Subgroups — C₂×C₃×#4699 #11874210

Order 1152.

1185 conjugacy classes of subgroups.

The subgroup lattice has 219 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₂	C₂	1	4	yes	maximal
2	C₂×C₃×#869	C₂	1	2	yes	maximal
3	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#4699	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#494	—	3	1	no	maximal
6	C₂×#4699	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×Dic₆	C₂²	1	4	yes	—
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	—	2	4	no	—
11	C₂×C₃×Dic₁₂	—	2	16	no	—
12	C₂×C₃×Dic₁₂	C₂²	1	16	yes	—
13	C₃×#869	—	2	4	no	—
14	C₃×#869	C₂²	1	8	yes	—
15	C₃²×C₄×C₈	C₂²	1	4	yes	—
16	C₂²×C₃×Q₁₆	—	3	4	no	—
17	C₂×C₃×C₄⋊Q₈	—	3	2	no	—
18	C₂²×Dic₁₂	C₂×C₃	1	4	yes	—
19	C₂×#869	C₂×C₃	1	2	yes	—
20	C₃×#494	—	3	8	no	—
21	C₂×C₃×C₄×C₈	—	2	1	no	—
22	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
23	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
24	#4699	C₂×C₃	1	8	yes	direct product factor
25	C₂²×C₃²×C₄	D₄	1	2	yes	—
26	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
27	C₂²×C₃×Dic₃	—	2	4	no	—
28	C₂×C₃×Dic₆	—	2	16	no	—
29	C₂×C₃×Dic₆	—	2	4	no	—
30	C₂×C₃×Dic₆	C₂³	1	8	yes	—
31	C₂×C₃²×C₈	D₄	1	8	yes	—
32	C₂×C₃²×C₈	C₂³	1	4	yes	—
33	C₃²×C₄²	D₄	1	2	yes	—
34	C₃²×C₄²	C₂³	1	2	yes	—
35	C₃×C₄⋊Dic₃	—	2	16	no	—
36	C₃×Dic₁₂	—	2	32	no	—
37	C₂×#494	—	3	1	no	Sylow
38	C₂²×C₃×Q₈	—	3	4	no	—
39	C₂²×Dic₆	C₂²×C₃	1	4	yes	—
40	C₂×C₃×Q₁₆	—	3	16	no	—
41	C₂×C₃×Q₁₆	—	6	16	no	—
42	C₂²×C₃×C₈	—	2	2	no	—
43	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
44	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
45	C₃×C₄⋊Q₈	—	3	8	no	—
46	C₃×C₄⋊Q₈	—	6	4	no	—
47	C₂×C₃×C₄⋊C₄	—	6	4	no	—
48	C₂×C₃×C₄²	—	2	1	no	—
49	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₃	—	2	4	no	—
52	C₂×Dic₁₂	—	2	16	no	—
53	C₂×Dic₁₂	C₂²×C₃	1	16	yes	—
54	#869	—	2	4	no	—
55	#869	C₂²×C₃	1	8	yes	—
56	C₃×C₄×C₈	—	2	4	no	—
57	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
58	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
59	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	Q₁₆	1	8	yes	complemented
61	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
63	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
64	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
65	C₂×C₃×Dic₃	—	2	8	no	—
66	C₂×C₃×Dic₃	—	4	8	no	—
67	C₃×Dic₆	—	2	16	no	—
68	C₃×Dic₆	—	4	8	no	—
69	C₃²×C₈	C₂×D₄	1	8	yes	—
70	C₂²×Q₁₆	—	3	4	no	—
71	C₂×C₄⋊Q₈	—	3	2	no	—
72	#494	—	3	8	no	—
73	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
74	C₂×C₃×Q₈	—	3	8	no	—
75	C₂×C₃×Q₈	—	6	4	no	—
76	C₂×C₃×Q₈	—	6	16	no	—
77	C₂²×C₃×C₄	—	2	2	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	6	4	no	—
80	C₂²×C₃×C₄	D₁₂	1	2	yes	—
81	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
82	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
84	C₂²×Dic₃	—	2	4	no	—
85	C₂×Dic₆	—	2	16	no	—
86	C₂×Dic₆	—	2	4	no	—
87	C₂×Dic₆	C₂³×C₃	1	8	yes	—
88	C₃×Q₁₆	—	6	32	no	—
89	C₂×C₃×C₈	—	2	8	no	—
90	C₂×C₃×C₈	—	2	4	no	—
91	C₂×C₃×C₈	D₁₂	1	8	yes	—
92	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
93	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
94	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
95	C₃×C₄⋊C₄	—	6	16	no	—
96	C₃×C₄²	—	2	2	no	—
97	C₃×C₄²	—	2	2	no	—
98	C₃×C₄²	D₁₂	1	2	yes	—
99	C₃×C₄²	C₃×D₄	1	2	yes	—
100	C₃×C₄²	C₂²×D₃	1	2	yes	—
101	C₃×C₄²	C₂³×C₃	1	2	yes	—
102	C₄⋊Dic₃	—	2	16	no	—
103	Dic₁₂	—	2	32	no	—
104	C₂²×C₃²	#127	1	2	yes	—
105	C₂²×C₃²	C₂×Q₁₆	1	4	yes	—
106	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
107	C₃²×C₄	#127	1	2	yes	—
108	C₃²×C₄	C₂×Q₁₆	1	8	yes	complemented
109	C₃²×C₄	C₂²×D₄	1	2	yes	—
110	C₃×Dic₃	—	4	8	no	—
111	C₂²×Q₈	—	3	4	no	—
112	C₂×Q₁₆	—	3	16	no	—
113	C₂×Q₁₆	—	6	16	no	—
114	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
115	C₄⋊Q₈	—	3	8	no	—
116	C₄⋊Q₈	—	6	4	no	—
117	C₂×C₄⋊C₄	—	6	4	no	—
118	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
119	C₄×C₈	C₂×C₃×D₃	1	4	yes	—
120	C₂³×C₃	—	2	1	no	—
121	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
