Subgroups — C₂²×C₃×#338 #10638946

Order 768.

2110 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×M₁₆	C₂	1	3	yes	maximal
2	C₂×C₃×#338	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#338	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
5	C₂²×C₃×M₁₆	—	2	6	no	—
6	C₂²×C₃×M₁₆	—	2	18	no	—
7	C₂²×C₃×M₁₆	C₄	1	18	yes	—
8	C₂²×C₃×M₁₆	C₂²	1	18	yes	—
9	C₂³×C₃×C₈	C₄	1	6	yes	—
10	C₃×#338	C₂²	1	16	yes	direct product factor
11	C₂³×M₁₆	C₂×C₃	1	3	yes	—
12	C₂×#338	C₂×C₃	1	12	yes	direct product factor
13	C₂⁵×C₃	Q₈	1	1	yes	characteristic
14	C₂³×C₃×C₄	C₂×C₄	1	9	yes	—
15	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
16	C₂³×C₃×C₄	D₄	1	3	yes	—
17	C₂³×C₃×C₄	D₄	1	9	yes	—
18	C₂³×C₃×C₄	Q₈	1	3	yes	—
19	C₂³×C₃×C₄	C₂³	1	3	yes	—
20	C₂×C₃×M₁₆	—	2	36	no	—
21	C₂×C₃×M₁₆	—	2	36	no	—
22	C₂×C₃×M₁₆	—	2	36	no	—
23	C₂×C₃×M₁₆	—	2	36	no	—
24	C₂×C₃×M₁₆	C₂×C₄	1	36	yes	—
25	C₂×C₃×M₁₆	C₂³	1	12	yes	—
26	C₂²×C₃×C₈	—	2	6	no	—
27	C₂²×C₃×C₈	—	2	18	no	—
28	C₂²×C₃×C₈	C₂×C₄	1	36	yes	—
29	C₂⁴×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₂²×M₁₆	—	2	6	no	—
31	C₂²×M₁₆	—	2	18	no	—
32	C₂²×M₁₆	C₃×C₄	1	18	yes	—
33	C₂²×M₁₆	C₂²×C₃	1	18	yes	—
34	C₂³×C₈	C₃×C₄	1	6	yes	—
35	#338	C₂²×C₃	1	16	yes	direct product factor
36	C₂⁴×C₃	—	4	4	no	—
37	C₂⁴×C₃	C₄⋊C₄	1	3	yes	—
38	C₂⁴×C₃	C₄⋊C₄	1	9	yes	—
39	C₂⁴×C₃	C₂×Q₈	1	3	yes	—
40	C₂²×C₃×C₄	C₄²	1	6	yes	—
41	C₂²×C₃×C₄	C₄²	1	9	yes	—
42	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂²⋊C₄	1	3	yes	—
44	C₂²×C₃×C₄	C₂²⋊C₄	1	9	yes	—
45	C₂²×C₃×C₄	C₂²⋊C₄	1	18	yes	—
46	C₂²×C₃×C₄	C₂²⋊C₄	1	18	yes	—
47	C₂²×C₃×C₄	C₄⋊C₄	1	9	yes	—
48	C₂²×C₃×C₄	C₄⋊C₄	1	18	yes	—
49	C₂²×C₃×C₄	C₄⋊C₄	1	9	yes	—
50	C₂²×C₃×C₄	C₂²×C₄	1	9	yes	—
51	C₂²×C₃×C₄	C₂²×C₄	1	9	yes	—
52	C₂²×C₃×C₄	C₂×D₄	1	9	yes	—
53	C₂²×C₃×C₄	C₂×D₄	1	9	yes	—
54	C₂²×C₃×C₄	C₂×Q₈	1	3	yes	—
55	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
56	C₃×M₁₆	—	2	72	no	—
57	C₃×M₁₆	—	2	24	no	—
58	C₂×C₃×C₈	—	2	36	no	—
59	C₂×C₃×C₈	—	2	36	no	—
60	C₂×C₃×C₈	C₂²×C₄	1	24	yes	—
61	C₂⁵	C₃×Q₈	1	1	yes	characteristic
62	C₂³×C₄	C₂×C₃×C₄	1	3	yes	—
63	C₂³×C₄	C₂×C₃×C₄	1	9	yes	—
64	C₂³×C₄	C₃×D₄	1	3	yes	—
65	C₂³×C₄	C₃×D₄	1	9	yes	—
66	C₂³×C₄	C₃×Q₈	1	3	yes	—
67	C₂³×C₄	C₂³×C₃	1	3	yes	—
