Subgroups — C₂²×C₃×#364 #10652006

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×M₃₂	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#364	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#364	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₃×M₃₂	—	2	6	no	—
9	C₂×C₃×M₃₂	C₂²	1	12	yes	—
10	C₂²×C₃×C₁₆	—	2	2	no	—
11	C₂×C₃×C₄×C₈	C₄	1	6	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	6	yes	—
13	C₃×#364	C₂²	1	16	yes	direct product factor
14	C₂²×M₃₂	C₂×C₃	1	2	yes	—
15	C₂²×C₄×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂×#364	C₂×C₃	1	12	yes	direct product factor
17	C₂³×C₃×C₄	—	2	1	no	—
18	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₈	C₈	1	6	yes	—
20	C₂²×C₃×C₈	C₈	1	2	yes	—
21	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
22	C₂²×C₃×C₈	C₂×C₄	1	6	yes	—
23	C₂²×C₃×C₈	D₄	1	3	yes	—
24	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₈	Q₈	1	1	yes	characteristic
26	C₂²×C₃×C₈	Q₈	1	3	yes	—
27	C₂²×C₃×C₈	C₂³	1	3	yes	—
28	C₂×C₃×C₄²	—	2	3	no	—
29	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
30	C₃×M₃₂	—	2	12	no	—
31	C₃×M₃₂	C₂³	1	8	yes	—
32	C₂×C₃×C₁₆	—	2	12	no	—
33	C₃×C₄×C₈	C₂×C₄	1	12	yes	—
34	C₃×C₄×C₈	C₂³	1	4	yes	—
35	C₂³×C₈	C₃×C₄	1	1	yes	characteristic
36	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
37	C₂²×C₄²	C₃×C₄	1	1	yes	characteristic
38	C₂×M₃₂	—	2	6	no	—
39	C₂×M₃₂	C₂²×C₃	1	12	yes	—
40	C₂²×C₁₆	—	2	2	no	—
41	C₂×C₄×C₈	C₃×C₄	1	6	yes	—
42	C₂×C₄×C₈	C₂²×C₃	1	6	yes	—
43	#364	C₂²×C₃	1	16	yes	direct product factor
44	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
45	C₂²×C₃×C₄	—	2	8	no	—
46	C₂²×C₃×C₄	—	2	6	no	—
47	C₂²×C₃×C₄	C₄⋊C₄	1	3	yes	—
48	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₃×C₄	C₂×C₈	1	3	yes	—
50	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₃×C₄	M₁₆	1	3	yes	—
52	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
53	C₂×C₃×C₈	C₄⋊C₄	1	3	yes	—
54	C₂×C₃×C₈	C₄⋊C₄	1	6	yes	—
55	C₂×C₃×C₈	C₄⋊C₄	1	3	yes	—
56	C₂×C₃×C₈	C₂×C₈	1	12	yes	—
57	C₂×C₃×C₈	C₂×C₈	1	12	yes	—
58	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
59	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
60	C₂×C₃×C₈	C₂×D₄	1	3	yes	—
61	C₂×C₃×C₈	C₂×D₄	1	3	yes	—
62	C₂×C₃×C₈	C₂×Q₈	1	3	yes	—
63	C₂×C₃×C₈	C₂×Q₈	1	3	yes	—
64	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
65	C₃×C₄²	—	2	6	no	—
66	C₃×C₄²	C₂²×C₄	1	4	yes	—
67	C₃×C₁₆	—	2	8	no	—
68	C₂³×C₄	—	2	1	no	—
69	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
70	C₂²×C₈	C₃×C₈	1	6	yes	—
71	C₂²×C₈	C₃×C₈	1	2	yes	—
72	C₂²×C₈	C₂×C₃×C₄	1	3	yes	—
73	C₂²×C₈	C₂×C₃×C₄	1	6	yes	—
74	C₂²×C₈	C₃×D₄	1	3	yes	—
75	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
76	C₂²×C₈	C₃×Q₈	1	1	yes	characteristic
77	C₂²×C₈	C₃×Q₈	1	3	yes	—
78	C₂²×C₈	C₂³×C₃	1	3	yes	—
79	C₂×C₄²	—	2	3	no	—
80	C₂×C₄²	C₂×C₃×C₄	1	6	yes	—
81	M₃₂	—	2	12	no	—
82	M₃₂	C₂³×C₃	1	8	yes	—
83	C₂×C₁₆	—	2	12	no	—
84	C₄×C₈	C₂×C₃×C₄	1	12	yes	—
85	C₄×C₈	C₂³×C₃	1	4	yes	—
86	C₂³×C₃	—	2	4	no	—
87	C₂³×C₃	C₄⋊C₈	1	3	yes	—
88	C₂³×C₃	C₄⋊C₈	1	1	yes	characteristic
89	C₂³×C₃	C₂×M₁₆	1	3	yes	—
90	C₂×C₃×C₄	—	2	24	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	C₄⋊C₈	1	6	yes	—
93	C₂×C₃×C₄	C₄⋊C₈	1	3	yes	—
94	C₂×C₃×C₄	C₄⋊C₈	1	3	yes	—
95	C₂×C₃×C₄	C₂×C₄⋊C₄	1	3	yes	—
96	C₂×C₃×C₄	C₂×C₄⋊C₄	1	3	yes	—
97	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
98	C₂×C₃×C₄	C₂²×C₈	1	3	yes	—
99	C₂×C₃×C₄	C₂×M₁₆	1	3	yes	—
100	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
101	C₃×C₈	C₂×C₄⋊C₄	1	3	yes	—
102	C₃×C₈	C₂×C₄⋊C₄	1	3	yes	—
