Subgroups — C₂×C₃×#4682 #11874947

Order 1152.

929 conjugacy classes of subgroups.

The subgroup lattice has 245 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×#818	C₂	1	4	yes	maximal
4	C₃×#4682	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₃×#500	—	3	1	no	maximal
6	C₂×#4682	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×#818	C₂²	1	16	yes	—
16	C₂×C₃×C₄⋊Q₈	—	3	2	no	—
17	C₂×#869	C₂×C₃	1	2	yes	—
18	C₃×#500	—	3	8	no	—
19	C₂×C₃×#107	—	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×#818	C₂×C₃	1	4	yes	—
24	#4682	C₂×C₃	1	8	yes	direct product factor
25	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	—	2	4	no	—
29	C₂×C₃×Dic₆	—	2	8	no	—
30	C₂×C₃×Dic₆	—	4	8	no	—
31	C₂×C₃²×C₈	Q₈	1	8	yes	complemented
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	8	no	—
36	C₃×C₄⋊Dic₃	—	2	4	no	—
37	C₃×C₄⋊Dic₃	C₂³	1	8	yes	—
38	C₂×#500	—	3	1	no	Sylow
39	C₂²×C₃×Q₈	—	6	2	no	—
40	C₂²×Dic₆	—	2	2	no	—
41	C₂²×C₃×C₈	—	2	2	no	—
42	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
43	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
44	C₃×C₄⋊Q₈	—	3	8	no	—
45	C₃×C₄⋊Q₈	—	6	4	no	—
46	C₂×C₃×C₄⋊C₄	—	3	4	no	—
47	C₂×C₃×C₄⋊C₄	—	6	2	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	2	no	—
52	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
53	#869	—	2	4	no	—
54	#869	C₂²×C₃	1	8	yes	—
55	C₃×#107	—	3	16	no	—
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	#818	C₂²×C₃	1	16	yes	—
60	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	D₈	1	4	yes	—
62	C₂×C₃²×C₄	Q₁₆	1	4	yes	—
63	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
66	C₂×C₃²×C₄	C₂×Q₈	1	4	yes	—
67	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
68	C₂×C₃×Dic₃	—	2	8	no	—
69	C₂×C₃×Dic₃	—	4	8	no	—
70	C₃×Dic₆	—	4	16	no	—
71	C₃²×C₈	C₂×Q₈	1	8	yes	complemented
72	C₂×C₄⋊Q₈	—	3	2	no	—
73	#500	—	3	8	no	—
74	C₂×#107	—	3	4	no	—
75	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
76	C₂×C₃×Q₈	—	6	8	no	—
77	C₂×C₃×Q₈	—	12	8	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	2	1	no	—
81	C₂²×C₃×C₄	—	6	4	no	—
82	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
83	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
88	C₂²×Dic₃	—	2	4	no	—
89	C₂×Dic₆	—	2	8	no	—
90	C₂×Dic₆	—	4	8	no	—
91	C₂×C₃×C₈	—	2	8	no	—
92	C₂×C₃×C₈	—	2	4	no	—
93	C₂×C₃×C₈	Dic₆	1	8	yes	complemented
94	C₂×C₃×C₈	C₃×Q₈	1	8	yes	complemented
95	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
96	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
97	C₃×C₄⋊C₄	—	3	8	no	—
98	C₃×C₄⋊C₄	—	6	4	no	—
99	C₃×C₄⋊C₄	—	6	8	no	—
100	C₃×C₄²	—	2	2	no	—
101	C₃×C₄²	—	2	2	no	—
102	C₃×C₄²	D₁₂	1	2	yes	—
103	C₃×C₄²	C₃×D₄	1	2	yes	—
104	C₃×C₄²	C₂²×D₃	1	2	yes	—
105	C₃×C₄²	C₂³×C₃	1	2	yes	—
106	C₄⋊Dic₃	—	2	4	no	—
107	C₄⋊Dic₃	—	2	8	no	—
108	C₄⋊Dic₃	C₂³×C₃	1	8	yes	—
109	C₂²×C₃²	C₄⋊Q₈	1	2	yes	—
110	C₂²×C₃²	C₂×D₈	1	2	yes	—
111	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
112	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
113	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
114	C₃²×C₄	C₂×D₈	1	4	yes	—
115	C₃²×C₄	C₂×Q₁₆	1	4	yes	—
116	C₃²×C₄	C₂²×Q₈	1	2	yes	—
117	C₃×Dic₃	—	4	8	no	—
118	C₂²×Q₈	—	6	2	no	—
119	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
120	C₄⋊Q₈	—	3	8	no	—
121	C₄⋊Q₈	—	6	4	no	—
122	C₂×C₄⋊C₄	—	3	4	no	—
123	C₂×C₄⋊C₄	—	6	2	no	—
124	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
125	#107	—	3	16	no	—
126	C₄×C₈	C₂×C₃×D₃	1	4	yes	—
127	C₂³×C₃	—	2	1	no	—
128	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
129	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
130	C₃×Q₈	—	12	16	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	2	no	—
136	C₂×C₃×C₄	—	2	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	6	8	no	—
139	C₂×C₃×C₄	—	12	8	no	—
140	C₂×C₃×C₄	D₂₄	1	4	yes	—
141	C₂×C₃×C₄	Dic₁₂	1	4	yes	—
142	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
143	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
144	C₂×C₃×C₄	C₂×Dic₆	1	4	yes	—
145	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
147	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
148	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
