Subgroups — C₂×C₃×#4680 #11874883

Order 1152.

929 conjugacy classes of subgroups.

The subgroup lattice has 215 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#869	C₂	1	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#817	C₂	1	4	yes	maximal
4	C₃×#4680	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#498	—	3	1	no	maximal
6	C₂×#4680	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×Dic₆	—	2	2	no	—
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
12	C₃×#869	—	2	4	no	—
13	C₃×#869	C₂²	1	8	yes	—
14	C₃²×C₄×C₈	C₂²	1	4	yes	—
15	C₃×#817	C₂²	1	16	yes	—
16	C₂×C₃×C₄⋊Q₈	—	3	2	no	—
17	C₂×#869	C₂×C₃	1	2	yes	—
18	C₃×#498	—	3	8	no	—
19	C₂×C₃×#106	—	3	4	no	—
20	C₂×C₃×C₄×C₈	—	2	1	no	—
21	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×#817	C₂×C₃	1	4	yes	—
24	#4680	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	8	no	—
29	C₂×C₃×Dic₆	—	4	8	no	—
30	C₂×C₃²×C₈	Q₈	1	8	yes	complemented
31	C₂×C₃²×C₈	C₂³	1	4	yes	—
32	C₃²×C₄²	D₄	1	2	yes	—
33	C₃²×C₄²	C₂³	1	2	yes	—
34	C₃×C₄⋊Dic₃	—	2	8	no	—
35	C₃×C₄⋊Dic₃	—	2	4	no	—
36	C₃×C₄⋊Dic₃	C₂³	1	8	yes	—
37	C₂×#498	—	3	1	no	Sylow
38	C₂²×C₃×Q₈	—	6	2	no	—
39	C₂²×Dic₆	—	2	2	no	—
40	C₂²×C₃×C₈	—	2	2	no	—
41	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
42	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
43	C₃×C₄⋊Q₈	—	3	8	no	—
44	C₃×C₄⋊Q₈	—	6	4	no	—
45	C₂×C₃×C₄⋊C₄	—	3	4	no	—
46	C₂×C₃×C₄⋊C₄	—	6	2	no	—
47	C₂×C₃×C₄²	—	2	1	no	—
48	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₄⋊Dic₃	—	2	2	no	—
51	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
52	#869	—	2	4	no	—
53	#869	C₂²×C₃	1	8	yes	—
54	C₃×#106	—	3	16	no	—
55	C₃×C₄×C₈	—	2	4	no	—
56	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
57	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
58	#817	C₂²×C₃	1	16	yes	—
59	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	SD₁₆	1	8	yes	—
61	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
63	C₂×C₃²×C₄	C₂×Q₈	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	—	4	16	no	—
68	C₃²×C₈	C₂×Q₈	1	8	yes	complemented
69	C₂×C₄⋊Q₈	—	3	2	no	—
70	#498	—	3	8	no	—
71	C₂×#106	—	3	4	no	—
72	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
73	C₂×C₃×Q₈	—	6	8	no	—
74	C₂×C₃×Q₈	—	12	8	no	—
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	—	6	4	no	—
78	C₂²×C₃×C₄	D₁₂	1	2	yes	—
79	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
80	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂²×Dic₃	—	2	4	no	—
83	C₂×Dic₆	—	2	8	no	—
84	C₂×Dic₆	—	4	8	no	—
85	C₂×C₃×C₈	—	2	8	no	—
86	C₂×C₃×C₈	—	2	4	no	—
87	C₂×C₃×C₈	Dic₆	1	8	yes	complemented
88	C₂×C₃×C₈	C₃×Q₈	1	8	yes	complemented
89	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
90	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
91	C₃×C₄⋊C₄	—	3	8	no	—
92	C₃×C₄⋊C₄	—	6	4	no	—
93	C₃×C₄⋊C₄	—	6	8	no	—
94	C₃×C₄²	—	2	2	no	—
95	C₃×C₄²	—	2	2	no	—
96	C₃×C₄²	D₁₂	1	2	yes	—
97	C₃×C₄²	C₃×D₄	1	2	yes	—
98	C₃×C₄²	C₂²×D₃	1	2	yes	—
99	C₃×C₄²	C₂³×C₃	1	2	yes	—
100	C₄⋊Dic₃	—	2	4	no	—
101	C₄⋊Dic₃	—	2	8	no	—
102	C₄⋊Dic₃	C₂³×C₃	1	8	yes	—
103	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
104	C₂²×C₃²	C₂×SD₁₆	1	4	yes	—
105	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
106	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
107	C₃²×C₄	C₂×SD₁₆	1	8	yes	—
108	C₃²×C₄	C₂²×Q₈	1	2	yes	—
109	C₃×Dic₃	—	4	8	no	—
110	C₂²×Q₈	—	6	2	no	—
111	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
112	C₄⋊Q₈	—	3	8	no	—
113	C₄⋊Q₈	—	6	4	no	—
114	C₂×C₄⋊C₄	—	3	4	no	—
115	C₂×C₄⋊C₄	—	6	2	no	—
116	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
117	#106	—	3	16	no	—
118	C₄×C₈	C₂×C₃×D₃	1	4	yes	—
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
121	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
122	C₃×Q₈	—	12	16	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	8	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	6	8	no	—
