Subgroups — C₂×C₃×#1846 #10639403

Order 768.

1182 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#107	C₂	1	2	yes	maximal
2	C₂²×C₃×#106	C₂	1	2	yes	maximal
3	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#382	C₂	1	2	yes	maximal
5	C₃×#1846	C₂	1	8	yes	maximal, direct product factor
6	C₂×#1846	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×#107	—	2	8	no	—
12	C₂×C₃×#107	C₂²	1	8	yes	—
13	C₂×C₃×#106	—	2	8	no	—
14	C₂×C₃×#106	C₂²	1	8	yes	—
15	C₂×C₃×C₄×C₈	C₄	1	8	yes	complemented
16	C₂×C₃×C₄×C₈	C₂²	1	4	yes	—
17	C₃×#382	—	2	4	no	—
18	C₃×#382	C₂²	1	8	yes	—
19	C₂×C₃×#95	—	2	4	no	—
20	C₂²×#107	C₂×C₃	1	2	yes	—
21	C₂²×#106	C₂×C₃	1	2	yes	—
22	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
23	C₂×#382	C₂×C₃	1	2	yes	—
24	#1846	C₂×C₃	1	8	yes	direct product factor
25	C₂³×C₃×C₄	—	2	4	no	—
26	C₂³×C₃×C₄	D₄	1	2	yes	—
27	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₃×C₈	D₄	1	4	yes	—
31	C₂²×C₃×C₈	D₄	1	4	yes	—
32	C₂²×C₃×C₈	Q₈	1	4	yes	—
33	C₂²×C₃×C₈	Q₈	1	4	yes	—
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₂²×C₃×C₈	C₂³	1	2	yes	—
36	C₂×C₃×C₄⋊C₄	—	2	8	no	—
37	C₂×C₃×C₄⋊C₄	—	2	8	no	—
38	C₂×C₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
40	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
41	C₂×C₃×C₄²	D₄	1	2	yes	—
42	C₂×C₃×C₄²	Q₈	1	4	yes	—
43	C₂×C₃×C₄²	C₂³	1	2	yes	—
44	C₃×#107	—	2	16	no	—
45	C₃×#106	—	2	16	no	—
46	C₃×C₄×C₈	C₂×C₄	1	16	yes	complemented
47	C₃×#95	—	2	16	no	—
48	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
52	C₂×#107	—	2	8	no	—
53	C₂×#107	C₂²×C₃	1	8	yes	—
54	C₂×#106	—	2	8	no	—
55	C₂×#106	C₂²×C₃	1	8	yes	—
56	C₂×C₄×C₈	C₃×C₄	1	8	yes	complemented
57	C₂×C₄×C₈	C₂²×C₃	1	4	yes	—
58	#382	—	2	4	no	—
59	#382	C₂²×C₃	1	8	yes	—
60	C₂×#95	—	2	4	no	—
61	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₃×C₄	—	2	8	no	—
63	C₂²×C₃×C₄	—	2	8	no	—
64	C₂²×C₃×C₄	—	4	8	no	—
65	C₂²×C₃×C₄	—	4	16	no	—
66	C₂²×C₃×C₄	C₄⋊C₄	1	8	yes	—
67	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
68	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
69	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
71	C₂²×C₃×C₄	C₂×D₄	1	2	yes	—
72	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
73	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
74	C₂²×C₃×C₄	C₂×Q₈	1	2	yes	—
75	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
76	C₂×C₃×C₈	C₄⋊C₄	1	16	yes	—
77	C₂×C₃×C₈	C₄⋊C₄	1	16	yes	—
78	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
81	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
82	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
83	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
84	C₃×C₄⋊C₄	—	2	8	no	—
85	C₃×C₄⋊C₄	—	2	8	no	—
86	C₃×C₄⋊C₄	—	4	8	no	—
87	C₃×C₄²	C₄⋊C₄	1	8	yes	—
88	C₃×C₄²	C₂²×C₄	1	4	yes	—
89	C₃×C₄²	C₂×Q₈	1	4	yes	—
90	C₂³×C₄	—	2	4	no	—
91	C₂³×C₄	C₃×D₄	1	2	yes	—
92	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
93	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
94	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
95	C₂²×C₈	C₃×D₄	1	4	yes	—
96	C₂²×C₈	C₃×D₄	1	4	yes	—
97	C₂²×C₈	C₃×Q₈	1	4	yes	—
98	C₂²×C₈	C₃×Q₈	1	4	yes	—
99	C₂²×C₈	C₂³×C₃	1	2	yes	—
100	C₂²×C₈	C₂³×C₃	1	2	yes	—
101	C₂×C₄⋊C₄	—	2	8	no	—
102	C₂×C₄⋊C₄	—	2	8	no	—
103	C₂×C₄⋊C₄	—	2	4	no	—
104	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
105	C₂×C₄²	C₂×C₃×C₄	1	4	yes	—
106	C₂×C₄²	C₃×D₄	1	2	yes	—
107	C₂×C₄²	C₃×Q₈	1	4	yes	—
108	C₂×C₄²	C₂³×C₃	1	2	yes	—
109	#107	—	2	16	no	—
110	#106	—	2	16	no	—
111	C₄×C₈	C₂×C₃×C₄	1	16	yes	complemented
112	#95	—	2	16	no	—
113	C₂³×C₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂³×C₃	#127	1	2	yes	—
115	C₂³×C₃	C₄⋊Q₈	1	2	yes	—
116	C₂³×C₃	#135	1	8	yes	—
117	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	—	2	16	no	—
120	C₂×C₃×C₄	—	4	8	no	—
121	C₂×C₃×C₄	—	4	16	no	—
122	C₂×C₃×C₄	—	8	16	no	—
123	C₂×C₃×C₄	C₂×C₄⋊C₄	1	8	yes	—
124	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₃×C₄	#127	1	2	yes	—
128	C₂×C₃×C₄	C₄⋊Q₈	1	4	yes	—
129	C₂×C₃×C₄	C₄⋊Q₈	1	4	yes	—
130	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
131	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
137	C₃×C₈	C₂×C₄⋊C₄	1	8	yes	—
138	C₃×C₈	C₂×C₄⋊C₄	1	8	yes	—
