Subgroups — C₃×#2985 #84849

Order 384.

322 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×#446	C₂	1	2	yes	maximal, complemented
3	C₃×#446	C₂	1	4	yes	maximal, complemented
4	C₃×C₈×Q₈	C₂	1	2	yes	maximal, complemented
5	C₃×#433	C₂	1	4	yes	maximal
6	C₃×#432	C₂	1	1	yes	maximal, characteristic
7	C₃×#431	C₂	1	1	yes	maximal, characteristic
8	#2985	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₃×#125	C₄	1	2	yes	—
12	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
13	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₃×C₄×Q₈	C₂²	1	2	yes	—
15	C₃×C₄×Q₈	C₂²	1	2	yes	—
16	C₃×#117	C₂²	1	2	yes	—
17	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊C₈	C₂²	1	2	yes	—
19	C₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₃×C₄⋊C₈	C₂²	1	4	yes	—
21	C₃×C₄⋊C₈	C₂²	1	4	yes	—
22	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₃×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₃×#97	C₂²	1	2	yes	—
25	C₃×#97	C₂²	1	4	yes	—
26	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
27	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
28	C₃×C₄×C₈	C₂²	1	4	yes	—
29	#533	C₂×C₃	1	1	yes	characteristic
30	#446	C₂×C₃	1	2	yes	complemented
31	#446	C₂×C₃	1	4	yes	complemented
32	C₈×Q₈	C₂×C₃	1	2	yes	complemented
33	#433	C₂×C₃	1	4	yes	—
34	#432	C₂×C₃	1	1	yes	characteristic
35	#431	C₂×C₃	1	1	yes	characteristic
36	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₈	C₂³	1	2	yes	—
42	C₂×C₃×C₈	C₂³	1	2	yes	—
43	C₂×C₃×C₈	C₂³	1	4	yes	—
44	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
47	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
51	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄²	C₂³	1	1	yes	characteristic
54	C₃×C₄²	C₂³	1	4	yes	—
55	C₃×C₄²	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
57	C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
58	#125	C₃×C₄	1	2	yes	—
59	C₂²⋊Q₈	C₃×C₄	1	2	yes	—
60	C₂²⋊Q₈	C₃×C₄	1	2	yes	—
61	C₄×Q₈	C₂²×C₃	1	2	yes	—
62	C₄×Q₈	C₂²×C₃	1	2	yes	—
63	#117	C₂²×C₃	1	2	yes	—
64	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
65	C₄⋊C₈	C₂²×C₃	1	2	yes	—
66	C₄⋊C₈	C₂²×C₃	1	2	yes	—
67	C₄⋊C₈	C₂²×C₃	1	4	yes	—
68	C₄⋊C₈	C₂²×C₃	1	4	yes	—
69	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
70	C₂²⋊C₈	C₂²×C₃	1	2	yes	—
71	#97	C₂²×C₃	1	2	yes	—
72	#97	C₂²×C₃	1	4	yes	—
73	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₄×C₈	C₂²×C₃	1	4	yes	—
76	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
77	C₃×Q₈	—	2	4	no	—
78	C₃×Q₈	—	4	2	no	—
79	C₂×C₃×C₄	—	2	2	no	—
80	C₂×C₃×C₄	—	2	2	no	—
81	C₂×C₃×C₄	—	4	1	no	—
82	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
87	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
88	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
90	C₃×C₈	—	2	2	no	—
91	C₃×C₈	—	2	2	no	—
92	C₃×C₈	—	2	4	no	—
93	C₂×Q₈	C₂×C₃×C₄	1	2	yes	—
94	C₂×Q₈	C₂×C₃×C₄	1	2	yes	—
95	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₈	C₂³×C₃	1	2	yes	—
99	C₂×C₈	C₂³×C₃	1	2	yes	—
100	C₂×C₈	C₂³×C₃	1	4	yes	—
101	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
102	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
103	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
104	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
105	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
106	C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
107	C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
108	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
109	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
110	C₄²	C₂³×C₃	1	1	yes	characteristic
111	C₄²	C₂³×C₃	1	4	yes	—
112	C₄²	C₂³×C₃	1	1	yes	characteristic
113	C₂²×C₃	—	2	1	no	—
114	C₂²×C₃	—	2	1	no	—
115	C₂²×C₃	—	2	1	no	—
116	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	2	1	no	—
119	C₃×C₄	—	2	4	no	—
120	C₃×C₄	—	2	1	no	—
121	C₃×C₄	—	2	1	no	—
122	C₃×C₄	—	2	4	no	—
123	C₃×C₄	#131	1	2	yes	—
124	C₃×C₄	2₋¹⁺⁴	1	2	yes	—
125	C₂³	C₂²×C₃×C₄	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
136	C₂×C₄	C₂²×C₃×C₄	1	4	yes	—
137	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	2	no	—
141	C₈	—	2	4	no	—
142	C₂×C₃	—	4	1	no	—
143	C₂×C₃	#519	1	1	yes	characteristic
144	C₂×C₃	C₂×#131	1	1	yes	characteristic
145	C₂×C₃	#568	1	1	yes	characteristic
146	C₂²	—	2	1	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	C₂³×C₃×C₄	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	4	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	4	no	—
155	C₄	—	2	1	no	—
156	C₄	C₃×#131	1	2	yes	—
157	C₄	C₃×2₋¹⁺⁴	1	2	yes	—
158	C₃	#2985	1	1	yes	characteristic, direct product factor, Sylow
159	C₂	—	4	1	no	—
160	C₂	C₃×#519	1	1	yes	characteristic
161	C₂	C₂×C₃×#131	1	1	yes	characteristic
162	C₂	C₃×#568	1	1	yes	characteristic

	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	C₂³×C₃	C₂²×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
2. C₃×#446	C₂	semidirect
3. C₃×#446	C₂	semidirect
4. C₃×C₈×Q₈	C₂	semidirect
8. #2985	C₃	direct (N × H)
30. #446	C₂×C₃	semidirect
31. #446	C₂×C₃	semidirect
32. C₈×Q₈	C₂×C₃	semidirect
158. C₃	#2985	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×#533	2
C₃×#446	2
C₃×C₈×Q₈	2
C₃×#433	2
C₃×#432	2
C₃×#431	2
#2985	3

Subgroups — C3×#2985 #84849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#2985 #84849