Subgroups — C₂²×Dic₃×C₃⋊C₈ #11857111

Order 1152.

1689 conjugacy classes of subgroups.

The subgroup lattice has 235 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃×C₃⋊C₈	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₈×Dic₃	—	3	1	no	maximal
6	C₂²×C₄×C₃⋊C₈	—	3	1	no	maximal
7	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂³×C₃×Dic₃	C₄	1	2	yes	—
9	C₂²×C₃⋊(C₃⋊C₈)	C₄	1	8	yes	complemented
10	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	6	yes	—
11	C₂×C₃×C₄×Dic₃	C₄	1	6	yes	—
12	C₂×C₃×C₄×Dic₃	C₂²	1	6	yes	—
13	C₂²×C₃×C₃⋊C₈	C₄	1	8	yes	complemented
14	C₂²×C₃×C₃⋊C₈	C₂²	1	6	yes	—
15	Dic₃×C₃⋊C₈	C₂²	1	16	yes	direct product factor
16	C₂³×C₃×C₈	—	3	1	no	—
17	C₂²×C₃×C₄²	—	3	1	no	—
18	C₂²×C₄×Dic₃	D₃	1	1	yes	characteristic
19	C₂³×C₃⋊C₈	—	3	1	no	—
20	C₂³×C₃⋊C₈	—	3	1	no	—
21	C₂³×C₃⋊C₈	—	6	1	no	—
22	C₂³×C₃⋊C₈	D₃	1	1	yes	characteristic
23	C₂×C₈×Dic₃	—	3	12	no	—
24	C₂×C₄×C₃⋊C₈	—	3	12	no	—
25	C₂⁴×C₃²	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
27	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
28	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
29	C₂²×C₃²×C₄	C₂³	1	3	yes	—
30	C₂²×C₃×Dic₃	C₈	1	16	yes	complemented
31	C₂²×C₃×Dic₃	C₂×C₄	1	12	yes	—
32	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	24	yes	complemented
33	C₂×C₃⋊(C₃⋊C₈)	C₂³	1	4	yes	—
34	C₃×C₄×Dic₃	C₂×C₄	1	12	yes	—
35	C₃×C₄×Dic₃	C₂³	1	4	yes	—
36	C₂×C₃×C₃⋊C₈	C₂×C₄	1	24	yes	complemented
37	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
38	C₂²×C₄×C₈	—	9	1	no	Sylow
39	C₂³×C₃×C₄	—	2	1	no	—
40	C₂³×C₃×C₄	—	3	2	no	—
41	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
43	C₂³×Dic₃	Dic₃	1	2	yes	—
44	C₂²×C₃×C₈	—	3	8	no	—
45	C₂²×C₃×C₈	—	3	6	no	—
46	C₂×C₃×C₄²	—	3	6	no	—
47	C₂×C₃×C₄²	—	3	6	no	—
48	C₂×C₄×Dic₃	Dic₃	1	6	yes	—
49	C₂×C₄×Dic₃	C₂×D₃	1	6	yes	—
50	C₂²×C₃⋊C₈	—	3	8	no	—
51	C₂²×C₃⋊C₈	—	3	8	no	—
52	C₂²×C₃⋊C₈	—	3	6	no	—
53	C₂²×C₃⋊C₈	—	3	6	no	—
54	C₂²×C₃⋊C₈	—	6	8	no	—
55	C₂²×C₃⋊C₈	—	6	6	no	—
56	C₂²×C₃⋊C₈	Dic₃	1	8	yes	direct product factor
57	C₂²×C₃⋊C₈	C₂×D₃	1	6	yes	—
58	C₈×Dic₃	—	3	16	no	—
59	C₄×C₃⋊C₈	—	3	16	no	—
60	C₂³×C₃²	C₄²	1	4	yes	—
61	C₂³×C₃²	C₂×C₈	1	8	yes	—
62	C₂³×C₃²	C₂²×C₄	1	3	yes	—
63	C₂×C₃²×C₄	C₄²	1	12	yes	—
64	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
65	C₂×C₃²×C₄	C₂²×C₄	1	3	yes	—
66	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
