Subgroups — C₂×C₃×#427 #84773

Order 384.

512 conjugacy classes of subgroups.

The subgroup lattice has 136 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	C₂×C₃×#117	C₂	1	2	yes	maximal
3	C₃×#427	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×#107	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#106	C₂	1	2	yes	maximal, complemented
6	C₂×#427	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₄	1	8	yes	complemented
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₈	C₂²	1	4	yes	—
12	C₃×#117	—	2	4	no	—
13	C₃×#117	C₂²	1	8	yes	—
14	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
16	C₂×C₃×C₄²	—	2	2	no	—
17	C₃×#107	C₂²	1	8	yes	complemented
18	C₃×#106	C₂²	1	8	yes	complemented
19	C₂³×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×#117	C₂×C₃	1	2	yes	—
21	#427	C₂×C₃	1	8	yes	direct product factor
22	C₂×#107	C₂×C₃	1	2	yes	complemented
23	C₂×#106	C₂×C₃	1	2	yes	complemented
24	C₂⁴×C₃	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	—	2	4	no	—
26	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₄	D₄	1	2	yes	—
28	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	Q₈	1	4	yes	—
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
33	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₈	C₂×C₄	1	16	yes	complemented
35	C₂×C₃×C₈	C₂³	1	2	yes	—
36	C₂×C₃×C₈	C₂³	1	2	yes	—
37	C₃×C₄⋊C₄	—	2	4	no	—
38	C₃×C₄⋊C₄	C₂³	1	8	yes	—
39	C₃×C₂²⋊C₄	—	2	8	no	—
40	C₃×C₄²	—	2	8	no	—
41	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
42	C₂²×C₈	C₃×C₄	1	8	yes	complemented
43	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₈	C₂²×C₃	1	4	yes	—
46	#117	—	2	4	no	—
47	#117	C₂²×C₃	1	8	yes	—
48	C₂×C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₂×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²	—	2	2	no	—
51	#107	C₂²×C₃	1	8	yes	complemented
52	#106	C₂²×C₃	1	8	yes	complemented
53	C₂³×C₃	—	2	4	no	—
54	C₂³×C₃	C₄⋊C₄	1	4	yes	—
55	C₂³×C₃	C₂×D₄	1	2	yes	—
56	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄	—	2	8	no	—
58	C₂×C₃×C₄	—	4	8	no	—
59	C₂×C₃×C₄	C₄⋊C₄	1	8	yes	—
60	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
61	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
62	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
63	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
66	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
68	C₃×C₈	C₂²×C₄	1	4	yes	—
69	C₃×C₈	C₂²×C₄	1	4	yes	—
70	C₂⁴	C₃×D₄	1	1	yes	characteristic
71	C₂²×C₄	—	2	4	no	—
72	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
73	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
74	C₂²×C₄	C₃×D₄	1	2	yes	—
75	C₂²×C₄	C₃×Q₈	1	4	yes	—
76	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₂²×C₄	C₂³×C₃	1	2	yes	—
78	C₂×C₈	C₂×C₃×C₄	1	4	yes	—
79	C₂×C₈	C₂×C₃×C₄	1	4	yes	—
80	C₂×C₈	C₂×C₃×C₄	1	16	yes	complemented
81	C₂×C₈	C₂³×C₃	1	2	yes	—
82	C₂×C₈	C₂³×C₃	1	2	yes	—
83	C₄⋊C₄	—	2	4	no	—
84	C₄⋊C₄	C₂³×C₃	1	8	yes	—
85	C₂²⋊C₄	—	2	8	no	—
86	C₄²	—	2	8	no	—
87	C₂²×C₃	—	2	4	no	—
88	C₂²×C₃	—	2	8	no	—
89	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
90	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₃	#135	1	4	yes	—
92	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
93	C₃×C₄	—	4	8	no	—
94	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
95	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
96	C₃×C₄	C₂³×C₄	1	1	yes	characteristic
97	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
98	C₂³	—	2	4	no	—
99	C₂³	C₃×C₄⋊C₄	1	4	yes	—
100	C₂³	C₂×C₃×D₄	1	2	yes	—
101	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	4	8	no	—
104	C₂×C₄	C₃×C₄⋊C₄	1	8	yes	—
105	C₂×C₄	C₃×C₄⋊C₄	1	4	yes	—
106	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
107	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
108	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
109	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
110	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
111	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
112	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
113	C₈	C₂²×C₃×C₄	1	4	yes	—
114	C₈	C₂²×C₃×C₄	1	4	yes	—
115	C₂×C₃	—	2	4	no	—
116	C₂×C₃	#427	1	4	yes	direct product factor
117	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₃	C₂×#135	1	2	yes	—
119	C₂²	—	2	4	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂×C₃×C₄⋊C₄	1	4	yes	—
122	C₂²	C₂×C₃×C₄⋊C₄	1	2	yes	—
123	C₂²	C₃×#135	1	4	yes	—
124	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
125	C₄	—	4	8	no	—
126	C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
127	C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
128	C₄	C₂³×C₃×C₄	1	1	yes	characteristic
129	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
130	C₃	C₂×#427	1	1	yes	characteristic, direct product factor, Sylow
131	C₂	—	2	4	no	—
132	C₂	C₃×#427	1	4	yes	direct product factor
133	C₂	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
134	C₂	C₂×C₃×#135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	D₄
commutator	C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂×D₄
prime radical	C₂⁴×C₃	D₄
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×#427	C₂	direct (N × H)
4. C₂×C₃×#107	C₂	semidirect
5. C₂×C₃×#106	C₂	semidirect
6. C₂×#427	C₃	direct (N × H)
8. C₂²×C₃×C₈	C₄	semidirect
17. C₃×#107	C₂²	semidirect
18. C₃×#106	C₂²	semidirect
21. #427	C₂×C₃	direct (N × H)
22. C₂×#107	C₂×C₃	semidirect
23. C₂×#106	C₂×C₃	semidirect
34. C₂×C₃×C₈	C₂×C₄	semidirect
42. C₂²×C₈	C₃×C₄	semidirect
51. #107	C₂²×C₃	semidirect
52. #106	C₂²×C₃	semidirect
80. C₂×C₈	C₂×C₃×C₄	semidirect
116. C₂×C₃	#427	direct (N × H)
130. C₃	C₂×#427	direct (N × H)
132. C₂	C₃×#427	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
C₂×C₃×#117	2
C₃×#427	2
C₂×C₃×#107	2
C₂×C₃×#106	2
C₂×#427	3

Subgroups — C2×C3×#427 #84773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#427 #84773