Subgroups — C₃×#3287 #85151

Order 384.

504 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#135	C₂	1	2	yes	maximal, complemented
2	C₃×#550	C₂	1	2	yes	maximal, complemented
3	C₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#466	C₂	1	2	yes	maximal, complemented
5	C₃×#460	C₂	1	2	yes	maximal, complemented
6	C₃×#449	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
9	#3287	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#41	C₂²	1	4	yes	—
11	C₃×#135	—	2	8	no	—
12	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×SD₁₆	C₂²	1	4	yes	—
14	C₂×C₃×D₈	C₂²	1	2	yes	—
15	C₂×C₃×D₈	C₂²	1	2	yes	—
16	C₂²×C₃×C₈	C₂²	1	2	yes	—
17	C₃×#127	—	2	2	no	—
18	C₃×#127	C₂²	1	2	yes	—
19	C₃×#121	—	2	4	no	—
20	C₃×#121	C₂²	1	4	yes	—
21	C₃×C₄×Q₈	C₂²	1	2	yes	—
22	C₃×C₄×D₄	—	2	2	no	—
23	C₃×C₄×D₄	C₂²	1	1	yes	characteristic
24	C₃×#107	C₂²	1	1	yes	characteristic
25	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₃×#103	C₂²	1	2	yes	—
27	C₃×#102	C₂²	1	4	yes	—
28	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₂×#135	C₂×C₃	1	2	yes	complemented
31	#550	C₂×C₃	1	2	yes	complemented
32	#493	C₂×C₃	1	1	yes	characteristic, complemented
33	#466	C₂×C₃	1	2	yes	complemented
34	#460	C₂×C₃	1	2	yes	complemented
35	#449	C₂×C₃	1	4	yes	complemented
36	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
37	C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
38	C₃×#41	—	2	8	no	—
39	C₃×#41	—	4	4	no	—
40	C₂×C₃×Q₈	C₂³	1	2	yes	—
41	C₂×C₃×D₄	—	2	4	no	—
42	C₂×C₃×D₄	—	2	4	no	—
43	C₂×C₃×D₄	—	2	2	no	—
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	D₄	1	1	yes	characteristic
46	C₂×C₃×D₄	C₂³	1	4	yes	—
47	C₂²×C₃×C₄	—	2	4	no	—
48	C₂²×C₃×C₄	C₂³	1	2	yes	—
49	C₃×Q₁₆	D₄	1	4	yes	complemented
50	C₃×SD₁₆	—	2	8	no	—
51	C₃×D₈	—	2	4	no	—
52	C₃×D₈	—	4	2	no	—
53	C₂×C₃×C₈	—	2	4	no	—
54	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₃×C₈	C₂³	1	2	yes	—
56	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
59	C₃×C₄⋊C₄	C₂³	1	2	yes	—
60	C₃×C₂²⋊C₄	—	2	4	no	—
61	C₃×C₂²⋊C₄	D₄	1	2	yes	—
62	C₃×C₄²	—	2	2	no	—
63	C₃×C₄²	C₂³	1	1	yes	characteristic
64	C₂×#41	C₂²×C₃	1	4	yes	—
65	#135	—	2	8	no	—
66	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₂×SD₁₆	C₂²×C₃	1	4	yes	—
68	C₂×D₈	C₂²×C₃	1	2	yes	—
69	C₂×D₈	C₂²×C₃	1	2	yes	—
70	C₂²×C₈	C₂²×C₃	1	2	yes	—
71	#127	—	2	2	no	—
72	#127	C₂²×C₃	1	2	yes	—
73	#121	—	2	4	no	—
74	#121	C₂²×C₃	1	4	yes	—
75	C₄×Q₈	C₂²×C₃	1	2	yes	—
76	C₄×D₄	—	2	2	no	—
77	C₄×D₄	C₂²×C₃	1	1	yes	characteristic
78	#107	C₂²×C₃	1	1	yes	characteristic
79	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
80	#103	C₂²×C₃	1	2	yes	—
81	#102	C₂²×C₃	1	4	yes	—
82	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
83	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
84	C₂³×C₃	—	2	4	no	—
85	C₂³×C₃	C₂×D₄	1	2	yes	—
86	C₃×Q₈	—	2	2	no	—
87	C₃×Q₈	C₂×D₄	1	4	yes	—
88	C₃×D₄	—	2	2	no	—
89	C₃×D₄	—	2	4	no	—
90	C₃×D₄	—	2	4	no	—
91	C₃×D₄	—	4	8	no	—
92	C₃×D₄	—	4	4	no	—
93	C₃×D₄	—	8	2	no	—
94	C₃×D₄	—	8	4	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	—	2	2	no	—
97	C₂×C₃×C₄	—	2	2	no	—
98	C₂×C₃×C₄	—	2	2	no	—
99	C₂×C₃×C₄	—	4	8	no	—
100	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
102	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
