Subgroups — C₂×C₃×#341 #72452

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 142 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	2	yes	maximal
2	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#341	C₂	1	4	yes	maximal, direct product factor
4	C₂×#341	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃×M₁₆	—	2	4	no	—
7	C₂×C₃×M₁₆	—	2	2	no	—
8	C₂×C₃×M₁₆	C₄	1	8	yes	—
9	C₂×C₃×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₃×C₈	—	2	4	no	—
11	C₂²×C₃×C₈	—	2	2	no	—
12	C₂²×C₃×C₈	C₄	1	2	yes	—
13	C₂²×C₃×C₈	C₄	1	2	yes	—
14	C₂²×C₃×C₈	C₂²	1	2	yes	—
15	C₂²×M₁₆	C₂×C₃	1	2	yes	—
16	C₂³×C₈	C₂×C₃	1	1	yes	characteristic
17	#341	C₂×C₃	1	4	yes	direct product factor
18	C₂⁴×C₃	Q₈	1	1	yes	characteristic
19	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
22	C₂²×C₃×C₄	D₄	1	4	yes	—
23	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	Q₈	1	1	yes	characteristic
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₃×M₁₆	—	2	8	no	—
28	C₃×M₁₆	—	2	4	no	—
29	C₃×M₁₆	C₂×C₄	1	8	yes	—
30	C₂×C₃×C₈	—	2	4	no	—
31	C₂×C₃×C₈	—	2	4	no	—
32	C₂×C₃×C₈	—	2	8	no	—
33	C₂×C₃×C₈	—	2	4	no	—
34	C₂×C₃×C₈	—	2	4	no	—
35	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
36	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₂×M₁₆	—	2	4	no	—
38	C₂×M₁₆	—	2	2	no	—
39	C₂×M₁₆	C₃×C₄	1	8	yes	—
40	C₂×M₁₆	C₂²×C₃	1	4	yes	—
41	C₂²×C₈	—	2	4	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₃×C₄	1	2	yes	—
44	C₂²×C₈	C₃×C₄	1	2	yes	—
45	C₂²×C₈	C₂²×C₃	1	2	yes	—
46	C₂³×C₃	—	2	4	no	—
47	C₂³×C₃	C₄⋊C₄	1	4	yes	—
48	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂³×C₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄	C₄²	1	2	yes	—
52	C₂×C₃×C₄	C₄²	1	2	yes	—
53	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
54	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
56	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
57	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
58	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
62	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₃×C₈	—	2	4	no	—
65	C₃×C₈	—	2	8	no	—
66	C₂⁴	C₃×Q₈	1	1	yes	characteristic
67	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
68	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
70	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
71	C₂²×C₄	C₃×D₄	1	4	yes	—
72	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
73	C₂²×C₄	C₃×Q₈	1	1	yes	characteristic
74	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
75	M₁₆	—	2	8	no	—
76	M₁₆	—	2	4	no	—
77	M₁₆	C₂×C₃×C₄	1	8	yes	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	8	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×C₃×C₄	1	4	yes	—
84	C₂²×C₃	—	2	8	no	—
85	C₂²×C₃	—	2	2	no	—
86	C₂²×C₃	—	2	2	no	—
87	C₂²×C₃	#95	1	2	yes	—
88	C₂²×C₃	#95	1	2	yes	—
89	C₂²×C₃	#108	1	4	yes	—
90	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₃×C₄	#95	1	2	yes	—
93	C₃×C₄	#95	1	2	yes	—
94	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
95	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₃×C₄	C₂×C₂²⋊C₄	1	2	yes	—
97	C₂³	—	2	4	no	—
98	C₂³	C₃×C₄⋊C₄	1	1	yes	characteristic
99	C₂³	C₃×C₄⋊C₄	1	4	yes	—
100	C₂³	C₃×C₄⋊C₄	1	1	yes	characteristic
101	C₂³	C₂×C₃×Q₈	1	1	yes	characteristic
102	C₂×C₄	C₃×C₄²	1	2	yes	—
103	C₂×C₄	C₃×C₄²	1	2	yes	—
104	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
106	C₂×C₄	C₃×C₂²⋊C₄	1	4	yes	—
107	C₂×C₄	C₃×C₂²⋊C₄	1	4	yes	—
108	C₂×C₄	C₃×C₄⋊C₄	1	4	yes	—
109	C₂×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
113	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
114	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
115	C₈	—	2	4	no	—
116	C₈	—	2	8	no	—
117	C₂×C₃	—	2	2	no	—
118	C₂×C₃	—	2	2	no	—
119	C₂×C₃	#341	1	4	yes	direct product factor
120	C₂×C₃	C₂×#95	1	1	yes	characteristic
121	C₂×C₃	C₂×#108	1	2	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	8	no	—
125	C₂²	C₃×#95	1	2	yes	—
126	C₂²	C₃×#95	1	2	yes	—
127	C₂²	C₃×#108	1	4	yes	—
128	C₂²	C₂×C₃×C₄⋊C₄	1	2	yes	—
129	C₂²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
130	C₄	C₃×#95	1	2	yes	—
131	C₄	C₃×#95	1	2	yes	—
132	C₄	C₂×C₃×C₄²	1	1	yes	characteristic
133	C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
134	C₄	C₂×C₃×C₂²⋊C₄	1	2	yes	—
135	C₃	C₂×#341	1	1	yes	characteristic, direct product factor, Sylow
136	C₂	—	2	2	no	—
137	C₂	—	2	2	no	—
138	C₂	C₃×#341	1	4	yes	direct product factor
139	C₂	C₂×C₃×#95	1	1	yes	characteristic
140	C₂	C₂×C₃×#108	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	D₄
commutator	C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂³×C₃	C₄⋊C₄
prime radical	C₂⁴×C₃	Q₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#341	C₂	direct (N × H)
4. C₂×#341	C₃	direct (N × H)
17. #341	C₂×C₃	direct (N × H)
119. C₂×C₃	#341	direct (N × H)
135. C₃	C₂×#341	direct (N × H)
138. C₂	C₃×#341	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
C₂³×C₃×C₈	2
C₃×#341	2
C₂×#341	3

Subgroups — C2×C3×#341 #72452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#341 #72452