67	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂×C₈	1	48	yes	complemented
69	C₂×C₃×Dic₃	C₂²×C₄	1	8	yes	—
70	C₃⋊(C₃⋊C₈)	C₂²×C₄	1	8	yes	complemented
71	C₃×C₃⋊C₈	C₂²×C₄	1	8	yes	complemented
72	C₂³×C₈	—	3	1	no	—
73	C₂³×C₈	—	9	1	no	—
74	C₂²×C₄²	—	3	1	no	—
75	C₂×C₄×C₈	—	9	12	no	—
76	C₂⁴×C₃	—	2	1	no	—
77	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
78	C₂⁴×C₃	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	—	2	3	no	—
80	C₂²×C₃×C₄	—	2	4	no	—
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	—	2	3	no	—
83	C₂²×C₃×C₄	—	3	16	no	—
84	C₂²×C₃×C₄	—	3	12	no	—
85	C₂²×C₃×C₄	C₄×D₃	1	3	yes	—
86	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
87	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
88	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
89	C₂²×C₃×C₄	C₂×Dic₃	1	3	yes	—
90	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
91	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
92	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
93	C₂²×Dic₃	C₃⋊C₈	1	16	yes	direct product factor
94	C₂²×Dic₃	C₂×Dic₃	1	12	yes	—
95	C₂×C₃×C₈	—	3	24	no	—
96	C₂×C₃×C₈	—	3	4	no	—
97	C₃×C₄²	—	3	12	no	—
98	C₃×C₄²	—	3	4	no	—
99	C₄×Dic₃	C₂×Dic₃	1	12	yes	—
100	C₄×Dic₃	C₂²×D₃	1	4	yes	—
101	C₂×C₃⋊C₈	—	3	24	no	—
102	C₂×C₃⋊C₈	—	3	24	no	—
103	C₂×C₃⋊C₈	—	3	4	no	—
104	C₂×C₃⋊C₈	—	3	4	no	—
105	C₂×C₃⋊C₈	—	6	24	no	—
106	C₂×C₃⋊C₈	—	6	4	no	—
107	C₂×C₃⋊C₈	C₂×Dic₃	1	24	yes	direct product factor
108	C₂×C₃⋊C₈	C₂²×D₃	1	4	yes	—
109	C₂²×C₃²	C₄×C₈	1	16	yes	complemented
110	C₂²×C₃²	C₂×C₄²	1	6	yes	—
111	C₂²×C₃²	C₂²×C₈	1	12	yes	—
112	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
113	C₃²×C₄	C₂×C₄²	1	6	yes	—
114	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
115	C₃²×C₄	C₂³×C₄	1	1	yes	characteristic
116	C₃×Dic₃	C₂²×C₈	1	16	yes	complemented
117	C₂³×C₄	—	3	2	no	—
118	C₂³×C₄	D₃²	1	1	yes	characteristic
119	C₂²×C₈	—	3	8	no	—
120	C₂²×C₈	—	3	6	no	—
121	C₂²×C₈	—	9	8	no	—
122	C₂²×C₈	—	9	6	no	—
123	C₂×C₄²	—	3	6	no	—
124	C₂×C₄²	—	3	6	no	—
125	C₄×C₈	—	9	16	no	—
126	C₂³×C₃	—	2	8	no	—
127	C₂³×C₃	—	2	4	no	—
128	C₂³×C₃	—	2	3	no	—
129	C₂³×C₃	D₃×C₈	1	8	yes	—
130	C₂³×C₃	C₂×C₃⋊C₈	1	8	yes	—
131	C₂³×C₃	C₄×Dic₃	1	4	yes	—
132	C₂³×C₃	C₄×Dic₃	1	4	yes	—
133	C₂³×C₃	C₂×C₄×D₃	1	3	yes	—
134	C₂³×C₃	C₂²×Dic₃	1	3	yes	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	12	no	—
137	C₂×C₃×C₄	—	2	6	no	—
