Subgroups — C₃×#3383 #85247

Order 384.

324 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#557	C₂	1	2	yes	maximal
2	C₃×#500	C₂	1	1	yes	maximal, characteristic
3	C₃×#499	C₂	1	2	yes	maximal
4	C₃×#479	C₂	1	4	yes	maximal
5	C₃×#475	C₂	1	2	yes	maximal
6	C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
7	C₃×C₄×Q₁₆	C₂	1	2	yes	maximal
8	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
9	#3383	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₃×C₄⋊Q₈	—	2	2	no	—
12	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×#125	—	2	4	no	—
14	C₃×#125	C₂²	1	4	yes	—
15	C₃×C₄×Q₈	—	2	2	no	—
16	C₃×C₄×Q₈	C₂²	1	2	yes	—
17	C₃×C₄×Q₈	C₂²	1	4	yes	—
18	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
19	C₃×#107	C₂²	1	1	yes	characteristic
20	C₃×#107	C₂²	1	2	yes	—
21	C₃×#107	C₂²	1	2	yes	—
22	C₃×#106	C₂²	1	4	yes	—
23	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊C₈	C₂²	1	2	yes	—
25	C₃×#103	C₂²	1	4	yes	—
26	C₃×#103	C₂²	1	2	yes	—
27	C₃×C₄×C₈	C₂²	1	2	yes	—
28	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	#557	C₂×C₃	1	2	yes	—
30	#500	C₂×C₃	1	1	yes	characteristic
31	#499	C₂×C₃	1	2	yes	—
32	#479	C₂×C₃	1	4	yes	—
33	#475	C₂×C₃	1	2	yes	—
34	C₈×Q₈	C₂×C₃	1	1	yes	characteristic
35	C₄×Q₁₆	C₂×C₃	1	2	yes	—
36	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
37	C₂×C₃×Q₈	—	2	2	no	—
38	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
39	C₂×C₃×Q₈	C₂³	1	2	yes	—
40	C₃×Q₁₆	Q₈	1	4	yes	complemented
41	C₂×C₃×C₈	C₂³	1	2	yes	—
42	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
44	C₃×C₄⋊C₄	—	2	4	no	—
45	C₃×C₄⋊C₄	—	2	2	no	—
46	C₃×C₄⋊C₄	—	2	4	no	—
47	C₃×C₄⋊C₄	—	2	2	no	—
48	C₃×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₃×C₄⋊C₄	D₄	1	2	yes	—
51	C₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₃×C₄⋊C₄	C₂³	1	4	yes	—
53	C₃×C₄²	—	2	4	no	—
54	C₃×C₄²	—	2	2	no	—
55	C₃×C₄²	C₂³	1	2	yes	—
56	C₃×C₄²	C₂³	1	1	yes	characteristic
57	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
60	#125	—	2	4	no	—
61	#125	C₂²×C₃	1	4	yes	—
62	C₄×Q₈	—	2	2	no	—
63	C₄×Q₈	C₂²×C₃	1	2	yes	—
64	C₄×Q₈	C₂²×C₃	1	4	yes	—
65	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
66	#107	C₂²×C₃	1	1	yes	characteristic
67	#107	C₂²×C₃	1	2	yes	—
68	#107	C₂²×C₃	1	2	yes	—
69	#106	C₂²×C₃	1	4	yes	—
70	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊C₈	C₂²×C₃	1	2	yes	—
72	#103	C₂²×C₃	1	4	yes	—
73	#103	C₂²×C₃	1	2	yes	—
74	C₄×C₈	C₂²×C₃	1	2	yes	—
75	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
76	C₃×Q₈	—	2	2	no	—
77	C₃×Q₈	—	2	2	no	—
78	C₃×Q₈	—	4	4	no	—
79	C₃×Q₈	C₂×Q₈	1	4	yes	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	2	2	no	—
82	C₂×C₃×C₄	—	2	4	no	—
83	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
88	C₃×C₈	—	2	2	no	—
89	C₃×C₈	—	2	1	no	—
90	C₃×C₈	C₂×Q₈	1	2	yes	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
93	C₂×Q₈	C₂³×C₃	1	2	yes	—
94	Q₁₆	C₃×Q₈	1	4	yes	complemented
95	C₂×C₈	C₂³×C₃	1	2	yes	—
96	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
97	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
104	C₄⋊C₄	C₃×D₄	1	2	yes	—
105	C₄⋊C₄	C₂³×C₃	1	2	yes	—
106	C₄⋊C₄	C₂³×C₃	1	4	yes	—
107	C₄²	—	2	4	no	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂³×C₃	1	2	yes	—
110	C₄²	C₂³×C₃	1	1	yes	characteristic
111	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
112	C₃×C₄	—	2	4	no	—
113	C₃×C₄	—	2	2	no	—
114	C₃×C₄	—	2	2	no	—
115	C₃×C₄	—	2	1	no	—
116	C₃×C₄	—	4	4	no	—
117	C₃×C₄	—	4	2	no	—
118	C₃×C₄	#135	1	2	yes	—
119	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
120	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
121	Q₈	—	2	2	no	—
122	Q₈	—	2	2	no	—
123	Q₈	—	4	4	no	—
124	Q₈	C₂×C₃×Q₈	1	4	yes	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
129	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
131	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	1	no	—
135	C₈	C₂×C₃×Q₈	1	2	yes	—
136	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
137	C₂×C₃	C₂×#135	1	1	yes	characteristic
138	C₂×C₃	#578	1	1	yes	characteristic
139	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	4	no	—
144	C₄	—	4	4	no	—
145	C₄	—	4	2	no	—
146	C₄	C₃×#135	1	2	yes	—
147	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
148	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
149	C₃	#3383	1	1	yes	characteristic, direct product factor, Sylow
150	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
151	C₂	C₂×C₃×#135	1	1	yes	characteristic
152	C₂	C₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
9. #3383	C₃	direct (N × H)
40. C₃×Q₁₆	Q₈	semidirect
94. Q₁₆	C₃×Q₈	semidirect
149. C₃	#3383	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#557	2
C₃×#500	2
C₃×#499	2
C₃×#479	2
C₃×#475	2
C₃×C₈×Q₈	2
C₃×C₄×Q₁₆	2
#3383	3

Subgroups — C3×#3383 #85247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3383 #85247