Subgroups — C₂×C₃²×#364 #11769201

Order 1152.

504 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×M₃₂	C₂	1	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#364	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#364	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃²×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
8	C₃²×M₃₂	—	2	2	no	—
9	C₃²×M₃₂	C₂²	1	4	yes	—
10	C₂×C₃²×C₁₆	—	2	2	no	—
11	C₃²×C₄×C₈	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	2	yes	—
13	C₂×C₃×M₃₂	C₂×C₃	1	8	yes	—
14	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
15	C₃×#364	C₂×C₃	1	16	yes	direct product factor
16	C₂²×C₃²×C₄	—	2	1	no	—
17	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₈	C₈	1	2	yes	—
19	C₂×C₃²×C₈	C₈	1	2	yes	—
20	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
23	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
25	C₂×C₃²×C₈	Q₈	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
27	C₃²×C₄²	—	2	1	no	—
28	C₃²×C₄²	C₂×C₄	1	2	yes	—
29	C₃²×C₁₆	—	2	4	no	—
30	C₂×#364	C₃²	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×C₃×C₈	C₃×C₄	1	4	yes	—
32	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
33	C₂×C₃×C₄²	C₃×C₄	1	4	yes	—
34	C₃×M₃₂	—	2	8	no	—
35	C₃×M₃₂	C₂²×C₃	1	16	yes	—
36	C₂×C₃×C₁₆	—	2	8	no	—
37	C₃×C₄×C₈	C₃×C₄	1	8	yes	—
38	C₃×C₄×C₈	C₂²×C₃	1	8	yes	—
39	C₂³×C₃²	M₁₆	1	1	yes	characteristic
40	C₂×C₃²×C₄	—	2	4	no	—
41	C₂×C₃²×C₄	—	2	2	no	—
42	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
46	C₂×C₃²×C₄	M₁₆	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
48	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₃²×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₃²×C₈	C₂×C₈	1	4	yes	—
51	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₃²×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×M₃₂	C₂×C₃²	1	2	yes	—
54	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
55	#364	C₂×C₃²	1	4	yes	direct product factor
56	C₂²×C₃×C₄	—	2	4	no	—
57	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
58	C₂×C₃×C₈	C₃×C₈	1	8	yes	—
59	C₂×C₃×C₈	C₃×C₈	1	8	yes	—
60	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
61	C₂×C₃×C₈	C₂×C₃×C₄	1	8	yes	—
62	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
63	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
64	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
65	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
66	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
67	C₃×C₄²	—	2	4	no	—
68	C₃×C₄²	C₂×C₃×C₄	1	8	yes	—
69	C₃×C₁₆	—	2	16	no	—
70	C₂²×C₃²	—	2	2	no	—
71	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
72	C₂²×C₃²	C₄⋊C₈	1	1	yes	characteristic
73	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
74	C₃²×C₄	—	2	4	no	—
75	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
76	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
77	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
79	C₂²×C₈	C₃²×C₄	1	1	yes	characteristic
80	C₂²×C₈	C₂²×C₃²	1	1	yes	characteristic
81	C₂×C₄²	C₃²×C₄	1	1	yes	characteristic
82	M₃₂	—	2	2	no	—
83	M₃₂	C₂²×C₃²	1	4	yes	—
84	C₂×C₁₆	—	2	2	no	—
85	C₄×C₈	C₃²×C₄	1	2	yes	—
86	C₄×C₈	C₂²×C₃²	1	2	yes	—
87	C₂³×C₃	C₃×M₁₆	1	4	yes	—
88	C₂×C₃×C₄	—	2	16	no	—
89	C₂×C₃×C₄	—	2	8	no	—
90	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
91	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
