Subgroups — C₂×C₃×#4683 #11874914

Order 1152.

801 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#870	C₂	1	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#818	C₂	1	2	yes	maximal
4	C₂×C₃×#817	C₂	1	2	yes	maximal
5	C₃×#4683	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×#499	—	3	1	no	maximal
7	C₂×#4683	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
12	C₂×C₃×Dic₃⋊C₄	—	2	4	no	—
13	C₃×#870	—	2	4	no	—
14	C₃×#870	C₂²	1	8	yes	—
15	C₃²×C₄×C₈	C₂²	1	4	yes	—
16	C₃×#818	C₂²	1	8	yes	—
17	C₃×#817	C₂²	1	8	yes	—
18	C₂×C₃×#125	—	3	2	no	—
19	C₂×#870	C₂×C₃	1	2	yes	—
20	C₃×#499	—	3	8	no	—
21	C₂×C₃×#107	—	3	2	no	—
22	C₂×C₃×#106	—	3	2	no	—
23	C₂×C₃×C₄×C₈	—	2	1	no	—
24	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
25	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
26	C₂×#818	C₂×C₃	1	2	yes	—
27	C₂×#817	C₂×C₃	1	2	yes	—
28	#4683	C₂×C₃	1	8	yes	direct product factor
29	C₂²×C₃²×C₄	D₄	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	4	no	—
32	C₂×C₃²×C₈	Q₈	1	4	yes	—
33	C₂×C₃²×C₈	Q₈	1	4	yes	—
34	C₂×C₃²×C₈	C₂³	1	2	yes	—
35	C₂×C₃²×C₈	C₂³	1	2	yes	—
36	C₃²×C₄²	D₄	1	2	yes	—
37	C₃²×C₄²	C₂³	1	2	yes	—
38	C₃×C₄⋊Dic₃	—	2	4	no	—
39	C₃×C₄⋊Dic₃	C₂³	1	8	yes	—
40	C₃×Dic₃⋊C₄	—	2	16	no	—
41	C₂×#499	—	3	1	no	Sylow
42	C₂²×C₃×C₈	—	2	1	no	—
43	C₂²×C₃×C₈	—	2	1	no	—
44	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₃×#125	—	3	8	no	—
49	C₃×#125	—	6	4	no	—
50	C₂×C₃×C₄⋊C₄	—	3	4	no	—
51	C₂×C₃×C₄⋊C₄	—	6	4	no	—
52	C₂×C₃×C₄²	—	2	1	no	—
53	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
56	C₂×Dic₃⋊C₄	—	2	4	no	—
57	#870	—	2	4	no	—
58	#870	C₂²×C₃	1	8	yes	—
59	C₃×#107	—	3	8	no	—
60	C₃×#106	—	3	8	no	—
61	C₃×C₄×C₈	—	2	4	no	—
62	C₃×C₄×C₈	C₂×D₃	1	4	yes	—
63	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
64	#818	C₂²×C₃	1	8	yes	—
65	#817	C₂²×C₃	1	8	yes	—
66	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	—	2	4	no	—
68	C₂×C₃²×C₄	C₂×D₄	1	4	yes	—
69	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
71	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
72	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
73	C₂×C₃×Dic₃	—	2	8	no	—
74	C₂×C₃×Dic₃	—	4	8	no	—
75	C₃²×C₈	C₂×Q₈	1	4	yes	—
76	C₃²×C₈	C₂×Q₈	1	4	yes	—
77	C₂×#125	—	3	2	no	—
78	#499	—	3	8	no	—
79	C₂×#107	—	3	2	no	—
80	C₂×#106	—	3	2	no	—
81	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	—	2	2	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	—	6	4	no	—
85	C₂²×C₃×C₄	D₁₂	1	2	yes	—
86	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
87	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
89	C₂²×Dic₃	—	2	4	no	—
90	C₂×C₃×C₈	—	2	4	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	Dic₆	1	4	yes	—
95	C₂×C₃×C₈	Dic₆	1	4	yes	—
96	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
97	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
98	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
99	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
100	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
101	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
102	C₃×C₄⋊C₄	—	3	8	no	—
103	C₃×C₄⋊C₄	—	6	16	no	—
104	C₃×C₄⋊C₄	—	6	4	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	—	2	2	no	—
107	C₃×C₄²	D₁₂	1	2	yes	—
108	C₃×C₄²	C₃×D₄	1	2	yes	—
109	C₃×C₄²	C₂²×D₃	1	2	yes	—
110	C₃×C₄²	C₂³×C₃	1	2	yes	—
111	C₄⋊Dic₃	—	2	4	no	—
112	C₄⋊Dic₃	C₂³×C₃	1	8	yes	—
113	Dic₃⋊C₄	—	2	16	no	—
114	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
115	C₂²×C₃²	#135	1	4	yes	—
116	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
117	C₃²×C₄	—	2	4	no	—
118	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
119	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
120	C₃²×C₄	C₂²×Q₈	1	1	yes	characteristic
121	C₃×Dic₃	—	4	8	no	—
122	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
124	#125	—	3	8	no	—
125	#125	—	6	4	no	—
126	C₂×C₄⋊C₄	—	3	4	no	—
127	C₂×C₄⋊C₄	—	6	4	no	—
128	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
129	#107	—	3	8	no	—
130	#106	—	3	8	no	—
131	C₄×C₈	C₂×C₃×D₃	1	4	yes	—
132	C₂³×C₃	—	2	1	no	—