129	C₂×C₃×C₄	—	12	8	no	—
130	C₂×C₃×C₄	#204	1	8	yes	—
131	C₂×C₃×C₄	C₃×SD₁₆	1	8	yes	—
132	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
133	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
135	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
137	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
138	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
140	C₂×Dic₃	—	2	8	no	—
141	C₂×Dic₃	—	4	8	no	—
142	Dic₆	—	4	16	no	—
143	C₃×C₈	—	2	8	no	—
144	C₃×C₈	C₂×Dic₆	1	8	yes	complemented
145	C₃×C₈	C₂×C₃×Q₈	1	8	yes	complemented
146	C₂×C₃²	#498	1	4	yes	complemented
147	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
148	C₂×C₃²	C₂²×SD₁₆	1	2	yes	—
149	C₂×Q₈	—	6	8	no	—
150	C₂×Q₈	—	12	8	no	—
151	C₂²×C₄	—	6	4	no	—
152	C₂²×C₄	C₃×D₁₂	1	2	yes	—
153	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
154	C₂×C₈	C₃×Dic₆	1	8	yes	complemented
155	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
156	C₄⋊C₄	—	3	8	no	—
157	C₄⋊C₄	—	6	4	no	—
158	C₄⋊C₄	—	6	8	no	—
159	C₄²	C₃×D₁₂	1	2	yes	—
160	C₄²	C₂²×C₃×D₃	1	2	yes	—
161	C₂²×C₃	—	2	4	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	#869	1	2	yes	—
165	C₂²×C₃	C₂×#204	1	4	yes	—
166	C₂²×C₃	C₃×C₄⋊Q₈	1	2	yes	—
167	C₂²×C₃	C₂×C₃×SD₁₆	1	4	yes	—
168	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
169	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
170	C₃×C₄	—	2	8	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	2	2	no	—
173	C₃×C₄	—	12	8	no	—
174	C₃×C₄	#869	1	2	yes	—
175	C₃×C₄	C₂×#204	1	8	yes	—
176	C₃×C₄	C₃×C₄⋊Q₈	1	2	yes	—
177	C₃×C₄	C₂×C₃×SD₁₆	1	8	yes	—
178	C₃×C₄	C₂²×Dic₆	1	2	yes	—
179	C₃×C₄	C₂²×C₃×Q₈	1	2	yes	—
180	Dic₃	—	4	8	no	—
181	C₃²	C₂×#498	1	1	yes	characteristic, complemented, Sylow
182	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
183	Q₈	—	12	16	no	—
184	C₂×C₄	—	6	8	no	—
185	C₂×C₄	—	12	8	no	—
186	C₂×C₄	C₃×#204	1	8	yes	—
187	C₂×C₄	C₂×C₃×Dic₆	1	4	yes	—
188	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
189	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
190	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
191	C₈	C₂×C₃×Dic₆	1	8	yes	complemented
192	C₂×C₃	—	2	4	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	—	2	2	no	—
195	C₂×C₃	#4680	1	4	yes	direct product factor
196	C₂×C₃	C₃×#498	1	4	yes	complemented
197	C₂×C₃	C₂×#869	1	1	yes	characteristic
198	C₂×C₃	C₂²×#204	1	2	yes	—
199	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
200	C₂×C₃	C₂²×C₃×SD₁₆	1	2	yes	—
201	C₂²	C₃×#869	1	2	yes	—
202	C₂²	C₂×C₃×#204	1	4	yes	—
203	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
204	C₄	—	12	8	no	—
205	C₄	C₃×#869	1	2	yes	—
206	C₄	C₂×C₃×#204	1	8	yes	—
207	C₄	C₂²×C₃×Dic₆	1	2	yes	—
208	C₃	—	2	1	no	—
209	C₃	C₂×#4680	1	1	yes	characteristic, direct product factor
210	C₃	C₂×C₃×#498	1	1	yes	characteristic, complemented
211	C₂	C₃×#4680	1	4	yes	direct product factor
212	C₂	C₂×C₃×#869	1	1	yes	characteristic
213	C₂	C₂²×C₃×#204	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×#498
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×#4680	C₂	direct (N × H)
6. C₂×#4680	C₃	direct (N × H)
24. #4680	C₂×C₃	direct (N × H)
30. C₂×C₃²×C₈	Q₈	semidirect
68. C₃²×C₈	C₂×Q₈	semidirect
87. C₂×C₃×C₈	Dic₆	semidirect
88. C₂×C₃×C₈	C₃×Q₈	semidirect
144. C₃×C₈	C₂×Dic₆	semidirect
145. C₃×C₈	C₂×C₃×Q₈	semidirect
146. C₂×C₃²	#498	semidirect
154. C₂×C₈	C₃×Dic₆	semidirect
181. C₃²	C₂×#498	semidirect
191. C₈	C₂×C₃×Dic₆	semidirect
195. C₂×C₃	#4680	direct (N × H)
196. C₂×C₃	C₃×#498	semidirect
209. C₃	C₂×#4680	direct (N × H)
210. C₃	C₂×C₃×#498	semidirect
211. C₂	C₃×#4680	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₂×C₃²×C₄×C₈	2
C₂×C₃×#817	2
C₃×#4680	2
C₂×C₃×#498	3
C₂×#4680	3

Subgroups — C2×C3×#4680 #11874883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4680 #11874883