68	C₂×M₁₆	—	2	36	no	—
69	C₂×M₁₆	—	2	36	no	—
70	C₂×M₁₆	—	2	36	no	—
71	C₂×M₁₆	—	2	36	no	—
72	C₂×M₁₆	C₂×C₃×C₄	1	36	yes	—
73	C₂×M₁₆	C₂³×C₃	1	12	yes	—
74	C₂²×C₈	—	2	18	no	—
75	C₂²×C₈	—	2	6	no	—
76	C₂²×C₈	C₂×C₃×C₄	1	36	yes	—
77	C₂³×C₃	—	2	12	no	—
78	C₂³×C₃	—	4	18	no	—
79	C₂³×C₃	—	4	6	no	—
80	C₂³×C₃	#95	1	9	yes	—
81	C₂³×C₃	#95	1	6	yes	—
82	C₂³×C₃	#95	1	1	yes	characteristic
83	C₂³×C₃	C₂×C₄⋊C₄	1	9	yes	—
84	C₂³×C₃	C₂×C₄⋊C₄	1	9	yes	—
85	C₂³×C₃	C₂²×Q₈	1	1	yes	characteristic
86	C₂×C₃×C₄	#95	1	18	yes	—
87	C₂×C₃×C₄	#95	1	18	yes	—
88	C₂×C₃×C₄	#95	1	9	yes	—
89	C₂×C₃×C₄	#95	1	3	yes	—
90	C₂×C₃×C₄	C₂×C₄²	1	9	yes	—
91	C₂×C₃×C₄	C₂×C₄²	1	3	yes	—
92	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	9	yes	—
93	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	9	yes	—
94	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	18	yes	—
95	C₂×C₃×C₄	C₂×C₄⋊C₄	1	9	yes	—
96	C₂×C₃×C₄	C₂×C₄⋊C₄	1	9	yes	—
97	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
98	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
99	C₃×C₈	—	2	24	no	—
100	C₂⁴	—	4	4	no	—
101	C₂⁴	C₃×C₄⋊C₄	1	3	yes	—
102	C₂⁴	C₃×C₄⋊C₄	1	9	yes	—
103	C₂⁴	C₂×C₃×Q₈	1	3	yes	—
104	C₂²×C₄	C₃×C₄²	1	9	yes	—
105	C₂²×C₄	C₃×C₄²	1	6	yes	—
106	C₂²×C₄	C₃×C₄²	1	1	yes	characteristic
107	C₂²×C₄	C₃×C₂²⋊C₄	1	9	yes	—
108	C₂²×C₄	C₃×C₂²⋊C₄	1	3	yes	—
109	C₂²×C₄	C₃×C₂²⋊C₄	1	18	yes	—
110	C₂²×C₄	C₃×C₂²⋊C₄	1	18	yes	—
111	C₂²×C₄	C₃×C₄⋊C₄	1	9	yes	—
112	C₂²×C₄	C₃×C₄⋊C₄	1	9	yes	—
113	C₂²×C₄	C₃×C₄⋊C₄	1	18	yes	—
114	C₂²×C₄	C₂²×C₃×C₄	1	9	yes	—
115	C₂²×C₄	C₂²×C₃×C₄	1	9	yes	—
116	C₂²×C₄	C₂×C₃×D₄	1	9	yes	—
117	C₂²×C₄	C₂×C₃×D₄	1	9	yes	—
118	C₂²×C₄	C₂×C₃×Q₈	1	3	yes	—
119	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
120	M₁₆	—	2	72	no	—
121	M₁₆	—	2	24	no	—
122	C₂×C₈	—	2	36	no	—
123	C₂×C₈	—	2	36	no	—
124	C₂×C₈	C₂²×C₃×C₄	1	24	yes	—
125	C₂²×C₃	—	2	18	no	—
126	C₂²×C₃	—	2	18	no	—
127	C₂²×C₃	—	4	6	no	—
128	C₂²×C₃	—	4	9	no	—
129	C₂²×C₃	—	4	1	no	—
130	C₂²×C₃	#338	1	4	yes	direct product factor
131	C₂²×C₃	C₂×#95	1	9	yes	—
132	C₂²×C₃	C₂×#95	1	3	yes	—
133	C₂²×C₃	C₂²×C₄⋊C₄	1	3	yes	—
134	C₃×C₄	C₂×#95	1	9	yes	—
135	C₃×C₄	C₂×#95	1	3	yes	—
136	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
