Subgroups — C₂×C₃×#2402 #10653157

Order 768.

2078 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2402	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃×#400	C₂	1	4	yes	maximal
3	C₂×C₃×#398	C₂	1	4	yes	maximal
4	C₂×C₃×C₄×C₄⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₃×C₄×C₄⋊C₄	C₂	1	4	yes	maximal
6	C₂×C₃×#376	C₂	1	1	yes	maximal, characteristic
7	C₂×#2402	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂²×C₃×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂²×C₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂²×C₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂²×C₃×C₄²	C₂²	1	2	yes	—
12	C₂²×C₃×C₄²	C₂²	1	4	yes	—
13	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₃×#400	C₂²	1	32	yes	—
15	C₃×#398	C₂²	1	32	yes	—
16	C₃×C₄×C₄⋊C₄	C₂²	1	16	yes	—
17	C₃×C₄×C₄⋊C₄	C₂²	1	32	yes	—
18	C₃×#376	C₂²	1	8	yes	—
19	C₂×C₃×#95	C₂²	1	2	yes	—
20	C₂×C₃×#95	C₂²	1	4	yes	—
21	C₂×C₃×#95	C₂²	1	4	yes	—
22	C₂×C₃×#95	C₂²	1	4	yes	—
23	C₂×C₃×#95	C₂²	1	2	yes	—
24	#2402	C₂×C₃	1	16	yes	direct product factor
25	C₂×#400	C₂×C₃	1	4	yes	—
26	C₂×#398	C₂×C₃	1	4	yes	—
27	C₂×C₄×C₄⋊C₄	C₂×C₃	1	4	yes	—
28	C₂×C₄×C₄⋊C₄	C₂×C₃	1	2	yes	—
29	C₂×#376	C₂×C₃	1	1	yes	characteristic
30	C₂³×C₃×C₄	C₂³	1	4	yes	—
31	C₂³×C₃×C₄	C₂³	1	4	yes	—
32	C₂³×C₃×C₄	C₂³	1	2	yes	—
33	C₂³×C₃×C₄	C₂³	1	4	yes	—
34	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×C₄⋊C₄	—	2	16	no	—
36	C₂×C₃×C₄⋊C₄	—	2	16	no	—
37	C₂×C₃×C₄⋊C₄	—	2	16	no	—
38	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	—
39	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	—
40	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	—
41	C₂×C₃×C₄²	—	2	16	no	—
42	C₂×C₃×C₄²	—	2	8	no	—
43	C₂×C₃×C₄²	Q₈	1	8	yes	—
44	C₂×C₃×C₄²	C₂³	1	8	yes	—
45	C₂×C₃×C₄²	C₂³	1	16	yes	—
46	C₂×C₃×C₄²	C₂³	1	4	yes	—
47	C₃×#95	C₂³	1	16	yes	—
48	C₃×#95	C₂³	1	8	yes	—
49	C₃×#95	C₂³	1	16	yes	—
50	C₃×#95	C₂³	1	16	yes	—
51	C₃×#95	C₂³	1	8	yes	—
52	C₂²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
53	C₂²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
54	C₂²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
55	C₂²×C₄²	C₂²×C₃	1	2	yes	—
56	C₂²×C₄²	C₂²×C₃	1	4	yes	—
57	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
58	#400	C₂²×C₃	1	32	yes	—
59	#398	C₂²×C₃	1	32	yes	—
60	C₄×C₄⋊C₄	C₂²×C₃	1	16	yes	—
61	C₄×C₄⋊C₄	C₂²×C₃	1	32	yes	—
62	#376	C₂²×C₃	1	8	yes	—
63	C₂×#95	C₂²×C₃	1	2	yes	—
64	C₂×#95	C₂²×C₃	1	4	yes	—
65	C₂×#95	C₂²×C₃	1	4	yes	—
66	C₂×#95	C₂²×C₃	1	4	yes	—
67	C₂×#95	C₂²×C₃	1	2	yes	—
