Subgroups — C₃×Q₈×Dic₁₈ #13152667

Order 1728.

706 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×Dic₉	C₂	1	2	yes	maximal
3	C₃×#63450	C₂	1	6	yes	maximal
4	C₃×#63431	C₂	1	3	yes	maximal
5	C₃×C₄×Dic₁₈	C₂	1	3	yes	maximal
6	C₃×Q₈×Dic₆	—	3	1	no	maximal
7	Q₈×Dic₁₈	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₁₈	C₂²	1	6	yes	—
10	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
11	C₃×C₉×C₄⋊C₄	C₂²	1	3	yes	—
12	C₃×C₄²×C₉	C₂²	1	3	yes	—
13	C₃×C₄⋊Dic₉	C₂²	1	3	yes	—
14	C₃×C₄⋊Dic₉	C₂²	1	6	yes	—
15	C₃×Dic₉⋊C₄	C₂²	1	6	yes	—
16	C₃×C₄×Dic₉	C₂²	1	6	yes	—
17	C₃²×C₄×Q₈	D₃	1	1	yes	characteristic
18	C₃×Q₈×Dic₃	—	3	2	no	—
19	C₃×#888	—	3	6	no	—
20	C₃×#869	—	3	3	no	—
21	C₃×C₄×Dic₆	—	3	3	no	—
22	C₄×Q₈×C₉	—	2	1	no	—
23	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
24	Q₈×Dic₉	C₂×C₃	1	2	yes	—
25	#63450	C₂×C₃	1	6	yes	—
26	#63431	C₂×C₃	1	3	yes	—
27	C₄×Dic₁₈	C₂×C₃	1	3	yes	—
28	C₃×Q₈×C₉	Q₈	1	4	yes	complemented
29	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
32	C₂×C₃×Dic₉	C₂³	1	6	yes	—
33	C₂×C₃×Dic₉	C₂³	1	2	yes	—
34	C₃×Dic₁₈	—	4	6	no	—
35	C₃×Dic₁₈	Q₈	1	4	yes	direct product factor
36	C₃×Q₈²	—	9	1	no	—
37	Q₈×Dic₆	—	3	1	no	—
38	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×Dic₆	—	3	1	no	—
40	C₂×C₃×Dic₆	—	3	6	no	—
41	C₃²×C₄⋊C₄	C₂×D₃	1	3	yes	—
42	C₃²×C₄²	C₂×D₃	1	3	yes	—
43	C₃×C₄⋊Dic₃	—	3	6	no	—
44	C₃×C₄⋊Dic₃	—	3	3	no	—
45	C₃×Dic₃⋊C₄	—	3	6	no	—
46	C₃×C₄×Dic₃	—	3	6	no	—
47	C₂×Q₈×C₉	—	2	1	no	—
48	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
49	C₂×Dic₁₈	C₂²×C₃	1	6	yes	—
50	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
51	C₉×C₄⋊C₄	—	2	3	no	—
52	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
53	C₄²×C₉	—	2	3	no	—
54	C₄²×C₉	C₂²×C₃	1	3	yes	—
55	C₄⋊Dic₉	C₂²×C₃	1	6	yes	—
56	C₄⋊Dic₉	C₂²×C₃	1	3	yes	—
57	Dic₉⋊C₄	C₂²×C₃	1	6	yes	—
58	C₄×Dic₉	C₂²×C₃	1	6	yes	—
59	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
60	C₃×C₄×C₉	—	2	3	no	—
61	C₃×C₄×C₉	C₂×Q₈	1	6	yes	—
62	C₃×C₄×C₉	C₂×Q₈	1	2	yes	—
63	C₃×Dic₉	—	2	6	no	—
64	C₃×Dic₉	C₂×Q₈	1	4	yes	—
65	C₃×C₄⋊Q₈	—	9	6	no	—
66	C₃×C₄⋊Q₈	—	9	3	no	—
67	C₃×C₄×Q₈	—	2	1	no	—
68	C₃×C₄×Q₈	—	9	2	no	—
69	C₃×C₄×Q₈	—	9	3	no	—
70	C₃×C₄×Q₈	D₉	1	1	yes	characteristic
71	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
72	Q₈×Dic₃	—	3	2	no	—
73	#888	—	3	6	no	—
74	#869	—	3	3	no	—
75	C₄×Dic₆	—	3	3	no	—
76	C₃²×Q₈	Dic₆	1	4	yes	—
77	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
78	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
79	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₃×Dic₃	—	3	2	no	—
81	C₂×C₃×Dic₃	—	3	6	no	—
82	C₃×Dic₆	—	3	4	no	—
83	C₃×Dic₆	—	12	6	no	—
84	Q₈×C₉	—	2	4	no	—
