Subgroups — C₃×#2905 #84769

Order 384.

484 conjugacy classes of subgroups.

The subgroup lattice has 128 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	C₂×C₃×#117	C₂	1	1	yes	maximal, characteristic
3	C₃×#424	C₂	1	8	yes	maximal, complemented
4	C₃×#423	C₂	1	4	yes	maximal, complemented
5	#2905	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×M₁₆	—	2	8	no	—
8	C₂×C₃×M₁₆	C₂²	1	4	yes	—
9	C₂×C₃×M₁₆	C₂²	1	4	yes	—
10	C₂²×C₃×C₈	C₂²	1	4	yes	—
11	C₃×#117	C₄	1	4	yes	—
12	C₃×#117	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊C₄	C₄	1	2	yes	—
14	C₂×C₃×C₂²⋊C₄	C₄	1	2	yes	—
15	C₂×C₃×C₄²	C₂²	1	2	yes	—
16	C₃×C₄⋊C₈	C₂²	1	16	yes	complemented
17	C₂²×M₁₆	C₂×C₃	1	2	yes	—
18	C₂×#117	C₂×C₃	1	1	yes	characteristic
19	#424	C₂×C₃	1	8	yes	complemented
20	#423	C₂×C₃	1	4	yes	complemented
21	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	C₂×C₄	1	4	yes	—
23	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄	D₄	1	4	yes	—
26	C₂²×C₃×C₄	Q₈	1	4	yes	—
27	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×C₄	C₂³	1	2	yes	—
29	C₃×M₁₆	—	2	8	no	—
30	C₃×M₁₆	—	2	8	no	—
31	C₂×C₃×C₈	—	2	8	no	—
32	C₂×C₃×C₈	C₂³	1	8	yes	—
33	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
34	C₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
35	C₃×C₄²	C₂×C₄	1	4	yes	—
36	C₃×C₄²	C₂³	1	4	yes	—
37	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
38	C₂×M₁₆	—	2	8	no	—
39	C₂×M₁₆	C₂²×C₃	1	4	yes	—
40	C₂×M₁₆	C₂²×C₃	1	4	yes	—
41	C₂²×C₈	C₂²×C₃	1	4	yes	—
42	#117	C₃×C₄	1	4	yes	—
43	#117	C₂²×C₃	1	4	yes	—
44	C₂×C₄⋊C₄	C₃×C₄	1	2	yes	—
45	C₂×C₂²⋊C₄	C₃×C₄	1	2	yes	—
46	C₂×C₄²	C₂²×C₃	1	2	yes	—
47	C₄⋊C₈	C₂²×C₃	1	16	yes	complemented
48	C₂³×C₃	—	4	2	no	—
49	C₂³×C₃	C₄⋊C₄	1	4	yes	—
50	C₂³×C₃	C₂²×C₄	1	2	yes	—
51	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄	—	4	4	no	—
53	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
54	C₂×C₃×C₄	C₄⋊C₄	1	8	yes	—
55	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
56	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
58	C₂×C₃×C₄	C₂×D₄	1	4	yes	—
59	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
60	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
61	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
62	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
63	C₃×C₈	—	2	8	no	—
64	C₂⁴	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂²×C₄	C₂×C₃×C₄	1	4	yes	—
66	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
68	C₂²×C₄	C₃×D₄	1	4	yes	—
69	C₂²×C₄	C₃×Q₈	1	4	yes	—
70	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
71	C₂²×C₄	C₂³×C₃	1	2	yes	—
72	M₁₆	—	2	8	no	—
73	M₁₆	—	2	8	no	—
74	C₂×C₈	—	2	8	no	—
75	C₂×C₈	C₂³×C₃	1	8	yes	—
76	C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
77	C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
78	C₄²	C₂×C₃×C₄	1	4	yes	—
79	C₄²	C₂³×C₃	1	4	yes	—
80	C₂²×C₃	—	2	2	no	—
81	C₂²×C₃	—	2	4	no	—
82	C₂²×C₃	—	4	4	no	—
83	C₂²×C₃	C₂×C₄⋊C₄	1	4	yes	—
84	C₂²×C₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
86	C₃×C₄	—	2	8	no	—
87	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
88	C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
89	C₃×C₄	C₂²×D₄	1	1	yes	characteristic
90	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
91	C₂³	—	4	2	no	—
92	C₂³	C₃×C₄⋊C₄	1	4	yes	—
93	C₂³	C₂²×C₃×C₄	1	2	yes	—
94	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
95	C₂×C₄	—	4	4	no	—
96	C₂×C₄	C₃×C₄⋊C₄	1	4	yes	—
97	C₂×C₄	C₃×C₄⋊C₄	1	8	yes	—
98	C₂×C₄	C₂²×C₃×C₄	1	8	yes	—
99	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₃×C₄	1	2	yes	—
101	C₂×C₄	C₂×C₃×D₄	1	4	yes	—
102	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
103	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
104	C₂×C₄	C₂×C₃×Q₈	1	4	yes	—
105	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
106	C₈	—	2	8	no	—
107	C₂×C₃	—	2	2	no	—
108	C₂×C₃	—	2	4	no	—
109	C₂×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₃	#568	1	2	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂×C₃×C₄⋊C₄	1	4	yes	—
115	C₂²	C₂×C₃×C₄⋊C₄	1	2	yes	—
116	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
117	C₄	—	2	8	no	—
118	C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
119	C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
120	C₄	C₂²×C₃×D₄	1	1	yes	characteristic
121	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
122	C₃	#2905	1	1	yes	characteristic, direct product factor, Sylow
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
126	C₂	C₃×#568	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂⁴
commutator	C₂²	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₄
prime radical	C₂⁴×C₃	C₂×C₄
LCS	G ▷ C₂² ▷ 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×#424	C₂	semidirect
4. C₃×#423	C₂	semidirect
5. #2905	C₃	direct (N × H)
16. C₃×C₄⋊C₈	C₂²	semidirect
19. #424	C₂×C₃	semidirect
20. #423	C₂×C₃	semidirect
47. C₄⋊C₈	C₂²×C₃	semidirect
122. C₃	#2905	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×#424	2
C₃×#423	2
#2905	3

Subgroups — C3×#2905 #84769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2905 #84769