150	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
151	C₂×C₃×C₄	C₂×C₃×Q₈	1	4	yes	—
152	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
154	C₂×Dic₃	—	2	8	no	—
155	C₂×Dic₃	—	4	8	no	—
156	Dic₆	—	4	16	no	—
157	C₃×C₈	—	2	8	no	—
158	C₃×C₈	C₂×Dic₆	1	8	yes	complemented
159	C₃×C₈	C₂×C₃×Q₈	1	8	yes	complemented
160	C₂×C₃²	#500	1	4	yes	complemented
161	C₂×C₃²	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₂×C₃²	C₂²×D₈	1	1	yes	characteristic
163	C₂×C₃²	C₂²×Q₁₆	1	1	yes	characteristic
164	C₂×Q₈	—	6	8	no	—
165	C₂×Q₈	—	12	8	no	—
166	C₂²×C₄	—	6	4	no	—
167	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
168	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
169	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
170	C₂×C₈	C₃×Dic₆	1	8	yes	complemented
171	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
172	C₄⋊C₄	—	3	8	no	—
173	C₄⋊C₄	—	6	4	no	—
174	C₄⋊C₄	—	6	8	no	—
175	C₄²	C₃×D₁₂	1	2	yes	—
176	C₄²	C₂²×C₃×D₃	1	2	yes	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	#869	1	2	yes	—
182	C₂²×C₃	C₂×D₂₄	1	2	yes	—
183	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
184	C₂²×C₃	C₃×C₄⋊Q₈	1	2	yes	—
185	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
186	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
187	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
188	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
189	C₃×C₄	—	2	4	no	—
190	C₃×C₄	—	2	4	no	—
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	12	8	no	—
194	C₃×C₄	#869	1	2	yes	—
195	C₃×C₄	C₂×D₂₄	1	4	yes	—
196	C₃×C₄	C₂×Dic₁₂	1	4	yes	—
197	C₃×C₄	C₃×C₄⋊Q₈	1	2	yes	—
198	C₃×C₄	C₂×C₃×D₈	1	4	yes	—
199	C₃×C₄	C₂×C₃×Q₁₆	1	4	yes	—
200	C₃×C₄	C₂²×Dic₆	1	2	yes	—
201	C₃×C₄	C₂²×C₃×Q₈	1	2	yes	—
202	Dic₃	—	4	8	no	—
203	C₃²	C₂×#500	1	1	yes	characteristic, complemented, Sylow
204	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
205	Q₈	—	12	16	no	—
206	C₂×C₄	—	6	8	no	—
207	C₂×C₄	—	12	8	no	—
208	C₂×C₄	C₃×D₂₄	1	4	yes	—
209	C₂×C₄	C₃×Dic₁₂	1	4	yes	—
210	C₂×C₄	C₂×C₃×Dic₆	1	4	yes	—
211	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
212	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
214	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
215	C₈	C₂×C₃×Dic₆	1	8	yes	complemented
216	C₂×C₃	—	2	4	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	#4682	1	4	yes	direct product factor
221	C₂×C₃	C₃×#500	1	4	yes	complemented
222	C₂×C₃	C₂×#869	1	1	yes	characteristic
223	C₂×C₃	C₂²×D₂₄	1	1	yes	characteristic
224	C₂×C₃	C₂²×Dic₁₂	1	1	yes	characteristic
225	C₂×C₃	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
226	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
227	C₂×C₃	C₂²×C₃×Q₁₆	1	1	yes	characteristic
228	C₂²	C₃×#869	1	2	yes	—
229	C₂²	C₂×C₃×D₂₄	1	2	yes	—
230	C₂²	C₂×C₃×Dic₁₂	1	2	yes	—
231	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
232	C₄	—	12	8	no	—
233	C₄	C₃×#869	1	2	yes	—
234	C₄	C₂×C₃×D₂₄	1	4	yes	—
235	C₄	C₂×C₃×Dic₁₂	1	4	yes	—
236	C₄	C₂²×C₃×Dic₆	1	2	yes	—
237	C₃	—	2	1	no	—
238	C₃	C₂×#4682	1	1	yes	characteristic, direct product factor
239	C₃	C₂×C₃×#500	1	1	yes	characteristic, complemented
240	C₂	C₃×#4682	1	4	yes	direct product factor
241	C₂	C₂×C₃×#869	1	1	yes	characteristic
242	C₂	C₂²×C₃×D₂₄	1	1	yes	characteristic
243	C₂	C₂²×C₃×Dic₁₂	1	1	yes	characteristic

	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×#500
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×#4682	C₂	direct (N × H)
6. C₂×#4682	C₃	direct (N × H)
24. #4682	C₂×C₃	direct (N × H)
31. C₂×C₃²×C₈	Q₈	semidirect
71. C₃²×C₈	C₂×Q₈	semidirect
93. C₂×C₃×C₈	Dic₆	semidirect
94. C₂×C₃×C₈	C₃×Q₈	semidirect
158. C₃×C₈	C₂×Dic₆	semidirect
159. C₃×C₈	C₂×C₃×Q₈	semidirect
160. C₂×C₃²	#500	semidirect
170. C₂×C₈	C₃×Dic₆	semidirect
203. C₃²	C₂×#500	semidirect
215. C₈	C₂×C₃×Dic₆	semidirect
220. C₂×C₃	#4682	direct (N × H)
221. C₂×C₃	C₃×#500	semidirect
238. C₃	C₂×#4682	direct (N × H)
239. C₃	C₂×C₃×#500	semidirect
240. C₂	C₃×#4682	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×#818	2
C₃×#4682	2
C₂×C₃×#500	3
C₂×#4682	3

Subgroups — C2×C3×#4682 #11874947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#4682 #11874947