Subgroups — C₃×#3250 #85114

Order 384.

360 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#529	C₂	1	2	yes	maximal, complemented
2	C₃×#479	C₂	1	2	yes	maximal, complemented
3	C₃×#478	C₂	1	2	yes	maximal, complemented
4	C₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#466	C₂	1	1	yes	maximal, characteristic
6	C₃×#465	C₂	1	2	yes	maximal, complemented
7	C₃×#464	C₂	1	2	yes	maximal, complemented
8	C₃×#463	C₂	1	2	yes	maximal, complemented
9	C₂×C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#3250	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×D₈	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
16	C₃×#126	—	2	2	no	—
17	C₃×#125	C₂²	1	2	yes	—
18	C₃×#124	C₂²	1	2	yes	—
19	C₃×#123	—	2	2	no	—
20	C₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₃×#121	C₂²	1	2	yes	—
22	C₃×C₄×Q₈	C₂²	1	2	yes	—
23	C₃×C₄×D₄	—	2	2	no	—
24	C₃×C₄×D₄	C₂²	1	2	yes	—
25	C₃×#117	—	2	2	no	—
26	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
27	C₃×#107	C₂²	1	1	yes	characteristic
28	C₃×#107	C₂²	1	1	yes	characteristic
29	C₃×#106	C₂²	1	2	yes	—
30	C₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₃×C₄⋊C₈	C₂²	1	2	yes	—
32	C₃×#103	C₂²	1	2	yes	—
33	C₃×#103	C₂²	1	2	yes	—
34	C₃×#102	C₂²	1	2	yes	—
35	C₃×#102	C₂²	1	2	yes	—
36	#529	C₂×C₃	1	2	yes	complemented
37	#479	C₂×C₃	1	2	yes	complemented
38	#478	C₂×C₃	1	2	yes	complemented
39	#467	C₂×C₃	1	1	yes	characteristic, complemented
40	#466	C₂×C₃	1	1	yes	characteristic
41	#465	C₂×C₃	1	2	yes	complemented
42	#464	C₂×C₃	1	2	yes	complemented
43	#463	C₂×C₃	1	2	yes	complemented
44	C₂×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
45	C₂×C₃×Q₈	C₂³	1	2	yes	—
46	C₂×C₃×D₄	—	2	2	no	—
47	C₂×C₃×D₄	C₂³	1	2	yes	—
48	C₂²×C₃×C₄	—	2	2	no	—
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×Q₁₆	—	4	1	no	—
53	C₃×SD₁₆	—	4	2	no	—
54	C₃×D₈	—	4	1	no	—
55	C₂×C₃×C₈	—	4	1	no	—
56	C₂×C₃×C₈	—	4	1	no	—
57	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₈	C₂³	1	2	yes	—
59	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
60	C₃×C₄⋊C₄	—	2	2	no	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	—	2	2	no	—
63	C₃×C₄⋊C₄	—	2	2	no	—
64	C₃×C₄⋊C₄	C₂³	1	2	yes	—
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×C₄²	—	2	2	no	—
72	C₃×C₄²	D₄	1	2	yes	—
73	C₃×C₄²	C₂³	1	2	yes	—
74	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
75	C₂×SD₁₆	C₂²×C₃	1	2	yes	—
76	C₂×D₈	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
78	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
79	#126	—	2	2	no	—
80	#125	C₂²×C₃	1	2	yes	—
81	#124	C₂²×C₃	1	2	yes	—
82	#123	—	2	2	no	—
83	C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
84	#121	C₂²×C₃	1	2	yes	—
85	C₄×Q₈	C₂²×C₃	1	2	yes	—
86	C₄×D₄	—	2	2	no	—
87	C₄×D₄	C₂²×C₃	1	2	yes	—
88	#117	—	2	2	no	—
89	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
90	#107	C₂²×C₃	1	1	yes	characteristic
91	#107	C₂²×C₃	1	1	yes	characteristic
92	#106	C₂²×C₃	1	2	yes	—
93	C₄⋊C₈	C₂²×C₃	1	2	yes	—
94	C₄⋊C₈	C₂²×C₃	1	2	yes	—
95	#103	C₂²×C₃	1	2	yes	—
96	#103	C₂²×C₃	1	2	yes	—
97	#102	C₂²×C₃	1	2	yes	—
98	#102	C₂²×C₃	1	2	yes	—
99	C₂³×C₃	—	2	2	no	—
100	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