68	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
69	C₂²×C₃×C₄	—	2	8	no	—
70	C₂²×C₃×C₄	—	2	8	no	—
71	C₂²×C₃×C₄	—	2	8	no	—
72	C₂²×C₃×C₄	—	2	8	no	—
73	C₂²×C₃×C₄	—	2	8	no	—
74	C₂²×C₃×C₄	—	2	8	no	—
75	C₂²×C₃×C₄	—	2	8	no	—
76	C₂²×C₃×C₄	—	2	8	no	—
77	C₂²×C₃×C₄	—	2	8	no	—
78	C₂²×C₃×C₄	C₂×Q₈	1	8	yes	—
79	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
80	C₂²×C₃×C₄	#41	1	16	yes	—
81	C₂²×C₃×C₄	#41	1	8	yes	—
82	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
83	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
84	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
85	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
86	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
87	C₃×C₄⋊C₄	—	2	32	no	—
88	C₃×C₄⋊C₄	—	2	32	no	—
89	C₃×C₄⋊C₄	—	2	32	no	—
90	C₃×C₄²	—	2	32	no	—
91	C₃×C₄²	—	2	16	no	—
92	C₃×C₄²	C₂×Q₈	1	16	yes	—
93	C₂³×C₄	C₂³×C₃	1	4	yes	—
94	C₂³×C₄	C₂³×C₃	1	4	yes	—
95	C₂³×C₄	C₂³×C₃	1	2	yes	—
96	C₂³×C₄	C₂³×C₃	1	4	yes	—
97	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	2	16	no	—
99	C₂×C₄⋊C₄	—	2	16	no	—
100	C₂×C₄⋊C₄	—	2	16	no	—
101	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	—
102	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	—
103	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	—
104	C₂×C₄²	—	2	16	no	—
105	C₂×C₄²	—	2	8	no	—
106	C₂×C₄²	C₃×Q₈	1	8	yes	—
107	C₂×C₄²	C₂³×C₃	1	8	yes	—
108	C₂×C₄²	C₂³×C₃	1	16	yes	—
109	C₂×C₄²	C₂³×C₃	1	4	yes	—
110	#95	C₂³×C₃	1	16	yes	—
111	#95	C₂³×C₃	1	8	yes	—
112	#95	C₂³×C₃	1	16	yes	—
113	#95	C₂³×C₃	1	16	yes	—
114	#95	C₂³×C₃	1	8	yes	—
115	C₂³×C₃	C₂²×Q₈	1	2	yes	—
116	C₂³×C₃	C₂×#41	1	4	yes	—
117	C₂³×C₃	C₂×#41	1	8	yes	—
118	C₂³×C₃	C₂⁵	1	1	yes	characteristic
119	C₂×C₃×C₄	—	2	32	no	—
120	C₂×C₃×C₄	—	2	16	no	—
121	C₂×C₃×C₄	—	2	8	no	—
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	8	no	—
124	C₂×C₃×C₄	—	2	8	no	—
125	C₂×C₃×C₄	—	2	8	no	—
126	C₂×C₃×C₄	—	2	8	no	—
127	C₂×C₃×C₄	—	2	8	no	—
128	C₂×C₃×C₄	—	2	8	no	—
129	C₂×C₃×C₄	—	2	8	no	—
130	C₂×C₃×C₄	—	4	16	no	—
131	C₂×C₃×C₄	—	4	16	no	—
132	C₂×C₃×C₄	#125	1	16	yes	—
133	C₂×C₃×C₄	C₂²×Q₈	1	8	yes	—
134	C₂×C₃×C₄	C₂²×Q₈	1	4	yes	—
135	C₂×C₃×C₄	C₂×#41	1	16	yes	—
136	C₂×C₃×C₄	C₂×#41	1	8	yes	—
137	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
138	C₂²×C₄	—	2	8	no	—
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	8	no	—
141	C₂²×C₄	—	2	8	no	—
142	C₂²×C₄	—	2	8	no	—
143	C₂²×C₄	—	2	8	no	—
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	8	no	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	C₂×C₃×Q₈	1	8	yes	—
148	C₂²×C₄	C₂×C₃×Q₈	1	4	yes	—