104	C₃×C₈	—	2	2	no	—
105	C₃×C₈	—	2	1	no	—
106	C₃×C₈	C₂×D₄	1	2	yes	—
107	#41	—	2	8	no	—
108	#41	—	4	4	no	—
109	C₂×Q₈	C₂³×C₃	1	2	yes	—
110	C₂×D₄	—	2	4	no	—
111	C₂×D₄	—	2	4	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	2	no	—
114	C₂×D₄	C₃×D₄	1	1	yes	characteristic
115	C₂×D₄	C₂³×C₃	1	4	yes	—
116	C₂²×C₄	—	2	4	no	—
117	C₂²×C₄	C₂³×C₃	1	2	yes	—
118	Q₁₆	C₃×D₄	1	4	yes	complemented
119	SD₁₆	—	2	8	no	—
120	D₈	—	2	4	no	—
121	D₈	—	4	2	no	—
122	C₂×C₈	—	2	4	no	—
123	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
124	C₂×C₈	C₂³×C₃	1	2	yes	—
125	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
128	C₄⋊C₄	C₂³×C₃	1	2	yes	—
129	C₂²⋊C₄	—	2	4	no	—
130	C₂²⋊C₄	C₃×D₄	1	2	yes	—
131	C₄²	—	2	2	no	—
132	C₄²	C₂³×C₃	1	1	yes	characteristic
133	C₂²×C₃	—	2	2	no	—
134	C₂²×C₃	—	2	2	no	—
135	C₂²×C₃	—	2	2	no	—
136	C₂²×C₃	—	4	4	no	—
137	C₂²×C₃	—	4	4	no	—
138	C₂²×C₃	—	4	4	no	—
139	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
140	C₃×C₄	—	2	2	no	—
141	C₃×C₄	—	2	4	no	—
142	C₃×C₄	—	2	1	no	—
143	C₃×C₄	—	4	2	no	—
144	C₃×C₄	#135	1	2	yes	complemented
145	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
146	C₃×C₄	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂³	—	2	4	no	—
148	C₂³	C₂×C₃×D₄	1	2	yes	—
149	Q₈	—	2	2	no	—
150	Q₈	C₂×C₃×D₄	1	4	yes	—
151	D₄	—	2	2	no	—
152	D₄	—	2	4	no	—
153	D₄	—	2	4	no	—
154	D₄	—	4	8	no	—
155	D₄	—	4	4	no	—
156	D₄	—	8	2	no	—
157	D₄	—	8	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	4	8	no	—
163	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
165	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	2	1	no	—
169	C₈	C₂×C₃×D₄	1	2	yes	—
170	C₂×C₃	—	4	2	no	—
171	C₂×C₃	—	8	4	no	—
172	C₂×C₃	D₄²	1	1	yes	characteristic
173	C₂×C₃	C₂×#135	1	1	yes	characteristic
174	C₂×C₃	#576	1	1	yes	characteristic
175	C₂²	—	2	2	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	4	no	—
179	C₂²	—	4	4	no	—
180	C₂²	—	4	4	no	—
181	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
182	C₄	—	2	1	no	—
183	C₄	—	2	2	no	—
184	C₄	—	2	4	no	—
185	C₄	—	4	2	no	—
186	C₄	C₃×#135	1	2	yes	complemented
187	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
189	C₃	#3287	1	1	yes	characteristic, direct product factor, Sylow
190	C₂	—	4	2	no	—
191	C₂	—	8	4	no	—
192	C₂	C₃×D₄²	1	1	yes	characteristic
193	C₂	C₂×C₃×#135	1	1	yes	characteristic
194	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×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#135	C₂	semidirect
2. C₃×#550	C₂	semidirect
3. C₃×#493	C₂	semidirect
4. C₃×#466	C₂	semidirect
5. C₃×#460	C₂	semidirect
6. C₃×#449	C₂	semidirect
7. C₃×C₄×Q₁₆	C₂	semidirect
8. C₃×C₈×D₄	C₂	semidirect
9. #3287	C₃	direct (N × H)
30. C₂×#135	C₂×C₃	semidirect
31. #550	C₂×C₃	semidirect
32. #493	C₂×C₃	semidirect
33. #466	C₂×C₃	semidirect
34. #460	C₂×C₃	semidirect
35. #449	C₂×C₃	semidirect
36. C₄×Q₁₆	C₂×C₃	semidirect
37. C₈×D₄	C₂×C₃	semidirect
49. C₃×Q₁₆	D₄	semidirect
118. Q₁₆	C₃×D₄	semidirect
144. C₃×C₄	#135	semidirect
186. C₄	C₃×#135	semidirect
189. C₃	#3287	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#135	2
C₃×#550	2
C₃×#493	2
C₃×#466	2
C₃×#460	2
C₃×#449	2
C₃×C₄×Q₁₆	2
C₃×C₈×D₄	2
#3287	3

Subgroups — C3×#3287 #85151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3287 #85151