139	C₂⁴	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	4	8	no	—
143	C₂²×C₄	—	4	16	no	—
144	C₂²×C₄	C₃×C₄⋊C₄	1	8	yes	—
145	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
147	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
149	C₂²×C₄	C₂×C₃×D₄	1	2	yes	—
150	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
151	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
152	C₂²×C₄	C₂×C₃×Q₈	1	2	yes	—
153	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
154	C₂×C₈	C₃×C₄⋊C₄	1	16	yes	—
155	C₂×C₈	C₃×C₄⋊C₄	1	16	yes	—
156	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
157	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
158	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
159	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
160	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
161	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
162	C₄⋊C₄	—	2	8	no	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	4	8	no	—
165	C₄²	C₃×C₄⋊C₄	1	8	yes	—
166	C₄²	C₂²×C₃×C₄	1	4	yes	—
167	C₄²	C₂×C₃×Q₈	1	4	yes	—
168	C₂²×C₃	#384	1	4	yes	—
169	C₂²×C₃	#427	1	8	yes	—
170	C₂²×C₃	#495	1	8	yes	—
171	C₂²×C₃	#499	1	8	yes	—
172	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₂²×C₃	C₂×#127	1	1	yes	characteristic
174	C₂²×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
175	C₂²×C₃	C₂×#135	1	4	yes	—
176	C₃×C₄	—	2	8	no	—
177	C₃×C₄	—	8	8	no	—
178	C₃×C₄	#384	1	4	yes	—
179	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
180	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
181	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
182	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
183	C₂³	C₂×C₃×C₄⋊C₄	1	2	yes	—
184	C₂³	C₃×#127	1	2	yes	—
185	C₂³	C₃×C₄⋊Q₈	1	2	yes	—
186	C₂³	C₃×#135	1	8	yes	—
187	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	2	16	no	—
190	C₂×C₄	—	4	16	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	—	8	16	no	—
193	C₂×C₄	C₂×C₃×C₄⋊C₄	1	8	yes	—
194	C₂×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
195	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
196	C₂×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
197	C₂×C₄	C₃×#127	1	2	yes	—
198	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
199	C₂×C₄	C₃×C₄⋊Q₈	1	4	yes	—
200	C₂×C₄	C₃×C₄⋊Q₈	1	4	yes	—
201	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
203	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
206	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
207	C₈	C₂×C₃×C₄⋊C₄	1	8	yes	—
208	C₈	C₂×C₃×C₄⋊C₄	1	8	yes	—
209	C₂×C₃	#1846	1	8	yes	direct product factor
210	C₂×C₃	C₂×#384	1	1	yes	characteristic
211	C₂×C₃	C₂×#427	1	2	yes	—
212	C₂×C₃	C₂×#495	1	2	yes	—
213	C₂×C₃	C₂×#499	1	2	yes	—
214	C₂²	C₃×#384	1	4	yes	—
215	C₂²	C₃×#427	1	8	yes	—
216	C₂²	C₃×#495	1	8	yes	—
217	C₂²	C₃×#499	1	8	yes	—
218	C₂²	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
219	C₂²	C₂×C₃×#127	1	1	yes	characteristic
220	C₂²	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
221	C₂²	C₂×C₃×#135	1	4	yes	—
222	C₄	—	2	8	no	—
223	C₄	—	8	8	no	—
224	C₄	C₃×#384	1	4	yes	—
225	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
226	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
227	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
228	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
229	C₃	C₂×#1846	1	1	yes	characteristic, direct product factor, Sylow
230	C₂	C₃×#1846	1	8	yes	direct product factor
231	C₂	C₂×C₃×#384	1	1	yes	characteristic
232	C₂	C₂×C₃×#427	1	2	yes	—
233	C₂	C₂×C₃×#495	1	2	yes	—
234	C₂	C₂×C₃×#499	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂²×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃×#1846	C₂	direct (N × H)
6. C₂×#1846	C₃	direct (N × H)
15. C₂×C₃×C₄×C₈	C₄	semidirect
24. #1846	C₂×C₃	direct (N × H)
46. C₃×C₄×C₈	C₂×C₄	semidirect
56. C₂×C₄×C₈	C₃×C₄	semidirect
111. C₄×C₈	C₂×C₃×C₄	semidirect
209. C₂×C₃	#1846	direct (N × H)
229. C₃	C₂×#1846	direct (N × H)
230. C₂	C₃×#1846	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×#107	2
C₂²×C₃×#106	2
C₂²×C₃×C₄×C₈	2
C₂×C₃×#382	2
C₃×#1846	2
C₂×#1846	3

Subgroups — C2×C3×#1846 #10639403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#1846 #10639403