Subgroups — C₂×C₃×Q₈×C₃⋊Dic₃ #13179559

Order 1728.

2412 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₃⋊Dic₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₄⋊(C₃⋊Dic₃)	C₂	1	3	yes	maximal
4	C₂×C₃×C₄×C₃⋊Dic₃	C₂	1	3	yes	maximal
5	C₂×Q₈×C₃⋊Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×Q₈×Dic₃	—	3	4	no	maximal
7	C₂×C₃³×Q₈	C₄	1	8	yes	complemented
8	C₂×C₃³×Q₈	C₂²	1	4	yes	—
9	C₂²×C₃³×C₄	C₂²	1	3	yes	—
10	C₂²×C₃×C₃⋊Dic₃	C₂²	1	3	yes	—
11	C₂²×C₃×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊(C₃⋊Dic₃)	C₂²	1	12	yes	—
13	C₃×C₄×C₃⋊Dic₃	C₂²	1	12	yes	—
14	C₂²×C₃²×Q₈	—	2	4	no	—
15	C₂²×C₃²×Q₈	D₃	1	4	yes	—
16	C₂²×C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
17	Q₈×C₃⋊Dic₃	C₂×C₃	1	8	yes	direct product factor
18	C₂×C₄⋊(C₃⋊Dic₃)	C₂×C₃	1	3	yes	—
19	C₂×C₄×C₃⋊Dic₃	C₂×C₃	1	3	yes	—
20	C₃×Q₈×Dic₃	—	3	32	no	—
21	C₂×C₃×C₄⋊Dic₃	—	3	12	no	—
22	C₂×C₃×C₄×Dic₃	—	3	12	no	—
23	C₂³×C₃³	C₂³	1	1	yes	characteristic
24	C₃³×Q₈	C₂×C₄	1	16	yes	complemented
25	C₂×C₃³×C₄	C₂×C₄	1	12	yes	—
26	C₂×C₃³×C₄	C₂³	1	6	yes	—
27	C₂×C₃×C₃⋊Dic₃	—	2	6	no	—
28	C₂×C₃×C₃⋊Dic₃	Q₈	1	4	yes	direct product factor
29	C₂×C₃×C₃⋊Dic₃	C₂³	1	6	yes	—
30	C₂×C₃×C₃⋊Dic₃	C₂³	1	2	yes	—
31	C₂×C₃×C₄×Q₈	—	9	1	no	—
32	C₂×Q₈×Dic₃	—	3	4	no	—
33	C₂×C₃²×Q₈	—	2	32	no	—
34	C₂×C₃²×Q₈	—	2	16	no	—
35	C₂×C₃²×Q₈	Dic₃	1	32	yes	complemented
36	C₂×C₃²×Q₈	C₃×C₄	1	8	yes	complemented
37	C₂×C₃²×Q₈	C₂×D₃	1	16	yes	—
38	C₂×C₃²×Q₈	C₂²×C₃	1	4	yes	—
39	C₂²×C₃²×C₄	—	2	12	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	12	yes	—
41	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
42	C₂²×C₃⋊Dic₃	C₂²×C₃	1	3	yes	—
43	C₂²×C₃⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃×Dic₃	—	3	4	no	—
45	C₂²×C₃×Dic₃	—	3	12	no	—
46	C₄⋊(C₃⋊Dic₃)	C₂²×C₃	1	12	yes	—
47	C₄×C₃⋊Dic₃	C₂²×C₃	1	12	yes	—
48	C₃×C₄⋊Dic₃	—	3	48	no	—
49	C₃×C₄×Dic₃	—	3	48	no	—
50	C₂²×C₃³	C₂²×C₄	1	2	yes	—
51	C₂²×C₃³	C₂×Q₈	1	2	yes	—
52	C₂²×C₃³	#41	1	2	yes	—
53	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
54	C₃³×C₄	C₂²×C₄	1	12	yes	—
55	C₃×C₃⋊Dic₃	—	2	6	no	—
56	C₃×C₃⋊Dic₃	C₂×Q₈	1	4	yes	direct product factor
57	C₂²×C₃×Q₈	—	2	4	no	—
58	C₂²×C₃×Q₈	C₃×D₃	1	4	yes	—
59	C₂²×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
60	C₃×C₄×Q₈	—	9	8	no	—
61	C₂×C₃×C₄⋊C₄	—	9	3	no	—
62	C₂×C₃×C₄²	—	9	3	no	—
63	Q₈×Dic₃	—	3	32	no	—
64	C₂×C₄⋊Dic₃	—	3	12	no	—
65	C₂×C₄×Dic₃	—	3	12	no	—