137	C₃×C₄	C₂²×C₂²⋊C₄	1	3	yes	—
138	C₂³	—	2	12	no	—
139	C₂³	—	4	18	no	—
140	C₂³	—	4	6	no	—
141	C₂³	C₃×#95	1	9	yes	—
142	C₂³	C₃×#95	1	6	yes	—
143	C₂³	C₃×#95	1	1	yes	characteristic
144	C₂³	C₂×C₃×C₄⋊C₄	1	9	yes	—
145	C₂³	C₂×C₃×C₄⋊C₄	1	9	yes	—
146	C₂³	C₂²×C₃×Q₈	1	1	yes	characteristic
147	C₂×C₄	C₃×#95	1	18	yes	—
148	C₂×C₄	C₃×#95	1	18	yes	—
149	C₂×C₄	C₃×#95	1	9	yes	—
150	C₂×C₄	C₃×#95	1	3	yes	—
151	C₂×C₄	C₂×C₃×C₄²	1	9	yes	—
152	C₂×C₄	C₂×C₃×C₄²	1	3	yes	—
153	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	9	yes	—
154	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	9	yes	—
155	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	18	yes	—
156	C₂×C₄	C₂×C₃×C₄⋊C₄	1	9	yes	—
157	C₂×C₄	C₂×C₃×C₄⋊C₄	1	9	yes	—
158	C₂×C₄	C₂³×C₃×C₄	1	3	yes	—
159	C₂×C₄	C₂²×C₃×D₄	1	3	yes	—
160	C₈	—	2	24	no	—
161	C₂×C₃	—	2	3	no	—
162	C₂×C₃	—	2	9	no	—
163	C₂×C₃	C₂×#338	1	6	yes	direct product factor
164	C₂×C₃	C₂²×#95	1	1	yes	characteristic
165	C₂²	—	2	18	no	—
166	C₂²	—	2	18	no	—
167	C₂²	—	4	6	no	—
168	C₂²	—	4	9	no	—
169	C₂²	—	4	1	no	—
170	C₂²	C₃×#338	1	4	yes	direct product factor
171	C₂²	C₂×C₃×#95	1	9	yes	—
172	C₂²	C₂×C₃×#95	1	3	yes	—
173	C₂²	C₂²×C₃×C₄⋊C₄	1	3	yes	—
174	C₄	C₂×C₃×#95	1	3	yes	—
175	C₄	C₂×C₃×#95	1	9	yes	—
176	C₄	C₂²×C₃×C₄²	1	1	yes	characteristic
177	C₄	C₂²×C₃×C₂²⋊C₄	1	3	yes	—
178	C₃	C₂²×#338	1	1	yes	characteristic, direct product factor, Sylow
179	C₂	—	2	9	no	—
180	C₂	—	2	3	no	—
181	C₂	C₂×C₃×#338	1	6	yes	direct product factor
182	C₂	C₂²×C₃×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₄²
commutator	C₄	C₂²×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂³×C₃	#95
prime radical	C₂⁵×C₃	Q₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#338	C₂	direct (N × H)
3. C₂²×#338	C₃	direct (N × H)
10. C₃×#338	C₂²	direct (N × H)
12. C₂×#338	C₂×C₃	direct (N × H)
35. #338	C₂²×C₃	direct (N × H)
130. C₂²×C₃	#338	direct (N × H)
163. C₂×C₃	C₂×#338	direct (N × H)
170. C₂²	C₃×#338	direct (N × H)
178. C₃	C₂²×#338	direct (N × H)
181. C₂	C₂×C₃×#338	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×M₁₆	2
C₂×C₃×#338	2
C₂²×#338	3

Subgroups — C22×C3×#338 #10638946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#338 #10638946