103	C₃×C₈	C₂²×C₈	1	8	yes	—
104	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
105	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
106	C₂⁴	C₃×M₁₆	1	1	yes	characteristic
107	C₂²×C₄	—	2	8	no	—
108	C₂²×C₄	—	2	6	no	—
109	C₂²×C₄	C₃×C₄⋊C₄	1	3	yes	—
110	C₂²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄	C₂×C₃×C₈	1	3	yes	—
112	C₂²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
113	C₂²×C₄	C₃×M₁₆	1	3	yes	—
114	C₂²×C₄	C₂²×C₃×C₄	1	3	yes	—
115	C₂×C₈	C₃×C₄⋊C₄	1	6	yes	—
116	C₂×C₈	C₃×C₄⋊C₄	1	3	yes	—
117	C₂×C₈	C₃×C₄⋊C₄	1	3	yes	—
118	C₂×C₈	C₂×C₃×C₈	1	12	yes	—
119	C₂×C₈	C₂×C₃×C₈	1	12	yes	—
120	C₂×C₈	C₂²×C₃×C₄	1	3	yes	—
121	C₂×C₈	C₂²×C₃×C₄	1	4	yes	—
122	C₂×C₈	C₂×C₃×D₄	1	3	yes	—
123	C₂×C₈	C₂×C₃×D₄	1	3	yes	—
124	C₂×C₈	C₂×C₃×Q₈	1	3	yes	—
125	C₂×C₈	C₂×C₃×Q₈	1	3	yes	—
126	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
127	C₄²	—	2	6	no	—
128	C₄²	C₂²×C₃×C₄	1	4	yes	—
129	C₁₆	—	2	8	no	—
130	C₂²×C₃	—	2	6	no	—
131	C₂²×C₃	—	2	6	no	—
132	C₂²×C₃	#364	1	4	yes	direct product factor
133	C₂²×C₃	C₂×C₄⋊C₈	1	3	yes	—
134	C₂²×C₃	C₂×C₄⋊C₈	1	3	yes	—
135	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
136	C₃×C₄	—	2	8	no	—
137	C₃×C₄	C₂×C₄⋊C₈	1	3	yes	—
138	C₃×C₄	C₂×C₄⋊C₈	1	3	yes	—
139	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
140	C₃×C₄	C₂³×C₈	1	1	yes	characteristic
141	C₂³	—	2	4	no	—
142	C₂³	C₃×C₄⋊C₈	1	3	yes	—
143	C₂³	C₃×C₄⋊C₈	1	1	yes	characteristic
144	C₂³	C₂×C₃×M₁₆	1	3	yes	—
145	C₂×C₄	—	2	24	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₃×C₄⋊C₈	1	6	yes	—
148	C₂×C₄	C₃×C₄⋊C₈	1	3	yes	—
149	C₂×C₄	C₃×C₄⋊C₈	1	3	yes	—
150	C₂×C₄	C₂×C₃×C₄⋊C₄	1	3	yes	—
151	C₂×C₄	C₂×C₃×C₄⋊C₄	1	3	yes	—
152	C₂×C₄	C₂²×C₃×C₈	1	3	yes	—
153	C₂×C₄	C₂²×C₃×C₈	1	3	yes	—
154	C₂×C₄	C₂×C₃×M₁₆	1	3	yes	—
155	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
156	C₈	C₂×C₃×C₄⋊C₄	1	3	yes	—
157	C₈	C₂×C₃×C₄⋊C₄	1	3	yes	—
158	C₈	C₂²×C₃×C₈	1	8	yes	—
159	C₈	C₂²×C₃×D₄	1	1	yes	characteristic
160	C₈	C₂²×C₃×Q₈	1	1	yes	characteristic
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	2	3	no	—
163	C₂×C₃	C₂×#364	1	6	yes	direct product factor
164	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
165	C₂²	—	2	6	no	—
166	C₂²	—	2	6	no	—
167	C₂²	C₃×#364	1	4	yes	direct product factor
168	C₂²	C₂×C₃×C₄⋊C₈	1	3	yes	—
169	C₂²	C₂×C₃×C₄⋊C₈	1	3	yes	—
170	C₂²	C₂²×C₃×M₁₆	1	1	yes	characteristic
171	C₄	—	2	8	no	—
172	C₄	C₂×C₃×C₄⋊C₈	1	3	yes	—
173	C₄	C₂×C₃×C₄⋊C₈	1	3	yes	—
174	C₄	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
175	C₄	C₂³×C₃×C₈	1	1	yes	characteristic
176	C₃	C₂²×#364	1	1	yes	characteristic, direct product factor, Sylow
177	C₂	—	2	1	no	—
178	C₂	—	2	3	no	—
179	C₂	C₂×C₃×#364	1	6	yes	direct product factor
180	C₂	C₂²×C₃×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₈	D₄
commutator	C₄	C₂³×C₃×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂³×C₃	C₄⋊C₈
prime radical	C₂⁴×C₃	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₈ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#364	C₂	direct (N × H)
4. C₂²×#364	C₃	direct (N × H)
13. C₃×#364	C₂²	direct (N × H)
16. C₂×#364	C₂×C₃	direct (N × H)
43. #364	C₂²×C₃	direct (N × H)
132. C₂²×C₃	#364	direct (N × H)
163. C₂×C₃	C₂×#364	direct (N × H)
167. C₂²	C₃×#364	direct (N × H)
176. C₃	C₂²×#364	direct (N × H)
179. C₂	C₂×C₃×#364	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×M₃₂	2
C₂²×C₃×C₄×C₈	2
C₂×C₃×#364	2
C₂²×#364	3

Subgroups — C22×C3×#364 #10652006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#364 #10652006