Subgroups — C₂×C₃×H(3,ℤ/4) #72445

Order 384.

404 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#117	C₂	1	2	yes	maximal
3	C₃×H(3,ℤ/4)	C₂	1	4	yes	maximal, direct product factor
4	C₂×H(3,ℤ/4)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃×M₁₆	—	2	2	no	—
7	C₂×C₃×M₁₆	—	2	2	no	—
8	C₂×C₃×M₁₆	C₄	1	2	yes	complemented
9	C₂×C₃×M₁₆	C₂²	1	2	yes	—
10	C₂²×C₃×C₈	C₄	1	2	yes	complemented
11	C₃×#117	—	2	2	no	—
12	C₃×#117	—	2	2	no	—
13	C₃×#117	C₄	1	4	yes	complemented
14	C₃×#117	C₂²	1	4	yes	—
15	C₂×C₃×C₄⋊C₄	—	2	2	no	—
16	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
17	C₂×C₃×C₄²	C₄	1	4	yes	complemented
18	C₂²×M₁₆	C₂×C₃	1	1	yes	characteristic
19	C₂×#117	C₂×C₃	1	2	yes	—
20	H(3,ℤ/4)	C₂×C₃	1	4	yes	direct product factor
21	C₂⁴×C₃	D₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	2	4	no	—
23	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄	D₄	1	2	yes	—
29	C₂²×C₃×C₄	D₄	1	2	yes	—
30	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
31	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₃×M₁₆	—	2	4	no	—
34	C₃×M₁₆	—	2	4	no	—
35	C₂×C₃×C₈	—	2	2	no	—
36	C₂×C₃×C₈	—	2	2	no	—
37	C₂×C₃×C₈	C₂×C₄	1	4	yes	complemented
38	C₃×C₄⋊C₄	—	2	4	no	—
39	C₃×C₄⋊C₄	—	2	4	no	—
40	C₃×C₂²⋊C₄	—	2	4	no	—
41	C₃×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₄²	—	2	4	no	—
43	C₃×C₄²	C₂×C₄	1	8	yes	complemented
44	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂×M₁₆	—	2	2	no	—
46	C₂×M₁₆	—	2	2	no	—
47	C₂×M₁₆	C₃×C₄	1	2	yes	complemented
48	C₂×M₁₆	C₂²×C₃	1	2	yes	—
49	C₂²×C₈	C₃×C₄	1	2	yes	complemented
50	#117	—	2	2	no	—
51	#117	—	2	2	no	—
52	#117	C₃×C₄	1	4	yes	complemented
53	#117	C₂²×C₃	1	4	yes	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₂²⋊C₄	—	2	2	no	—
56	C₂×C₄²	C₃×C₄	1	4	yes	complemented
57	C₂³×C₃	—	4	2	no	—
58	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
59	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
60	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄	—	2	8	no	—
64	C₂×C₃×C₄	—	4	8	no	—
65	C₂×C₃×C₄	C₄²	1	2	yes	—
66	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
67	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
68	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
76	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
77	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
78	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
79	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
83	C₃×C₈	—	2	4	no	—
84	C₂⁴	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
89	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₃×D₄	1	2	yes	—
92	C₂²×C₄	C₃×D₄	1	2	yes	—
93	C₂²×C₄	C₃×Q₈	1	1	yes	characteristic
94	C₂²×C₄	C₃×Q₈	1	1	yes	characteristic
95	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
96	M₁₆	—	2	4	no	—
97	M₁₆	—	2	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₃×C₄	1	4	yes	complemented
101	C₄⋊C₄	—	2	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₂²⋊C₄	—	2	4	no	—
104	C₂²⋊C₄	—	2	4	no	—
105	C₄²	—	2	4	no	—
106	C₄²	C₂×C₃×C₄	1	8	yes	complemented
107	C₂²×C₃	—	2	4	no	—
108	C₂²×C₃	—	2	2	no	—
109	C₂²×C₃	—	4	2	no	—
110	C₂²×C₃	—	4	1	no	—
111	C₂²×C₃	—	4	1	no	—
112	C₂²×C₃	#95	1	1	yes	characteristic