92	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
93	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
94	C₂×C₃×C₄	C₃×M₁₆	1	4	yes	—
95	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
96	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
97	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	—
98	C₃×C₈	C₂×C₃×C₈	1	16	yes	—
99	C₃×C₈	C₂×C₃×D₄	1	4	yes	—
100	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
101	C₂×C₃²	—	2	1	no	—
102	C₂×C₃²	—	2	1	no	—
103	C₂×C₃²	#364	1	2	yes	direct product factor
104	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
107	C₂×C₈	C₃²×C₈	1	2	yes	—
108	C₂×C₈	C₃²×C₈	1	2	yes	—
109	C₂×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
110	C₂×C₈	C₂×C₃²×C₄	1	2	yes	—
111	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
112	C₂×C₈	C₃²×D₄	1	1	yes	characteristic
113	C₂×C₈	C₃²×Q₈	1	1	yes	characteristic
114	C₂×C₈	C₃²×Q₈	1	1	yes	characteristic
115	C₂×C₈	C₂³×C₃²	1	1	yes	characteristic
116	C₄²	—	2	1	no	—
117	C₄²	C₂×C₃²×C₄	1	2	yes	—
118	C₁₆	—	2	4	no	—
119	C₂²×C₃	—	2	8	no	—
120	C₂²×C₃	C₃×C₄⋊C₈	1	4	yes	—
121	C₂²×C₃	C₃×C₄⋊C₈	1	4	yes	—
122	C₂²×C₃	C₂×C₃×M₁₆	1	4	yes	—
123	C₃×C₄	—	2	16	no	—
124	C₃×C₄	C₃×C₄⋊C₈	1	4	yes	—
125	C₃×C₄	C₃×C₄⋊C₈	1	4	yes	—
126	C₃×C₄	C₂×C₃×C₄⋊C₄	1	4	yes	—
127	C₃×C₄	C₂²×C₃×C₈	1	4	yes	—
128	C₃²	C₂×#364	1	1	yes	characteristic, direct product factor, Sylow
129	C₂³	C₃²×M₁₆	1	1	yes	characteristic
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄	C₃²×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
135	C₂×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
136	C₂×C₄	C₃²×M₁₆	1	1	yes	characteristic
137	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
138	C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
139	C₈	C₃²×C₄⋊C₄	1	1	yes	characteristic
140	C₈	C₂×C₃²×C₈	1	4	yes	—
141	C₈	C₂×C₃²×D₄	1	1	yes	characteristic
142	C₈	C₂×C₃²×Q₈	1	1	yes	characteristic
143	C₂×C₃	—	2	4	no	—
144	C₂×C₃	—	2	4	no	—
145	C₂×C₃	C₃×#364	1	8	yes	direct product factor
146	C₂×C₃	C₂×C₃×C₄⋊C₈	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₃²×C₄⋊C₈	1	1	yes	characteristic
149	C₂²	C₃²×C₄⋊C₈	1	1	yes	characteristic
150	C₂²	C₂×C₃²×M₁₆	1	1	yes	characteristic
151	C₄	—	2	4	no	—
152	C₄	C₃²×C₄⋊C₈	1	1	yes	characteristic
153	C₄	C₃²×C₄⋊C₈	1	1	yes	characteristic
154	C₄	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
155	C₄	C₂²×C₃²×C₈	1	1	yes	characteristic
156	C₃	C₂×C₃×#364	1	4	yes	direct product factor
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	C₃²×#364	1	2	yes	direct product factor
160	C₂	C₂×C₃²×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₈	D₄
commutator	C₄	C₂²×C₃²×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃²
socle	C₂²×C₃²	C₄⋊C₈
prime radical	C₂³×C₃²	M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₈ ◁ C₂²×C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃²×#364	C₂	direct (N × H)
4. C₂×C₃×#364	C₃	direct (N × H)
15. C₃×#364	C₂×C₃	direct (N × H)
30. C₂×#364	C₃²	direct (N × H)
55. #364	C₂×C₃²	direct (N × H)
103. C₂×C₃²	#364	direct (N × H)
128. C₃²	C₂×#364	direct (N × H)
145. C₂×C₃	C₃×#364	direct (N × H)
156. C₃	C₂×C₃×#364	direct (N × H)
159. C₂	C₃²×#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
C₂×C₃²×M₃₂	2
C₂×C₃²×C₄×C₈	2
C₃²×#364	2
C₂×C₃×#364	3

Subgroups — C2×C32×#364 #11769201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#364 #11769201