C₃×#447	2
C₃×#431	2
#3205	3

Subgroups — C3×#3205 #85069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3205 #85069