133	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
134	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	—	2	8	no	—
136	C₂×C₃×C₄	—	2	4	no	—
137	C₂×C₃×C₄	—	2	4	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	2	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	6	8	no	—
144	C₂×C₃×C₄	—	12	8	no	—
145	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
146	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
147	C₂×C₃×C₄	C₂×D₁₂	1	4	yes	—
148	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
150	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
152	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
153	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
155	C₂×Dic₃	—	2	8	no	—
156	C₂×Dic₃	—	4	8	no	—
157	C₃×C₈	—	2	4	no	—
158	C₃×C₈	—	2	4	no	—
159	C₃×C₈	C₂×Dic₆	1	4	yes	—
160	C₃×C₈	C₂×Dic₆	1	4	yes	—
161	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
162	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
163	C₂×C₃²	#499	1	4	yes	complemented
164	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₃²	C₂×#135	1	2	yes	—
166	C₂²×C₄	—	6	4	no	—
167	C₂²×C₄	C₃×D₁₂	1	2	yes	—
168	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
169	C₂×C₈	C₃×Dic₆	1	4	yes	—
170	C₂×C₈	C₃×Dic₆	1	4	yes	—
171	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
172	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
173	C₄⋊C₄	—	3	8	no	—
174	C₄⋊C₄	—	6	4	no	—
175	C₄⋊C₄	—	6	16	no	—
176	C₄²	C₃×D₁₂	1	2	yes	—
177	C₄²	C₂²×C₃×D₃	1	2	yes	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	#869	1	2	yes	—
182	C₂²×C₃	#904	1	4	yes	—
183	C₂²×C₃	C₃×C₄⋊Q₈	1	2	yes	—
184	C₂²×C₃	C₃×#135	1	4	yes	—
185	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
186	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
187	C₃×C₄	—	2	8	no	—
188	C₃×C₄	—	2	4	no	—
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	12	8	no	—
194	C₃×C₄	#869	1	2	yes	—
195	C₃×C₄	C₃×C₄⋊Q₈	1	2	yes	—
196	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
197	C₃×C₄	C₂²×Dic₆	1	1	yes	characteristic
198	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
199	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
200	Dic₃	—	4	8	no	—
201	C₃²	C₂×#499	1	1	yes	characteristic, complemented, Sylow
202	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	6	8	no	—
205	C₂×C₄	—	12	8	no	—
206	C₂×C₄	C₂×C₃×Dic₆	1	2	yes	—
207	C₂×C₄	C₂×C₃×Dic₆	1	2	yes	—
208	C₂×C₄	C₂×C₃×D₁₂	1	4	yes	—
209	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
210	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
211	C₈	C₂×C₃×Dic₆	1	4	yes	—
212	C₈	C₂×C₃×Dic₆	1	4	yes	—
213	C₂×C₃	—	2	4	no	—
214	C₂×C₃	—	2	1	no	—
215	C₂×C₃	—	2	2	no	—
216	C₂×C₃	#4683	1	4	yes	direct product factor
217	C₂×C₃	C₃×#499	1	4	yes	complemented
218	C₂×C₃	C₂×#869	1	1	yes	characteristic
219	C₂×C₃	C₂×#904	1	2	yes	—
220	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
221	C₂×C₃	C₂×C₃×#135	1	2	yes	—
222	C₂²	C₃×#869	1	2	yes	—
223	C₂²	C₃×#904	1	4	yes	—
224	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
225	C₄	—	2	4	no	—
226	C₄	—	12	8	no	—
227	C₄	C₃×#869	1	2	yes	—
228	C₄	C₂²×C₃×Dic₆	1	1	yes	characteristic
229	C₄	C₂²×C₃×Dic₆	1	1	yes	characteristic
230	C₃	—	2	1	no	—
231	C₃	C₂×#4683	1	1	yes	characteristic, direct product factor
232	C₃	C₂×C₃×#499	1	1	yes	characteristic, complemented
233	C₂	C₃×#4683	1	4	yes	direct product factor
234	C₂	C₂×C₃×#869	1	1	yes	characteristic
235	C₂	C₂×C₃×#904	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×#499
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
5. C₃×#4683	C₂	direct (N × H)
7. C₂×#4683	C₃	direct (N × H)
28. #4683	C₂×C₃	direct (N × H)
163. C₂×C₃²	#499	semidirect
201. C₃²	C₂×#499	semidirect
216. C₂×C₃	#4683	direct (N × H)
217. C₂×C₃	C₃×#499	semidirect
231. C₃	C₂×#4683	direct (N × H)
232. C₃	C₂×C₃×#499	semidirect
233. C₂	C₃×#4683	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×#870	2
C₂×C₃²×C₄×C₈	2
C₂×C₃×#818	2
C₂×C₃×#817	2
C₃×#4683	2
C₂×C₃×#499	3
C₂×#4683	3

Subgroups — C2×C3×#4683 #11874914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4683 #11874914