Subgroups — C₃×#3486 #87370

Order 384.

1492 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄×Q₈	C₂	1	8	yes	maximal, complemented
4	C₃×#548	C₂	1	8	yes	maximal, complemented
5	C₃×#536	C₂	1	6	yes	maximal, complemented
6	C₃×#525	C₂	1	6	yes	maximal, complemented
7	C₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
8	#3486	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×2₋¹⁺⁴	—	2	8	no	—
10	C₂×C₃×#41	C₂²	1	6	yes	—
11	C₂×C₃×#41	C₂²	1	4	yes	complemented
12	C₂²×C₃×Q₈	—	2	8	no	—
13	C₂²×C₃×Q₈	C₂²	1	3	yes	—
14	C₂²×C₃×Q₈	C₂²	1	8	yes	—
15	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Q₈	C₂²	1	4	yes	complemented
17	C₂³×C₃×C₄	C₂²	1	3	yes	—
18	C₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
19	C₃×#124	C₂²	1	12	yes	complemented
20	C₃×#123	C₂²	1	12	yes	—
21	C₃×C₂²⋊Q₈	C₂²	1	12	yes	—
22	C₃×C₂²⋊Q₈	C₂²	1	24	yes	complemented
23	C₃×#121	C₂²	1	12	yes	—
24	C₃×#120	C₂²	1	4	yes	—
25	C₃×C₄×Q₈	C₂²	1	8	yes	complemented
26	C₃×C₄×D₄	C₂²	1	24	yes	—
27	C₃×#117	C₂²	1	6	yes	—
28	C₂×2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
30	D₄×Q₈	C₂×C₃	1	8	yes	complemented
31	#548	C₂×C₃	1	8	yes	complemented
32	#536	C₂×C₃	1	6	yes	complemented
33	#525	C₂×C₃	1	6	yes	complemented
34	#519	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂⁴×C₃	C₂³	1	1	yes	characteristic
36	C₃×#41	—	2	16	no	—
37	C₃×#41	—	2	24	no	—
38	C₂×C₃×Q₈	—	2	32	no	—
39	C₂×C₃×Q₈	—	2	16	no	—
40	C₂×C₃×Q₈	—	2	12	no	—
41	C₂×C₃×Q₈	D₄	1	8	yes	complemented
42	C₂×C₃×Q₈	C₂³	1	6	yes	—
43	C₂×C₃×Q₈	C₂³	1	12	yes	—
44	C₂×C₃×Q₈	C₂³	1	4	yes	—
45	C₂×C₃×D₄	C₂³	1	8	yes	—
46	C₂×C₃×D₄	C₂³	1	4	yes	—
47	C₂×C₃×D₄	C₂³	1	6	yes	—
48	C₂²×C₃×C₄	—	2	12	no	—
49	C₂²×C₃×C₄	C₂³	1	12	yes	—
50	C₂²×C₃×C₄	C₂³	1	12	yes	—
51	C₂²×C₃×C₄	C₂³	1	3	yes	—
52	C₂²×C₃×C₄	C₂³	1	3	yes	—
53	C₃×C₄⋊C₄	C₂³	1	12	yes	—
54	C₃×C₄⋊C₄	C₂³	1	24	yes	—
55	C₃×C₂²⋊C₄	C₂³	1	4	yes	—
56	C₃×C₂²⋊C₄	C₂³	1	8	yes	—
57	C₃×C₂²⋊C₄	C₂³	1	24	yes	—
58	C₃×C₄²	C₂³	1	12	yes	—
59	2₋¹⁺⁴	—	2	8	no	—
60	C₂×#41	C₂²×C₃	1	6	yes	—
61	C₂×#41	C₂²×C₃	1	4	yes	complemented
62	C₂²×Q₈	—	2	8	no	—
63	C₂²×Q₈	C₂²×C₃	1	3	yes	—
64	C₂²×Q₈	C₂²×C₃	1	8	yes	—