85	Q₈×C₉	C₃×Q₈	1	4	yes	complemented
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	2	3	no	—
88	C₂×C₄×C₉	—	2	3	no	—
89	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
90	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
91	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
92	C₂×Dic₉	C₂³×C₃	1	6	yes	—
93	C₂×Dic₉	C₂³×C₃	1	2	yes	—
94	Dic₁₈	—	4	6	no	—
95	Dic₁₈	C₃×Q₈	1	4	yes	direct product factor
96	Q₈²	—	9	1	no	Sylow
97	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
100	C₂×C₃×Q₈	—	2	1	no	—
101	C₂×C₃×Q₈	—	9	1	no	—
102	C₂×C₃×Q₈	—	9	6	no	—
103	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
104	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
105	C₂×Dic₆	—	3	1	no	—
106	C₂×Dic₆	—	3	6	no	—
107	C₃×C₄⋊C₄	—	2	3	no	—
108	C₃×C₄⋊C₄	—	9	6	no	—
109	C₃×C₄⋊C₄	—	9	6	no	—
110	C₃×C₄⋊C₄	—	9	3	no	—
111	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
112	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
113	C₃×C₄²	—	2	3	no	—
114	C₃×C₄²	—	9	6	no	—
115	C₃×C₄²	C₂×D₉	1	3	yes	—
116	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
117	C₄⋊Dic₃	—	3	3	no	—
118	C₄⋊Dic₃	—	3	6	no	—
119	Dic₃⋊C₄	—	3	6	no	—
120	C₄×Dic₃	—	3	6	no	—
121	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
122	C₃²×C₄	—	2	3	no	—
123	C₃²×C₄	C₂×Dic₆	1	6	yes	—
124	C₃²×C₄	D₃×Q₈	1	2	yes	—
125	C₃×Dic₃	—	3	4	no	—
126	C₃×Dic₃	—	6	6	no	—
127	C₂²×C₉	—	2	1	no	—
128	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
129	C₄×C₉	—	2	3	no	—
130	C₄×C₉	—	2	6	no	—
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	4	3	no	—
133	C₄×C₉	C₂×C₃×Q₈	1	6	yes	—
134	C₄×C₉	C₂×C₃×Q₈	1	2	yes	—
135	Dic₉	—	2	6	no	—
136	Dic₉	C₂×C₃×Q₈	1	4	yes	—
137	C₄⋊Q₈	—	9	6	no	—
138	C₄⋊Q₈	—	9	3	no	—
139	C₄×Q₈	—	9	2	no	—
140	C₄×Q₈	—	9	3	no	—
141	C₄×Q₈	C₃×D₉	1	1	yes	characteristic
142	C₃×C₉	Q₈²	1	1	yes	characteristic, complemented, Sylow
143	C₃×Q₈	—	2	4	no	—
144	C₃×Q₈	—	9	4	no	—
145	C₃×Q₈	—	36	6	no	—
146	C₃×Q₈	Dic₁₈	1	4	yes	direct product factor
147	C₃×Q₈	C₃×Dic₆	1	4	yes	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	—	2	3	no	—
150	C₂×C₃×C₄	—	2	3	no	—
151	C₂×C₃×C₄	—	9	2	no	—
152	C₂×C₃×C₄	—	9	6	no	—
153	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
154	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
155	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
157	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
159	C₂×Dic₃	—	3	2	no	—
160	C₂×Dic₃	—	3	6	no	—
161	Dic₆	—	3	4	no	—
162	Dic₆	—	12	6	no	—
163	C₂×C₃²	C₂²×Dic₆	1	1	yes	characteristic
164	C₂×C₃²	C₂×D₃×Q₈	1	1	yes	characteristic
165	C₂×C₃²	#1009	1	1	yes	characteristic
166	C₂×C₉	—	2	1	no	—
167	C₂×C₉	—	2	1	no	—