149	C₂²×C₄	C₃×#41	1	16	yes	—
150	C₂²×C₄	C₃×#41	1	8	yes	—
151	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
152	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
153	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
154	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
155	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
156	C₄⋊C₄	—	2	32	no	—
157	C₄⋊C₄	—	2	32	no	—
158	C₄⋊C₄	—	2	32	no	—
159	C₄²	—	2	32	no	—
160	C₄²	—	2	16	no	—
161	C₄²	C₂×C₃×Q₈	1	16	yes	—
162	C₂²×C₃	C₂×#125	1	4	yes	—
163	C₂²×C₃	#529	1	16	yes	—
164	C₂²×C₃	#533	1	8	yes	—
165	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
166	C₂²×C₃	C₂²×#41	1	2	yes	—
167	C₂²×C₃	C₂²×#41	1	4	yes	—
168	C₃×C₄	—	2	16	no	—
169	C₃×C₄	—	2	8	no	—
170	C₃×C₄	—	4	8	no	—
171	C₃×C₄	—	4	8	no	—
172	C₃×C₄	C₂×#125	1	8	yes	—
173	C₂³	C₂²×C₃×Q₈	1	2	yes	—
174	C₂³	C₂×C₃×#41	1	4	yes	—
175	C₂³	C₂×C₃×#41	1	8	yes	—
176	C₂³	C₂⁵×C₃	1	1	yes	characteristic
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	32	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	8	no	—
188	C₂×C₄	—	4	16	no	—
189	C₂×C₄	—	4	16	no	—
190	C₂×C₄	C₃×#125	1	16	yes	—
191	C₂×C₄	C₂²×C₃×Q₈	1	8	yes	—
192	C₂×C₄	C₂²×C₃×Q₈	1	4	yes	—
193	C₂×C₄	C₂×C₃×#41	1	16	yes	—
194	C₂×C₄	C₂×C₃×#41	1	8	yes	—
195	C₂×C₃	#2402	1	8	yes	direct product factor
196	C₂×C₃	C₂²×#125	1	1	yes	characteristic
197	C₂×C₃	C₂×#529	1	4	yes	—
198	C₂×C₃	C₂×#533	1	2	yes	—
199	C₂²	C₂×C₃×#125	1	4	yes	—
200	C₂²	C₃×#529	1	16	yes	—
201	C₂²	C₃×#533	1	8	yes	—
202	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
203	C₂²	C₂²×C₃×#41	1	2	yes	—
204	C₂²	C₂²×C₃×#41	1	4	yes	—
205	C₄	—	2	16	no	—
206	C₄	—	2	8	no	—
207	C₄	—	4	8	no	—
208	C₄	—	4	8	no	—
209	C₄	C₂×C₃×#125	1	8	yes	—
210	C₃	C₂×#2402	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	C₃×#2402	1	8	yes	direct product factor
212	C₂	C₂²×C₃×#125	1	1	yes	characteristic
213	C₂	C₂×C₃×#529	1	4	yes	—
214	C₂	C₂×C₃×#533	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃×C₄	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
socle, prime radical	C₂⁴×C₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#2402	C₂	direct (N × H)
7. C₂×#2402	C₃	direct (N × H)
24. #2402	C₂×C₃	direct (N × H)
195. C₂×C₃	#2402	direct (N × H)
210. C₃	C₂×#2402	direct (N × H)
211. C₂	C₃×#2402	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#2402	2
C₂×C₃×#400	2
C₂×C₃×#398	2
C₂×C₃×C₄×C₄⋊C₄	2
C₂×C₃×#376	2
C₂×#2402	3

Subgroups — C2×C3×#2402 #10653157

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2402 #10653157