Subgroups — C₃×#3390 #85254

Order 384.

320 conjugacy classes of subgroups.

The subgroup lattice has 182 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×#501	C₂	1	2	yes	maximal
3	C₃×#499	C₂	1	1	yes	maximal, characteristic
4	C₃×#479	C₂	1	2	yes	maximal
5	C₃×#479	C₂	1	2	yes	maximal
6	C₃×#477	C₂	1	2	yes	maximal
7	C₃×#446	C₂	1	1	yes	maximal, characteristic
8	C₃×#441	C₂	1	2	yes	maximal
9	C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
10	#3390	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×Q₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊Q₈	—	2	2	no	—
13	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₃×#125	—	2	2	no	—
15	C₃×#125	—	2	2	no	—
16	C₃×#125	C₂²	1	2	yes	—
17	C₃×#125	C₂²	1	2	yes	—
18	C₃×C₄×Q₈	—	2	2	no	—
19	C₃×C₄×Q₈	C₂²	1	2	yes	—
20	C₃×C₄×Q₈	C₂²	1	2	yes	—
21	C₃×C₄×Q₈	C₂²	1	2	yes	—
22	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
23	C₃×#107	C₂²	1	1	yes	characteristic
24	C₃×#107	C₂²	1	1	yes	characteristic
25	C₃×#107	C₂²	1	2	yes	—
26	C₃×#107	C₂²	1	1	yes	characteristic
27	C₃×#106	C₂²	1	2	yes	—
28	C₃×#106	C₂²	1	2	yes	—
29	C₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃×#103	C₂²	1	2	yes	—
32	C₃×#103	C₂²	1	2	yes	—
33	C₃×#103	C₂²	1	2	yes	—
34	C₃×#97	C₂²	1	2	yes	—
35	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
36	#557	C₂×C₃	1	2	yes	—
37	#501	C₂×C₃	1	2	yes	—
38	#499	C₂×C₃	1	1	yes	characteristic
39	#479	C₂×C₃	1	2	yes	—
40	#479	C₂×C₃	1	2	yes	—
41	#477	C₂×C₃	1	2	yes	—
42	#446	C₂×C₃	1	1	yes	characteristic
43	#441	C₂×C₃	1	2	yes	—
44	C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
45	C₂×C₃×Q₈	—	2	2	no	—
46	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
47	C₂×C₃×Q₈	C₂³	1	2	yes	—
48	C₃×Q₁₆	Q₈	1	4	yes	complemented
49	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₃×C₈	C₂³	1	2	yes	—
51	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
52	C₃×C₄⋊C₄	—	2	2	no	—
53	C₃×C₄⋊C₄	—	2	2	no	—
54	C₃×C₄⋊C₄	—	2	2	no	—
55	C₃×C₄⋊C₄	—	2	2	no	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	2	2	no	—
59	C₃×C₄⋊C₄	—	2	2	no	—
60	C₃×C₄⋊C₄	D₄	1	2	yes	—
61	C₃×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₃×C₄²	—	2	2	no	—
66	C₃×C₄²	—	2	2	no	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	C₂³	1	1	yes	characteristic
69	C₃×C₄²	C₂³	1	2	yes	—
70	C₂×Q₁₆	C₂²×C₃	1	1	yes	characteristic
71	C₄⋊Q₈	—	2	2	no	—
72	C₄⋊Q₈	C₂²×C₃	1	2	yes	—
73	#125	—	2	2	no	—
74	#125	—	2	2	no	—
75	#125	C₂²×C₃	1	2	yes	—
76	#125	C₂²×C₃	1	2	yes	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	C₂²×C₃	1	2	yes	—
79	C₄×Q₈	C₂²×C₃	1	2	yes	—
80	C₄×Q₈	C₂²×C₃	1	2	yes	—
81	C₄×Q₈	C₂²×C₃	1	1	yes	characteristic
82	#107	C₂²×C₃	1	1	yes	characteristic
83	#107	C₂²×C₃	1	1	yes	characteristic
84	#107	C₂²×C₃	1	2	yes	—
85	#107	C₂²×C₃	1	1	yes	characteristic
86	#106	C₂²×C₃	1	2	yes	—
87	#106	C₂²×C₃	1	2	yes	—
88	C₄⋊C₈	C₂²×C₃	1	2	yes	—
89	C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
90	#103	C₂²×C₃	1	2	yes	—
91	#103	C₂²×C₃	1	2	yes	—
92	#103	C₂²×C₃	1	2	yes	—
93	#97	C₂²×C₃	1	2	yes	—
94	C₄×C₈	C₂²×C₃	1	1	yes	characteristic
95	C₃×Q₈	—	2	2	no	—
96	C₃×Q₈	—	4	1	no	—
97	C₃×Q₈	—	4	4	no	—
98	C₃×Q₈	C₂×Q₈	1	4	yes	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
105	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
106	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
108	C₃×C₈	—	2	1	no	—
109	C₃×C₈	—	2	2	no	—
110	C₃×C₈	C₂×Q₈	1	2	yes	—
111	C₂×Q₈	—	2	2	no	—
112	C₂×Q₈	C₃×D₄	1	1	yes	characteristic
113	C₂×Q₈	C₂³×C₃	1	2	yes	—
114	Q₁₆	C₃×Q₈	1	4	yes	complemented
115	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
116	C₂×C₈	C₂³×C₃	1	2	yes	—
117	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	C₃×D₄	1	2	yes	—
127	C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
128	C₄⋊C₄	C₂³×C₃	1	2	yes	—
129	C₄⋊C₄	C₂³×C₃	1	2	yes	—
130	C₄⋊C₄	C₂³×C₃	1	2	yes	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	C₂³×C₃	1	1	yes	characteristic
135	C₄²	C₂³×C₃	1	2	yes	—
136	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
137	C₃×C₄	—	2	4	no	—
138	C₃×C₄	—	2	1	no	—
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	—	2	1	no	—
141	C₃×C₄	—	2	2	no	—
142	C₃×C₄	—	4	2	no	—
143	C₃×C₄	—	4	2	no	—
144	C₃×C₄	—	4	2	no	—
145	C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
146	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
147	Q₈	—	2	2	no	—
148	Q₈	—	4	1	no	—
149	Q₈	—	4	4	no	—
150	Q₈	C₂×C₃×Q₈	1	4	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
157	C₂×C₄	C₂×C₃×D₄	1	2	yes	—
158	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₄	C₂⁴×C₃	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	2	2	no	—
162	C₈	C₂×C₃×Q₈	1	2	yes	—
163	C₂×C₃	D₄×Q₈	1	1	yes	characteristic
164	C₂×C₃	#575	1	1	yes	characteristic
165	C₂×C₃	#578	1	1	yes	characteristic
166	C₂²	C₂²×C₃×D₄	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	4	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
176	C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
177	C₃	#3390	1	1	yes	characteristic, direct product factor, Sylow
178	C₂	C₃×D₄×Q₈	1	1	yes	characteristic
179	C₂	C₃×#575	1	1	yes	characteristic
180	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
10. #3390	C₃	direct (N × H)
48. C₃×Q₁₆	Q₈	semidirect
114. Q₁₆	C₃×Q₈	semidirect
177. C₃	#3390	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×#501	2
C₃×#499	2
C₃×#479	2
C₃×#477	2
C₃×#446	2
C₃×#441	2
C₃×C₄×Q₁₆	2
#3390	3

Subgroups — C3×#3390 #85254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3390 #85254