66	C₂³×C₃²	—	2	4	no	—
67	C₂³×C₃²	C₂²×D₃	1	4	yes	—
68	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
69	C₃²×Q₈	—	2	64	no	—
70	C₃²×Q₈	C₂×Dic₃	1	64	yes	complemented
71	C₃²×Q₈	C₂×C₃×C₄	1	16	yes	complemented
72	C₂×C₃²×C₄	—	2	48	no	—
73	C₂×C₃²×C₄	—	2	24	no	—
74	C₂×C₃²×C₄	C₂×Dic₃	1	48	yes	—
75	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
76	C₂×C₃²×C₄	C₂²×D₃	1	24	yes	—
77	C₂×C₃²×C₄	C₂³×C₃	1	6	yes	—
78	C₂×C₃⋊Dic₃	—	2	6	no	—
79	C₂×C₃⋊Dic₃	C₃×Q₈	1	4	yes	direct product factor
80	C₂×C₃⋊Dic₃	C₂³×C₃	1	6	yes	—
81	C₂×C₃⋊Dic₃	C₂³×C₃	1	2	yes	—
82	C₂×C₃×Dic₃	—	3	16	no	—
83	C₂×C₃×Dic₃	—	3	8	no	—
84	C₂×C₃×Dic₃	—	3	24	no	—
85	C₂×C₃×Dic₃	—	6	24	no	—
86	C₂×C₄×Q₈	—	9	1	no	Sylow
87	C₂×C₃³	C₄×Q₈	1	4	yes	complemented
88	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
90	C₂×C₃³	C₂×#41	1	1	yes	characteristic
91	C₂×C₃×Q₈	—	2	32	no	—
92	C₂×C₃×Q₈	—	2	16	no	—
93	C₂×C₃×Q₈	C₃×Dic₃	1	32	yes	complemented
94	C₂×C₃×Q₈	C₃⋊Dic₃	1	8	yes	direct product factor
95	C₂×C₃×Q₈	C₂×C₃×D₃	1	16	yes	—
96	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	4	yes	—
97	C₂²×C₃×C₄	—	2	12	no	—
98	C₂²×C₃×C₄	—	9	1	no	—
99	C₂²×C₃×C₄	—	9	3	no	—
100	C₂²×C₃×C₄	C₂×C₃×D₃	1	12	yes	—
101	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
102	C₂²×Dic₃	—	3	4	no	—
103	C₂²×Dic₃	—	3	12	no	—
104	C₃×C₄⋊C₄	—	9	12	no	—
105	C₃×C₄²	—	9	12	no	—
106	C₄⋊Dic₃	—	3	48	no	—
107	C₄×Dic₃	—	3	48	no	—
108	C₂²×C₃²	—	2	4	no	—
109	C₂²×C₃²	—	2	8	no	—
110	C₂²×C₃²	—	2	8	no	—
111	C₂²×C₃²	—	2	8	no	—
112	C₂²×C₃²	D₃×Q₈	1	8	yes	—
113	C₂²×C₃²	#239	1	8	yes	—
114	C₂²×C₃²	C₂²×Dic₃	1	8	yes	—
115	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
116	C₂²×C₃²	C₂×C₃×Q₈	1	2	yes	—
117	C₂²×C₃²	C₃×#41	1	2	yes	—
118	C₂²×C₃²	C₂³×D₃	1	4	yes	—
119	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
120	C₃²×C₄	—	2	48	no	—
121	C₃²×C₄	C₂²×Dic₃	1	48	yes	—
122	C₃²×C₄	C₂²×C₃×C₄	1	12	yes	—
123	C₃⋊Dic₃	—	2	6	no	—
124	C₃⋊Dic₃	C₂×C₃×Q₈	1	4	yes	direct product factor
125	C₃×Dic₃	—	3	16	no	—
126	C₃×Dic₃	—	6	24	no	—
127	C₂²×Q₈	C₃×C₃⋊D₃	1	1	yes	characteristic
128	C₄×Q₈	—	9	8	no	—
129	C₂×C₄⋊C₄	—	9	3	no	—
130	C₂×C₄²	—	9	3	no	—
131	C₃³	C₂×C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
132	C₂³×C₃	—	2	4	no	—
133	C₂³×C₃	C₂²×C₃×D₃	1	4	yes	—
134	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
135	C₃×Q₈	—	2	64	no	—
136	C₃×Q₈	C₂×C₃×Dic₃	1	64	yes	complemented