138	C₂×C₃×C₄	—	2	3	no	—
139	C₂×C₃×C₄	—	2	6	no	—
140	C₂×C₃×C₄	—	3	48	no	—
141	C₂×C₃×C₄	—	3	8	no	—
142	C₂×C₃×C₄	C₄×Dic₃	1	12	yes	—
143	C₂×C₃×C₄	C₄×Dic₃	1	12	yes	—
144	C₂×C₃×C₄	C₂×C₄×D₃	1	3	yes	—
145	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
146	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
147	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
148	C₂×C₃×C₄	C₂²×Dic₃	1	3	yes	—
149	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
150	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
152	C₂×Dic₃	C₂×C₃⋊C₈	1	48	yes	direct product factor
153	C₂×Dic₃	C₂²×Dic₃	1	8	yes	—
154	C₃×C₈	—	3	8	no	—
155	C₃⋊C₈	—	3	8	no	—
156	C₃⋊C₈	—	3	8	no	—
157	C₃⋊C₈	—	6	8	no	—
158	C₃⋊C₈	C₂²×Dic₃	1	8	yes	direct product factor
159	C₂×C₃²	C₂×C₄×C₈	1	12	yes	complemented
160	C₂×C₃²	C₂²×C₄²	1	1	yes	characteristic
161	C₂×C₃²	C₂³×C₈	1	2	yes	—
162	C₂⁴	D₃×Dic₃	1	1	yes	characteristic
163	C₂²×C₄	—	3	16	no	—
164	C₂²×C₄	—	3	12	no	—
165	C₂²×C₄	D₃×Dic₃	1	3	yes	—
166	C₂²×C₄	D₃×Dic₃	1	4	yes	—
167	C₂²×C₄	#624	1	4	yes	—
168	C₂²×C₄	C₂×D₃²	1	3	yes	—
169	C₂×C₈	—	3	4	no	—
170	C₂×C₈	—	3	24	no	—
171	C₂×C₈	—	9	4	no	—
172	C₂×C₈	—	9	24	no	—
173	C₄²	—	3	4	no	—
174	C₄²	—	3	12	no	—
175	C₂²×C₃	—	2	16	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	12	no	—
178	C₂²×C₃	—	2	6	no	—
179	C₂²×C₃	C₄×C₃⋊C₈	1	16	yes	complemented
180	C₂²×C₃	C₈×Dic₃	1	16	yes	complemented
181	C₂²×C₃	C₂×D₃×C₈	1	12	yes	—
182	C₂²×C₃	C₂²×C₃⋊C₈	1	12	yes	—
183	C₂²×C₃	C₂×C₄×Dic₃	1	6	yes	—
184	C₂²×C₃	C₂×C₄×Dic₃	1	6	yes	—
185	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
186	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	6	no	—
190	C₃×C₄	—	3	16	no	—
191	C₃×C₄	C₂×C₄×Dic₃	1	6	yes	—
192	C₃×C₄	C₂×C₄×Dic₃	1	6	yes	—
193	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
194	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
195	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
196	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
197	Dic₃	C₂²×C₃⋊C₈	1	16	yes	direct product factor
198	C₃²	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
199	C₂³	D₃×C₃⋊C₈	1	8	yes	—
200	C₂³	Dic₃²	1	4	yes	—
201	C₂³	C₂×D₃×Dic₃	1	3	yes	—
202	C₂×C₄	—	3	8	no	—
203	C₂×C₄	—	3	48	no	—
204	C₂×C₄	Dic₃²	1	12	yes	—
205	C₂×C₄	C₂×D₃×Dic₃	1	6	yes	—
206	C₂×C₄	C₂×D₃×Dic₃	1	3	yes	—
207	C₂×C₄	C₂×#624	1	6	yes	—
208	C₂×C₄	C₂²×D₃²	1	1	yes	characteristic