65	C₂²×Q₈	C₂²×C₃	1	1	yes	characteristic
66	C₂²×Q₈	C₂²×C₃	1	4	yes	complemented
67	C₂³×C₄	C₂²×C₃	1	3	yes	—
68	C₄⋊Q₈	C₂²×C₃	1	12	yes	complemented
69	#124	C₂²×C₃	1	12	yes	complemented
70	#123	C₂²×C₃	1	12	yes	—
71	C₂²⋊Q₈	C₂²×C₃	1	12	yes	—
72	C₂²⋊Q₈	C₂²×C₃	1	24	yes	complemented
73	#121	C₂²×C₃	1	12	yes	—
74	#120	C₂²×C₃	1	4	yes	—
75	C₄×Q₈	C₂²×C₃	1	8	yes	complemented
76	C₄×D₄	C₂²×C₃	1	24	yes	—
77	#117	C₂²×C₃	1	6	yes	—
78	C₂³×C₃	—	2	2	no	—
79	C₂³×C₃	—	2	4	no	—
80	C₂³×C₃	C₂⁴	1	4	yes	—
81	C₂³×C₃	C₂⁴	1	1	yes	characteristic
82	C₂³×C₃	C₂⁴	1	2	yes	—
83	C₃×Q₈	—	2	24	no	—
84	C₃×Q₈	—	2	12	no	—
85	C₃×Q₈	C₂×D₄	1	16	yes	—
86	C₃×D₄	—	2	8	no	—
87	C₃×D₄	—	2	16	no	—
88	C₃×D₄	—	2	12	no	—
89	C₂×C₃×C₄	—	2	6	no	—
90	C₂×C₃×C₄	—	2	24	no	—
91	C₂×C₃×C₄	—	2	24	no	—
92	C₂×C₃×C₄	C₂×D₄	1	12	yes	—
93	C₂×C₃×C₄	C₂⁴	1	8	yes	—
94	C₂×C₃×C₄	C₂⁴	1	6	yes	—
95	C₂×C₃×C₄	C₂⁴	1	4	yes	—
96	C₂×C₃×C₄	C₂⁴	1	6	yes	—
97	C₂⁴	C₂³×C₃	1	1	yes	characteristic
98	#41	—	2	16	no	—
99	#41	—	2	24	no	—
100	C₂×Q₈	—	2	32	no	—
101	C₂×Q₈	—	2	16	no	—
102	C₂×Q₈	—	2	12	no	—
103	C₂×Q₈	C₃×D₄	1	8	yes	complemented
104	C₂×Q₈	C₂³×C₃	1	6	yes	—
105	C₂×Q₈	C₂³×C₃	1	12	yes	—
106	C₂×Q₈	C₂³×C₃	1	4	yes	—
107	C₂×D₄	C₂³×C₃	1	8	yes	—
108	C₂×D₄	C₂³×C₃	1	4	yes	—
109	C₂×D₄	C₂³×C₃	1	6	yes	—
110	C₂²×C₄	—	2	12	no	—
111	C₂²×C₄	C₂³×C₃	1	12	yes	—
112	C₂²×C₄	C₂³×C₃	1	12	yes	—
113	C₂²×C₄	C₂³×C₃	1	3	yes	—
114	C₂²×C₄	C₂³×C₃	1	3	yes	—
115	C₄⋊C₄	C₂³×C₃	1	12	yes	—
116	C₄⋊C₄	C₂³×C₃	1	24	yes	—
117	C₂²⋊C₄	C₂³×C₃	1	4	yes	—
118	C₂²⋊C₄	C₂³×C₃	1	8	yes	—
119	C₂²⋊C₄	C₂³×C₃	1	24	yes	—
120	C₄²	C₂³×C₃	1	12	yes	—
121	C₂²×C₃	—	2	8	no	—
122	C₂²×C₃	—	2	2	no	—
123	C₂²×C₃	—	2	2	no	—
124	C₂²×C₃	—	2	2	no	—
125	C₂²×C₃	—	2	4	no	—
126	C₂²×C₃	—	2	8	no	—
127	C₂²×C₃	C₂²×D₄	1	2	yes	—
128	C₂²×C₃	2₋¹⁺⁴	1	4	yes	complemented
129	C₂²×C₃	C₂⁵	1	1	yes	characteristic
130	C₃×C₄	—	2	4	no	—
131	C₃×C₄	—	2	6	no	—
132	C₃×C₄	—	2	8	no	—
133	C₃×C₄	C₂²×D₄	1	12	yes	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	4	no	—
136	C₂³	C₂⁴×C₃	1	4	yes	—
137	C₂³	C₂⁴×C₃	1	1	yes	characteristic
138	C₂³	C₂⁴×C₃	1	2	yes	—
139	Q₈	—	2	24	no	—
140	Q₈	—	2	12	no	—
141	Q₈	C₂×C₃×D₄	1	16	yes	—
