Subgroups — C₂×C₃×#340 #72449

Order 384.

512 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₃×#340	C₂	1	4	yes	maximal, direct product factor
4	C₂×#340	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₄	—	2	2	no	—
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₈	—	2	4	no	—
8	C₂²×C₃×C₈	C₄	1	4	yes	complemented
9	C₂²×C₃×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊C₄	—	2	2	no	—
11	C₂×C₃×C₄⋊C₄	—	2	4	no	—
12	C₂×C₃×C₄⋊C₄	C₄	1	8	yes	complemented
13	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂³×C₈	C₂×C₃	1	1	yes	characteristic
15	C₂²×C₄⋊C₄	C₂×C₃	1	2	yes	—
16	#340	C₂×C₃	1	4	yes	direct product factor
17	C₂⁴×C₃	D₄	1	1	yes	characteristic
18	C₂²×C₃×C₄	—	2	8	no	—
19	C₂²×C₃×C₄	—	2	4	no	—
20	C₂²×C₃×C₄	—	4	4	no	—
21	C₂²×C₃×C₄	—	4	4	no	—
22	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
24	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	D₄	1	4	yes	—
26	C₂²×C₃×C₄	Q₈	1	2	yes	—
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₈	—	2	8	no	—
29	C₂×C₃×C₈	—	2	4	no	—
30	C₂×C₃×C₈	C₂×C₄	1	4	yes	complemented
31	C₃×C₄⋊C₄	—	2	8	no	—
32	C₃×C₄⋊C₄	—	2	4	no	—
33	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	complemented
34	C₂³×C₄	—	2	2	no	—
35	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
36	C₂²×C₈	—	2	4	no	—
37	C₂²×C₈	C₃×C₄	1	4	yes	complemented
38	C₂²×C₈	C₂²×C₃	1	2	yes	—
39	C₂×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄⋊C₄	—	2	4	no	—
41	C₂×C₄⋊C₄	C₃×C₄	1	8	yes	complemented
42	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
43	C₂³×C₃	C₂²⋊C₄	1	4	yes	—
44	C₂³×C₃	C₄⋊C₄	1	2	yes	—
45	C₂³×C₃	D₈	1	2	yes	—
46	C₂³×C₃	SD₁₆	1	4	yes	—
47	C₂³×C₃	Q₁₆	1	2	yes	—
48	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄	—	2	8	no	—
50	C₂×C₃×C₄	—	4	16	no	—
51	C₂×C₃×C₄	—	4	4	no	—
52	C₂×C₃×C₄	—	4	4	no	—
53	C₂×C₃×C₄	C₄²	1	4	yes	—
54	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
55	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
56	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
57	C₂×C₃×C₄	C₄⋊C₄	1	8	yes	—
58	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
60	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
61	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₈	—	2	4	no	—
63	C₂⁴	C₃×D₄	1	1	yes	characteristic
64	C₂²×C₄	—	2	8	no	—
65	C₂²×C₄	—	2	4	no	—
66	C₂²×C₄	—	4	4	no	—
67	C₂²×C₄	—	4	4	no	—
68	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
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×Q₈	1	2	yes	—
73	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
74	C₂×C₈	—	2	8	no	—
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	C₂×C₃×C₄	1	4	yes	complemented
77	C₄⋊C₄	—	2	8	no	—
78	C₄⋊C₄	—	2	4	no	—
79	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	complemented
80	C₂²×C₃	#95	1	4	yes	—
81	C₂²×C₃	#102	1	8	yes	—
82	C₂²×C₃	#103	1	8	yes	—
83	C₂²×C₃	#106	1	4	yes	—
84	C₂²×C₃	#107	1	4	yes	—
85	C₂²×C₃	C₂×C₂²⋊C₄	1	2	yes	—
86	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
88	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
89	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
90	C₃×C₄	—	4	8	no	—
91	C₃×C₄	#95	1	4	yes	—
92	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
93	C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³	C₃×C₂²⋊C₄	1	4	yes	—
96	C₂³	C₃×C₄⋊C₄	1	2	yes	—
97	C₂³	C₃×D₈	1	2	yes	—
98	C₂³	C₃×SD₁₆	1	4	yes	—
99	C₂³	C₃×Q₁₆	1	2	yes	—
100	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
101	C₂×C₄	—	2	8	no	—
102	C₂×C₄	—	4	16	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₃×C₄²	1	4	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	2	yes	—
109	C₂×C₄	C₃×C₄⋊C₄	1	8	yes	—
110	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
112	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
113	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
114	C₈	—	2	4	no	—
115	C₂×C₃	#340	1	8	yes	direct product factor
116	C₂×C₃	C₂×#95	1	1	yes	characteristic
117	C₂×C₃	C₂×#102	1	2	yes	—
118	C₂×C₃	C₂×#103	1	2	yes	—
119	C₂×C₃	C₂×#106	1	1	yes	characteristic
120	C₂×C₃	C₂×#107	1	1	yes	characteristic
121	C₂²	C₃×#95	1	4	yes	—
122	C₂²	C₃×#102	1	8	yes	—
123	C₂²	C₃×#103	1	8	yes	—
124	C₂²	C₃×#106	1	4	yes	—
125	C₂²	C₃×#107	1	4	yes	—
126	C₂²	C₂×C₃×C₂²⋊C₄	1	2	yes	—
127	C₂²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
128	C₂²	C₂×C₃×D₈	1	1	yes	characteristic
129	C₂²	C₂×C₃×SD₁₆	1	2	yes	—
130	C₂²	C₂×C₃×Q₁₆	1	1	yes	characteristic
131	C₄	—	4	8	no	—
132	C₄	C₃×#95	1	4	yes	—
133	C₄	C₂×C₃×C₄²	1	1	yes	characteristic
134	C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
135	C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
136	C₃	C₂×#340	1	1	yes	characteristic, direct product factor, Sylow
137	C₂	C₃×#340	1	8	yes	direct product factor
138	C₂	C₂×C₃×#95	1	1	yes	characteristic
139	C₂	C₂×C₃×#102	1	2	yes	—
140	C₂	C₂×C₃×#103	1	2	yes	—
141	C₂	C₂×C₃×#106	1	1	yes	characteristic
142	C₂	C₂×C₃×#107	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃×#340	C₂	direct (N × H)
4. C₂×#340	C₃	direct (N × H)
8. C₂²×C₃×C₈	C₄	semidirect
12. C₂×C₃×C₄⋊C₄	C₄	semidirect
16. #340	C₂×C₃	direct (N × H)
30. C₂×C₃×C₈	C₂×C₄	semidirect
33. C₃×C₄⋊C₄	C₂×C₄	semidirect
37. C₂²×C₈	C₃×C₄	semidirect
41. C₂×C₄⋊C₄	C₃×C₄	semidirect
76. C₂×C₈	C₂×C₃×C₄	semidirect
79. C₄⋊C₄	C₂×C₃×C₄	semidirect
115. C₂×C₃	#340	direct (N × H)
136. C₃	C₂×#340	direct (N × H)
137. C₂	C₃×#340	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₈	2
C₂²×C₃×C₄⋊C₄	2
C₃×#340	2
C₂×#340	3

Subgroups — C2×C3×#340 #72449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#340 #72449