137	C₃×Q₈	C₂×C₃⋊Dic₃	1	16	yes	direct product factor
138	C₂×C₃×C₄	—	2	48	no	—
139	C₂×C₃×C₄	—	2	24	no	—
140	C₂×C₃×C₄	—	9	2	no	—
141	C₂×C₃×C₄	—	9	4	no	—
142	C₂×C₃×C₄	—	9	6	no	—
143	C₂×C₃×C₄	—	18	6	no	—
144	C₂×C₃×C₄	C₂×C₃×Dic₃	1	48	yes	—
145	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	12	yes	—
146	C₂×C₃×C₄	C₂²×C₃×D₃	1	24	yes	—
147	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	6	yes	—
148	C₂×Dic₃	—	3	8	no	—
149	C₂×Dic₃	—	3	16	no	—
150	C₂×Dic₃	—	3	24	no	—
151	C₂×Dic₃	—	6	24	no	—
152	C₂×C₃²	—	2	16	no	—
153	C₂×C₃²	—	2	4	no	—
154	C₂×C₃²	—	2	4	no	—
155	C₂×C₃²	—	2	4	no	—
156	C₂×C₃²	Q₈×Dic₃	1	16	yes	complemented
157	C₂×C₃²	C₃×C₄×Q₈	1	4	yes	complemented
158	C₂×C₃²	C₂×D₃×Q₈	1	4	yes	—
159	C₂×C₃²	C₂×#239	1	4	yes	—
160	C₂×C₃²	C₂³×Dic₃	1	4	yes	—
161	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
162	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
163	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
164	C₂×Q₈	C₃×C₃⋊Dic₃	1	8	yes	direct product factor
165	C₂×Q₈	C₂×C₃×C₃⋊D₃	1	4	yes	—
166	C₂²×C₄	—	9	1	no	—
167	C₂²×C₄	—	9	3	no	—
168	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	3	yes	—
169	C₄⋊C₄	—	9	12	no	—
170	C₄²	—	9	12	no	—
171	C₂²×C₃	—	2	4	no	—
172	C₂²×C₃	—	2	8	no	—
173	C₂²×C₃	—	2	8	no	—
174	C₂²×C₃	—	2	8	no	—
175	C₂²×C₃	C₃×D₃×Q₈	1	8	yes	—
176	C₂²×C₃	C₃×#239	1	8	yes	—
177	C₂²×C₃	C₂²×C₃×Dic₃	1	8	yes	—
178	C₂²×C₃	Q₈×C₃⋊D₃	1	2	yes	—
179	C₂²×C₃	#3832	1	2	yes	—
180	C₂²×C₃	C₂²×C₃⋊Dic₃	1	2	yes	—
181	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
182	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
183	C₃×C₄	—	2	48	no	—
184	C₃×C₄	—	9	4	no	—
185	C₃×C₄	—	18	6	no	—
186	C₃×C₄	C₂²×C₃×Dic₃	1	48	yes	—
187	C₃×C₄	C₂²×C₃⋊Dic₃	1	12	yes	—
188	Dic₃	—	3	16	no	—
189	Dic₃	—	6	24	no	—
190	C₃²	—	2	4	no	—
191	C₃²	C₂×Q₈×Dic₃	1	4	yes	complemented
192	C₃²	C₂×C₃×C₄×Q₈	1	1	yes	characteristic, complemented
193	C₂³	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
194	Q₈	C₂×C₃×C₃⋊Dic₃	1	16	yes	direct product factor
195	C₂×C₄	—	9	2	no	—
196	C₂×C₄	—	9	4	no	—
197	C₂×C₄	—	9	6	no	—
198	C₂×C₄	—	18	6	no	—
199	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	12	yes	—
200	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	6	yes	—
201	C₂×C₃	—	2	4	no	—
202	C₂×C₃	—	2	4	no	—
203	C₂×C₃	—	2	16	no	—
204	C₂×C₃	—	2	4	no	—
205	C₂×C₃	C₃×Q₈×Dic₃	1	16	yes	complemented