122	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
123	C₃×Q₈	—	6	16	no	—
124	C₃×Q₈	—	12	8	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	4	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	6	8	no	—
131	C₂×C₃×C₄	—	12	8	no	—
132	C₂×C₃×C₄	Dic₁₂	1	8	yes	complemented
133	C₂×C₃×C₄	C₃×Q₁₆	1	8	yes	complemented
134	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
136	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
137	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
139	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
140	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
142	C₂×Dic₃	—	2	8	no	—
143	C₂×Dic₃	—	4	8	no	—
144	Dic₆	—	2	16	no	—
145	Dic₆	—	4	8	no	—
146	C₃×C₈	—	2	8	no	—
147	C₃×C₈	C₂×D₁₂	1	8	yes	—
148	C₃×C₈	C₂×C₃×D₄	1	8	yes	—
149	C₂×C₃²	#494	1	4	yes	complemented
150	C₂×C₃²	C₂×#127	1	1	yes	characteristic
151	C₂×C₃²	C₂²×Q₁₆	1	2	yes	—
152	C₂×Q₈	—	3	8	no	—
153	C₂×Q₈	—	6	4	no	—
154	C₂×Q₈	—	6	16	no	—
155	C₂²×C₄	—	6	4	no	—
156	C₂²×C₄	C₃×D₁₂	1	2	yes	—
157	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
158	Q₁₆	—	6	32	no	—
159	C₂×C₈	C₃×D₁₂	1	8	yes	—
160	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
161	C₄⋊C₄	—	6	16	no	—
162	C₄²	C₃×D₁₂	1	2	yes	—
163	C₄²	C₂²×C₃×D₃	1	2	yes	—
164	C₂²×C₃	—	2	4	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	#874	1	2	yes	—
168	C₂²×C₃	C₂×Dic₁₂	1	4	yes	—
169	C₂²×C₃	C₃×#127	1	2	yes	—
170	C₂²×C₃	C₂×C₃×Q₁₆	1	4	yes	—
171	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
172	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
173	C₃×C₄	—	2	8	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	12	8	no	—
177	C₃×C₄	#874	1	2	yes	—
178	C₃×C₄	C₂×Dic₁₂	1	8	yes	complemented
179	C₃×C₄	C₃×#127	1	2	yes	—
180	C₃×C₄	C₂×C₃×Q₁₆	1	8	yes	complemented
181	C₃×C₄	C₂²×D₁₂	1	2	yes	—
182	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
183	Dic₃	—	4	8	no	—
184	C₃²	C₂×#494	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
186	Q₈	—	6	16	no	—
187	Q₈	—	12	8	no	—
188	C₂×C₄	—	6	8	no	—
189	C₂×C₄	—	12	8	no	—
190	C₂×C₄	C₃×Dic₁₂	1	8	yes	complemented
191	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
192	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
193	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
194	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
195	C₈	C₂×C₃×D₁₂	1	8	yes	—
196	C₂×C₃	—	2	4	no	—
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	2	no	—
199	C₂×C₃	#4699	1	4	yes	direct product factor
200	C₂×C₃	C₃×#494	1	4	yes	complemented
201	C₂×C₃	C₂×#874	1	1	yes	characteristic
202	C₂×C₃	C₂²×Dic₁₂	1	2	yes	—
203	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
204	C₂×C₃	C₂²×C₃×Q₁₆	1	2	yes	—
205	C₂²	C₃×#874	1	2	yes	—
206	C₂²	C₂×C₃×Dic₁₂	1	4	yes	—
207	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
208	C₄	—	12	8	no	—
209	C₄	C₃×#874	1	2	yes	—
210	C₄	C₂×C₃×Dic₁₂	1	8	yes	complemented
211	C₄	C₂²×C₃×D₁₂	1	2	yes	—
212	C₃	—	2	1	no	—
213	C₃	C₂×#4699	1	1	yes	characteristic, direct product factor
214	C₃	C₂×C₃×#494	1	1	yes	characteristic, complemented
215	C₂	C₃×#4699	1	4	yes	direct product factor
216	C₂	C₂×C₃×#874	1	1	yes	characteristic
217	C₂	C₂²×C₃×Dic₁₂	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₂×C₃×#494
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#4699	C₂	direct (N × H)
6. C₂×#4699	C₃	direct (N × H)
24. #4699	C₂×C₃	direct (N × H)
60. C₂×C₃²×C₄	Q₁₆	semidirect
108. C₃²×C₄	C₂×Q₁₆	semidirect
132. C₂×C₃×C₄	Dic₁₂	semidirect
133. C₂×C₃×C₄	C₃×Q₁₆	semidirect
149. C₂×C₃²	#494	semidirect
178. C₃×C₄	C₂×Dic₁₂	semidirect
180. C₃×C₄	C₂×C₃×Q₁₆	semidirect
184. C₃²	C₂×#494	semidirect
190. C₂×C₄	C₃×Dic₁₂	semidirect
199. C₂×C₃	#4699	direct (N × H)
200. C₂×C₃	C₃×#494	semidirect
210. C₄	C₂×C₃×Dic₁₂	semidirect
213. C₃	C₂×#4699	direct (N × H)
214. C₃	C₂×C₃×#494	semidirect
215. C₂	C₃×#4699	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×C₃×Dic₁₂	2
C₂×C₃×#869	2
C₂×C₃²×C₄×C₈	2
C₃×#4699	2
C₂×C₃×#494	3
C₂×#4699	3

Subgroups — C2×C3×#4699 #11874210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4699 #11874210