142	D₄	—	2	8	no	—
143	D₄	—	2	16	no	—
144	D₄	—	2	12	no	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	2	24	no	—
147	C₂×C₄	—	2	24	no	—
148	C₂×C₄	C₂×C₃×D₄	1	12	yes	—
149	C₂×C₄	C₂⁴×C₃	1	8	yes	—
150	C₂×C₄	C₂⁴×C₃	1	6	yes	—
151	C₂×C₄	C₂⁴×C₃	1	4	yes	—
152	C₂×C₄	C₂⁴×C₃	1	6	yes	—
153	C₂×C₃	—	2	2	no	—
154	C₂×C₃	—	2	4	no	—
155	C₂×C₃	—	4	4	no	—
156	C₂×C₃	C₂³×D₄	1	1	yes	characteristic
157	C₂×C₃	C₂×2₋¹⁺⁴	1	2	yes	—
158	C₂²	—	2	8	no	—
159	C₂²	—	2	2	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	2	2	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	C₂²×C₃×D₄	1	2	yes	—
165	C₂²	C₃×2₋¹⁺⁴	1	4	yes	complemented
166	C₂²	C₂⁵×C₃	1	1	yes	characteristic
167	C₄	—	2	4	no	—
168	C₄	—	2	6	no	—
169	C₄	—	2	8	no	—
170	C₄	C₂²×C₃×D₄	1	12	yes	—
171	C₃	#3486	1	1	yes	characteristic, direct product factor, Sylow
172	C₂	—	2	2	no	—
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂³×C₃×D₄	1	1	yes	characteristic
176	C₂	C₂×C₃×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×2₋¹⁺⁴	C₂	semidirect
2. C₂³×C₃×Q₈	C₂	semidirect
3. C₃×D₄×Q₈	C₂	semidirect
4. C₃×#548	C₂	semidirect
5. C₃×#536	C₂	semidirect
6. C₃×#525	C₂	semidirect
7. C₃×#519	C₂	semidirect
8. #3486	C₃	direct (N × H)
11. C₂×C₃×#41	C₂²	semidirect
16. C₂²×C₃×Q₈	C₂²	semidirect
18. C₃×C₄⋊Q₈	C₂²	semidirect
19. C₃×#124	C₂²	semidirect
22. C₃×C₂²⋊Q₈	C₂²	semidirect
25. C₃×C₄×Q₈	C₂²	semidirect
28. C₂×2₋¹⁺⁴	C₂×C₃	semidirect
29. C₂³×Q₈	C₂×C₃	semidirect
30. D₄×Q₈	C₂×C₃	semidirect
31. #548	C₂×C₃	semidirect
32. #536	C₂×C₃	semidirect
33. #525	C₂×C₃	semidirect
34. #519	C₂×C₃	semidirect
41. C₂×C₃×Q₈	D₄	semidirect
61. C₂×#41	C₂²×C₃	semidirect
66. C₂²×Q₈	C₂²×C₃	semidirect
68. C₄⋊Q₈	C₂²×C₃	semidirect
69. #124	C₂²×C₃	semidirect
72. C₂²⋊Q₈	C₂²×C₃	semidirect
75. C₄×Q₈	C₂²×C₃	semidirect
103. C₂×Q₈	C₃×D₄	semidirect
128. C₂²×C₃	2₋¹⁺⁴	semidirect
165. C₂²	C₃×2₋¹⁺⁴	semidirect
171. C₃	#3486	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×C₃×2₋¹⁺⁴	2
C₂³×C₃×Q₈	2
C₃×D₄×Q₈	2
C₃×#548	2
C₃×#536	2
C₃×#525	2
C₃×#519	2
#3486	3

Subgroups — C3×#3486 #87370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3486 #87370