209	C₈	—	3	8	no	—
210	C₈	—	9	8	no	—
211	C₂×C₃	—	2	12	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	C₂×C₄×C₃⋊C₈	1	12	yes	complemented
215	C₂×C₃	C₂×C₈×Dic₃	1	12	yes	complemented
216	C₂×C₃	C₂²×D₃×C₈	1	2	yes	—
217	C₂×C₃	C₂³×C₃⋊C₈	1	2	yes	—
218	C₂×C₃	C₂²×C₄×Dic₃	1	1	yes	characteristic
219	C₂×C₃	C₂²×C₄×Dic₃	1	1	yes	characteristic
220	C₂²	Dic₃×C₃⋊C₈	1	16	yes	direct product factor
221	C₂²	C₂×D₃×C₃⋊C₈	1	12	yes	—
222	C₂²	C₂×Dic₃²	1	6	yes	—
223	C₂²	C₂²×D₃×Dic₃	1	1	yes	characteristic
224	C₄	—	3	16	no	—
225	C₄	C₂×Dic₃²	1	6	yes	—
226	C₄	C₂²×D₃×Dic₃	1	1	yes	characteristic
227	C₄	C₂²×#624	1	1	yes	characteristic
228	C₃	—	2	1	no	—
229	C₃	C₂²×C₄×C₃⋊C₈	1	1	yes	characteristic, complemented
230	C₃	C₂²×C₈×Dic₃	1	1	yes	characteristic, complemented
231	C₂	C₂×Dic₃×C₃⋊C₈	1	12	yes	direct product factor
232	C₂	C₂²×D₃×C₃⋊C₈	1	2	yes	—
233	C₂	C₂²×Dic₃²	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃²
commutator, hypocenter	C₃²	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂²×D₃²
Fitting	C₂³×C₃²×C₄	C₂²
socle, prime radical	C₂⁴×C₃²	C₂×C₄
cosocle	C₂×C₃²×C₄	C₂⁴
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₃×C₃⋊C₈	C₂	direct (N × H)
9. C₂²×C₃⋊(C₃⋊C₈)	C₄	semidirect
13. C₂²×C₃×C₃⋊C₈	C₄	semidirect
15. Dic₃×C₃⋊C₈	C₂²	direct (N × H)
30. C₂²×C₃×Dic₃	C₈	semidirect
32. C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	semidirect
36. C₂×C₃×C₃⋊C₈	C₂×C₄	semidirect
56. C₂²×C₃⋊C₈	Dic₃	direct (N × H)
68. C₂×C₃×Dic₃	C₂×C₈	semidirect
70. C₃⋊(C₃⋊C₈)	C₂²×C₄	semidirect
71. C₃×C₃⋊C₈	C₂²×C₄	semidirect
93. C₂²×Dic₃	C₃⋊C₈	direct (N × H)
107. C₂×C₃⋊C₈	C₂×Dic₃	direct (N × H)
109. C₂²×C₃²	C₄×C₈	semidirect
116. C₃×Dic₃	C₂²×C₈	semidirect
152. C₂×Dic₃	C₂×C₃⋊C₈	direct (N × H)
158. C₃⋊C₈	C₂²×Dic₃	direct (N × H)
159. C₂×C₃²	C₂×C₄×C₈	semidirect
179. C₂²×C₃	C₄×C₃⋊C₈	semidirect
180. C₂²×C₃	C₈×Dic₃	semidirect
197. Dic₃	C₂²×C₃⋊C₈	direct (N × H)
198. C₃²	C₂²×C₄×C₈	semidirect
214. C₂×C₃	C₂×C₄×C₃⋊C₈	semidirect
215. C₂×C₃	C₂×C₈×Dic₃	semidirect
220. C₂²	Dic₃×C₃⋊C₈	direct (N × H)
229. C₃	C₂²×C₄×C₃⋊C₈	semidirect
230. C₃	C₂²×C₈×Dic₃	semidirect
231. C₂	C₂×Dic₃×C₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃⋊(C₃⋊C₈)	2
C₂²×C₃×C₄×Dic₃	2
C₂³×C₃×C₃⋊C₈	2
C₂×Dic₃×C₃⋊C₈	2
C₂²×C₈×Dic₃	3
C₂²×C₄×C₃⋊C₈	3

Subgroups — C22×Dic3×C3⋊C8 #11857111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₃×C₃⋊C₈ #11857111