101	C₃×Q₈	—	2	2	no	—
102	C₃×Q₈	—	2	2	no	—
103	C₃×D₄	—	2	2	no	—
104	C₃×D₄	—	2	2	no	—
105	C₃×D₄	—	4	4	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	2	1	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	—	4	1	no	—
113	C₂×C₃×C₄	—	4	4	no	—
114	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
115	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
117	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
118	C₃×C₈	—	2	1	no	—
119	C₃×C₈	—	2	2	no	—
120	C₃×C₈	—	2	1	no	—
121	C₂×Q₈	C₂³×C₃	1	2	yes	—
122	C₂×D₄	—	2	2	no	—
123	C₂×D₄	C₂³×C₃	1	2	yes	—
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
126	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
128	Q₁₆	—	4	1	no	—
129	SD₁₆	—	4	2	no	—
130	D₈	—	4	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
134	C₂×C₈	C₂³×C₃	1	2	yes	—
135	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	2	no	—
139	C₄⋊C₄	—	2	2	no	—
140	C₄⋊C₄	C₂³×C₃	1	2	yes	—
141	C₄⋊C₄	C₂³×C₃	1	2	yes	—
142	C₂²⋊C₄	—	2	2	no	—
143	C₂²⋊C₄	—	2	2	no	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	C₃×D₄	1	2	yes	—
149	C₄²	C₂³×C₃	1	2	yes	—
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	—	2	1	no	—
152	C₂²×C₃	—	4	2	no	—
153	C₂²×C₃	—	4	2	no	—
154	C₂²×C₃	—	4	2	no	—
155	C₂²×C₃	#135	1	2	yes	—
156	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	4	2	no	—
161	C₃×C₄	—	4	2	no	—
162	C₃×C₄	—	4	2	no	—
163	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
164	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
165	C₂³	—	2	2	no	—
166	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
167	Q₈	—	2	2	no	—
168	Q₈	—	2	2	no	—
169	D₄	—	2	2	no	—
170	D₄	—	2	2	no	—
171	D₄	—	4	4	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	2	no	—
186	C₈	—	2	1	no	—
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	—	8	2	no	—
189	C₂×C₃	#537	1	1	yes	characteristic
190	C₂×C₃	C₂×#135	1	1	yes	characteristic
191	C₂×C₃	#575	1	1	yes	characteristic
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₃×#135	1	2	yes	—
198	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	1	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
206	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
207	C₃	#3250	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	C₃×#537	1	1	yes	characteristic
211	C₂	C₂×C₃×#135	1	1	yes	characteristic
212	C₂	C₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃
prime radical	C₃×#466	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×#529	C₂	semidirect
2. C₃×#479	C₂	semidirect
3. C₃×#478	C₂	semidirect
4. C₃×#467	C₂	semidirect
6. C₃×#465	C₂	semidirect
7. C₃×#464	C₂	semidirect
8. C₃×#463	C₂	semidirect
9. C₂×C₃×C₄⋊C₈	C₂	semidirect
10. #3250	C₃	direct (N × H)
36. #529	C₂×C₃	semidirect
37. #479	C₂×C₃	semidirect
38. #478	C₂×C₃	semidirect
39. #467	C₂×C₃	semidirect
41. #465	C₂×C₃	semidirect
42. #464	C₂×C₃	semidirect
43. #463	C₂×C₃	semidirect
44. C₂×C₄⋊C₈	C₂×C₃	semidirect
207. C₃	#3250	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×#529	2
C₃×#479	2
C₃×#478	2
C₃×#467	2
C₃×#466	2
C₃×#465	2
C₃×#464	2
C₃×#463	2
C₂×C₃×C₄⋊C₈	2
#3250	3

Subgroups — C3×#3250 #85114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3250 #85114