206	C₂×C₃	Q₈×C₃⋊Dic₃	1	4	yes	direct product factor
207	C₂×C₃	C₂×C₃×D₃×Q₈	1	4	yes	—
208	C₂×C₃	C₂×C₃×#239	1	4	yes	—
209	C₂×C₃	C₂³×C₃×Dic₃	1	4	yes	—
210	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	1	yes	characteristic
211	C₂×C₃	C₂×#3832	1	1	yes	characteristic
212	C₂×C₃	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
213	C₂²	C₃×Q₈×C₃⋊D₃	1	2	yes	—
214	C₂²	C₃×#3832	1	2	yes	—
215	C₂²	C₂²×C₃×C₃⋊Dic₃	1	2	yes	—
216	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
217	C₄	—	9	4	no	—
218	C₄	—	18	6	no	—
219	C₄	C₂²×C₃×C₃⋊Dic₃	1	12	yes	—
220	C₃	—	2	4	no	—
221	C₃	C₂×C₃×Q₈×Dic₃	1	4	yes	complemented
222	C₃	C₂×Q₈×C₃⋊Dic₃	1	1	yes	characteristic, direct product factor
223	C₂	C₃×Q₈×C₃⋊Dic₃	1	4	yes	direct product factor
224	C₂	C₂×C₃×Q₈×C₃⋊D₃	1	1	yes	characteristic
225	C₂	C₂×C₃×#3832	1	1	yes	characteristic
226	C₂	C₂³×C₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×Q₈	C₂
socle, prime radical	C₂³×C₃³	C₂³
cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₂²×C₃×Q₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×C₄×Q₈
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₃⋊Dic₃	C₂	direct (N × H)
5. C₂×Q₈×C₃⋊Dic₃	C₃	direct (N × H)
7. C₂×C₃³×Q₈	C₄	semidirect
17. Q₈×C₃⋊Dic₃	C₂×C₃	direct (N × H)
24. C₃³×Q₈	C₂×C₄	semidirect
28. C₂×C₃×C₃⋊Dic₃	Q₈	direct (N × H)
35. C₂×C₃²×Q₈	Dic₃	semidirect
36. C₂×C₃²×Q₈	C₃×C₄	semidirect
56. C₃×C₃⋊Dic₃	C₂×Q₈	direct (N × H)
70. C₃²×Q₈	C₂×Dic₃	semidirect
71. C₃²×Q₈	C₂×C₃×C₄	semidirect
79. C₂×C₃⋊Dic₃	C₃×Q₈	direct (N × H)
87. C₂×C₃³	C₄×Q₈	semidirect
93. C₂×C₃×Q₈	C₃×Dic₃	semidirect
94. C₂×C₃×Q₈	C₃⋊Dic₃	direct (N × H)
124. C₃⋊Dic₃	C₂×C₃×Q₈	direct (N × H)
131. C₃³	C₂×C₄×Q₈	semidirect
136. C₃×Q₈	C₂×C₃×Dic₃	semidirect
137. C₃×Q₈	C₂×C₃⋊Dic₃	direct (N × H)
156. C₂×C₃²	Q₈×Dic₃	semidirect
157. C₂×C₃²	C₃×C₄×Q₈	semidirect
164. C₂×Q₈	C₃×C₃⋊Dic₃	direct (N × H)
191. C₃²	C₂×Q₈×Dic₃	semidirect
192. C₃²	C₂×C₃×C₄×Q₈	semidirect
194. Q₈	C₂×C₃×C₃⋊Dic₃	direct (N × H)
205. C₂×C₃	C₃×Q₈×Dic₃	semidirect
206. C₂×C₃	Q₈×C₃⋊Dic₃	direct (N × H)
221. C₃	C₂×C₃×Q₈×Dic₃	semidirect
222. C₃	C₂×Q₈×C₃⋊Dic₃	direct (N × H)
223. C₂	C₃×Q₈×C₃⋊Dic₃	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×C₃³×Q₈	2
C₃×Q₈×C₃⋊Dic₃	2
C₂×C₃×C₄⋊(C₃⋊Dic₃)	2
C₂×C₃×C₄×C₃⋊Dic₃	2
C₂×Q₈×C₃⋊Dic₃	3
C₂×C₃×Q₈×Dic₃	3

Subgroups — C2×C3×Q8×C3⋊Dic3 #13179559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Q₈×C₃⋊Dic₃ #13179559