168	C₂×C₉	—	2	1	no	—
169	C₂×C₉	C₂²×C₃×Q₈	1	1	yes	characteristic
170	C₂×C₉	C₂²×C₃×Q₈	1	1	yes	characteristic
171	C₂×C₉	C₃×H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×Q₈	—	9	1	no	—
173	C₂×Q₈	—	9	6	no	—
174	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
175	C₄⋊C₄	—	9	6	no	—
176	C₄⋊C₄	—	9	6	no	—
177	C₄⋊C₄	—	9	3	no	—
178	C₄⋊C₄	C₂×C₃×D₉	1	3	yes	—
179	C₄²	—	9	6	no	—
180	C₄²	C₂×C₃×D₉	1	3	yes	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
183	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
184	C₃×C₄	—	2	3	no	—
185	C₃×C₄	—	2	3	no	—
186	C₃×C₄	—	2	6	no	—
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	4	3	no	—
189	C₃×C₄	—	9	4	no	—
190	C₃×C₄	—	18	6	no	—
191	C₃×C₄	C₂×Dic₁₈	1	6	yes	—
192	C₃×C₄	Q₈×D₉	1	2	yes	—
193	C₃×C₄	C₂×C₃×Dic₆	1	6	yes	—
194	C₃×C₄	C₃×D₃×Q₈	1	2	yes	—
195	Dic₃	—	3	4	no	—
196	Dic₃	—	6	6	no	—
197	C₃²	Q₈×Dic₆	1	1	yes	characteristic
198	C₉	—	2	1	no	—
199	C₉	C₃×Q₈²	1	1	yes	characteristic, complemented
200	Q₈	—	9	4	no	—
201	Q₈	—	36	6	no	—
202	Q₈	C₃×Dic₁₈	1	4	yes	direct product factor
203	C₂×C₄	—	9	2	no	—
204	C₂×C₄	—	9	6	no	—
205	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
206	C₂×C₄	C₂²×C₃×D₉	1	3	yes	—
207	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	2	1	no	—
211	C₂×C₃	C₂²×Dic₁₈	1	1	yes	characteristic
212	C₂×C₃	C₂×Q₈×D₉	1	1	yes	characteristic
213	C₂×C₃	#63715	1	1	yes	characteristic
214	C₂×C₃	C₂²×C₃×Dic₆	1	1	yes	characteristic
215	C₂×C₃	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
216	C₂×C₃	C₃×#1009	1	1	yes	characteristic
217	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
218	C₄	—	2	3	no	—
219	C₄	—	9	4	no	—
220	C₄	—	18	6	no	—
221	C₄	C₂×C₃×Dic₁₈	1	6	yes	—
222	C₄	C₃×Q₈×D₉	1	2	yes	—
223	C₃	—	2	1	no	—
224	C₃	Q₈×Dic₁₈	1	1	yes	characteristic, direct product factor
225	C₃	C₃×Q₈×Dic₆	1	1	yes	characteristic
226	C₂	C₂²×C₃×Dic₁₈	1	1	yes	characteristic
227	C₂	C₂×C₃×Q₈×D₉	1	1	yes	characteristic
228	C₂	C₃×#63715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂³×D₃
hypercenter	C₃×C₄×Q₈	D₉
hypocenter	C₉	C₃×Q₈²
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
7. Q₈×Dic₁₈	C₃	direct (N × H)
28. C₃×Q₈×C₉	Q₈	semidirect
35. C₃×Dic₁₈	Q₈	direct (N × H)
85. Q₈×C₉	C₃×Q₈	semidirect
95. Dic₁₈	C₃×Q₈	direct (N × H)
142. C₃×C₉	Q₈²	semidirect
146. C₃×Q₈	Dic₁₈	direct (N × H)
199. C₉	C₃×Q₈²	semidirect
202. Q₈	C₃×Dic₁₈	direct (N × H)
224. C₃	Q₈×Dic₁₈	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×C₄×Q₈×C₉	2
C₃×Q₈×Dic₉	2
C₃×#63450	2
C₃×#63431	2
C₃×C₄×Dic₁₈	2
C₃×Q₈×Dic₆	3
Q₈×Dic₁₈	3

Subgroups — C3×Q8×Dic18 #13152667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×Dic₁₈ #13152667