113	C₂²×C₃	#95	1	2	yes	—
114	C₂²×C₃	#95	1	1	yes	characteristic
115	C₂²×C₃	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₃×C₄	—	4	8	no	—
118	C₃×C₄	#95	1	2	yes	—
119	C₃×C₄	#95	1	1	yes	characteristic
120	C₃×C₄	#95	1	1	yes	characteristic
121	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
122	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
124	C₂³	—	4	2	no	—
125	C₂³	C₃×C₂²⋊C₄	1	2	yes	—
126	C₂³	C₃×C₂²⋊C₄	1	2	yes	—
127	C₂³	C₃×C₄⋊C₄	1	1	yes	characteristic
128	C₂³	C₃×C₄⋊C₄	1	1	yes	characteristic
129	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₄	—	2	8	no	—
131	C₂×C₄	—	4	8	no	—
132	C₂×C₄	C₃×C₄²	1	1	yes	characteristic
133	C₂×C₄	C₃×C₄²	1	2	yes	—
134	C₂×C₄	C₃×C₄²	1	1	yes	characteristic
135	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
137	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
143	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
144	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
145	C₂×C₄	C₃×C₄⋊C₄	1	2	yes	—
146	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
147	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
148	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
149	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
150	C₈	—	2	4	no	—
151	C₂×C₃	—	2	2	no	—
152	C₂×C₃	—	2	2	no	—
153	C₂×C₃	—	2	1	no	—
154	C₂×C₃	—	2	1	no	—
155	C₂×C₃	H(3,ℤ/4)	1	2	yes	direct product factor
156	C₂×C₃	C₂×#95	1	1	yes	characteristic
157	C₂²	—	2	4	no	—
158	C₂²	—	2	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	C₃×#95	1	1	yes	characteristic
163	C₂²	C₃×#95	1	2	yes	—
164	C₂²	C₃×#95	1	1	yes	characteristic
165	C₂²	C₂×C₃×C₂²⋊C₄	1	2	yes	—
166	C₂²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
167	C₄	—	4	8	no	—
168	C₄	C₃×#95	1	2	yes	—
169	C₄	C₃×#95	1	1	yes	characteristic
170	C₄	C₃×#95	1	1	yes	characteristic
171	C₄	C₂×C₃×C₄²	1	1	yes	characteristic
172	C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
173	C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
174	C₃	C₂×H(3,ℤ/4)	1	1	yes	characteristic, direct product factor, Sylow
175	C₂	—	2	2	no	—
176	C₂	—	2	2	no	—
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	C₃×H(3,ℤ/4)	1	2	yes	direct product factor
180	C₂	C₂×C₃×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄²
commutator	C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂²×C₃	#95
prime radical	C₂⁴×C₃	D₄
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×H(3,ℤ/4)	C₂	direct (N × H)
4. C₂×H(3,ℤ/4)	C₃	direct (N × H)
8. C₂×C₃×M₁₆	C₄	semidirect
10. C₂²×C₃×C₈	C₄	semidirect
13. C₃×#117	C₄	semidirect
17. C₂×C₃×C₄²	C₄	semidirect
20. H(3,ℤ/4)	C₂×C₃	direct (N × H)
37. C₂×C₃×C₈	C₂×C₄	semidirect
43. C₃×C₄²	C₂×C₄	semidirect
47. C₂×M₁₆	C₃×C₄	semidirect
49. C₂²×C₈	C₃×C₄	semidirect
52. #117	C₃×C₄	semidirect
56. C₂×C₄²	C₃×C₄	semidirect
100. C₂×C₈	C₂×C₃×C₄	semidirect
106. C₄²	C₂×C₃×C₄	semidirect
155. C₂×C₃	H(3,ℤ/4)	direct (N × H)
174. C₃	C₂×H(3,ℤ/4)	direct (N × H)
179. C₂	C₃×H(3,ℤ/4)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×M₁₆	2
C₂×C₃×#117	2
C₃×H(3,ℤ/4)	2
C₂×H(3,ℤ/4)	3

Subgroups — C2×C3×H(3,ℤ/4) #72445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×H(3,ℤ/4) #72445