Subgroups — C₂×C₃×#2252 #10652330

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#500	C₂	1	2	yes	maximal
2	C₃×#2252	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×#366	C₂	1	4	yes	maximal
4	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#2252	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×C₄⋊Q₈	—	2	2	no	—
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₃×#500	—	2	4	no	—
9	C₃×#500	C₂²	1	8	yes	—
10	C₂×C₃×#107	—	2	2	no	—
11	C₂×C₃×#107	C₂²	1	4	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#366	C₂²	1	16	yes	—
14	C₃×C₄×C₁₆	C₂²	1	4	yes	—
15	C₂×#500	C₂×C₃	1	2	yes	—
16	#2252	C₂×C₃	1	8	yes	direct product factor
17	C₂×#366	C₂×C₃	1	4	yes	—
18	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₃×Q₈	—	4	2	no	—
20	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
21	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄⋊Q₈	—	2	8	no	—
23	C₃×C₄⋊Q₈	—	4	4	no	—
24	C₂×C₃×C₄⋊C₄	—	2	4	no	—
25	C₂×C₃×C₄⋊C₄	—	4	2	no	—
26	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
27	C₂×C₃×C₁₆	Q₈	1	8	yes	complemented
28	C₂×C₃×C₁₆	C₂³	1	4	yes	—
29	C₃×#107	—	2	8	no	—
30	C₃×#107	—	2	4	no	—
31	C₃×#107	C₂³	1	8	yes	—
32	C₃×C₄×C₈	D₄	1	2	yes	—
33	C₃×C₄×C₈	C₂³	1	2	yes	—
34	C₂×C₄⋊Q₈	—	2	2	no	—
35	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
36	#500	—	2	4	no	—
37	#500	C₂²×C₃	1	8	yes	—
38	C₂×#107	—	2	2	no	—
39	C₂×#107	C₂²×C₃	1	4	yes	—
40	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
41	#366	C₂²×C₃	1	16	yes	—
42	C₄×C₁₆	C₂²×C₃	1	4	yes	—
43	C₂×C₃×Q₈	—	4	8	no	—
44	C₂×C₃×Q₈	—	8	8	no	—
45	C₂²×C₃×C₄	—	4	4	no	—
46	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
47	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
48	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₈	Q₁₆	1	4	yes	—
50	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
52	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
53	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
54	C₃×C₄⋊C₄	—	2	8	no	—
55	C₃×C₄⋊C₄	—	4	4	no	—
56	C₃×C₄⋊C₄	—	4	8	no	—
57	C₃×C₄²	D₈	1	2	yes	—
58	C₃×C₄²	C₂×D₄	1	2	yes	—
59	C₃×C₁₆	C₂×Q₈	1	8	yes	complemented
60	C₂²×Q₈	—	4	2	no	—
61	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
62	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
63	C₄⋊Q₈	—	2	8	no	—
64	C₄⋊Q₈	—	4	4	no	—
65	C₂×C₄⋊C₄	—	2	4	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
68	C₂×C₁₆	C₃×Q₈	1	8	yes	complemented
69	C₂×C₁₆	C₂³×C₃	1	4	yes	—
70	#107	—	2	8	no	—
71	#107	—	2	4	no	—
72	#107	C₂³×C₃	1	8	yes	—
73	C₄×C₈	C₃×D₄	1	2	yes	—
74	C₄×C₈	C₂³×C₃	1	2	yes	—
75	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
76	C₃×Q₈	—	8	16	no	—
77	C₂×C₃×C₄	—	4	8	no	—
78	C₂×C₃×C₄	—	8	8	no	—
79	C₂×C₃×C₄	D₁₆	1	4	yes	—
80	C₂×C₃×C₄	Q₃₂	1	4	yes	—
81	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
82	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
84	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
85	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
86	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
87	C₃×C₈	C₄⋊Q₈	1	2	yes	—
88	C₃×C₈	C₂×Q₁₆	1	4	yes	—
89	C₃×C₈	C₂²×Q₈	1	2	yes	—
90	C₂×Q₈	—	4	8	no	—
91	C₂×Q₈	—	8	8	no	—
92	C₂²×C₄	—	4	4	no	—
93	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
94	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
96	C₂×C₈	C₃×Q₁₆	1	4	yes	—
97	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
98	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
99	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
100	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
101	C₄⋊C₄	—	2	8	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	4	8	no	—
104	C₄²	C₃×D₈	1	2	yes	—
105	C₄²	C₂×C₃×D₄	1	2	yes	—
106	C₁₆	C₂×C₃×Q₈	1	8	yes	complemented
107	C₂²×C₃	#500	1	2	yes	—
108	C₂²×C₃	C₂×D₁₆	1	2	yes	—
109	C₂²×C₃	C₂×Q₃₂	1	2	yes	—
110	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
111	C₃×C₄	—	8	8	no	—
112	C₃×C₄	#500	1	2	yes	—
113	C₃×C₄	C₂×D₁₆	1	4	yes	—
114	C₃×C₄	C₂×Q₃₂	1	4	yes	—
115	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
117	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
118	Q₈	—	8	16	no	—
119	C₂×C₄	—	4	8	no	—
120	C₂×C₄	—	8	8	no	—
121	C₂×C₄	C₃×D₁₆	1	4	yes	—
122	C₂×C₄	C₃×Q₃₂	1	4	yes	—
123	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
124	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
125	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
126	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
127	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
128	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
129	C₈	C₃×C₄⋊Q₈	1	2	yes	—
130	C₈	C₂×C₃×Q₁₆	1	4	yes	—
131	C₈	C₂²×C₃×Q₈	1	2	yes	—
132	C₂×C₃	#2252	1	4	yes	direct product factor
133	C₂×C₃	C₂×#500	1	1	yes	characteristic
134	C₂×C₃	C₂²×D₁₆	1	1	yes	characteristic
135	C₂×C₃	C₂²×Q₃₂	1	1	yes	characteristic
136	C₂²	C₃×#500	1	2	yes	—
137	C₂²	C₂×C₃×D₁₆	1	2	yes	—
138	C₂²	C₂×C₃×Q₃₂	1	2	yes	—
139	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
140	C₄	—	8	8	no	—
141	C₄	C₃×#500	1	2	yes	—
142	C₄	C₂×C₃×D₁₆	1	4	yes	—
143	C₄	C₂×C₃×Q₃₂	1	4	yes	—
144	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
145	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
146	C₃	C₂×#2252	1	1	yes	characteristic, direct product factor, Sylow
147	C₂	C₃×#2252	1	4	yes	direct product factor
148	C₂	C₂×C₃×#500	1	1	yes	characteristic
149	C₂	C₂²×C₃×D₁₆	1	1	yes	characteristic
150	C₂	C₂²×C₃×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2252	C₂	direct (N × H)
5. C₂×#2252	C₃	direct (N × H)
16. #2252	C₂×C₃	direct (N × H)
27. C₂×C₃×C₁₆	Q₈	semidirect
59. C₃×C₁₆	C₂×Q₈	semidirect
68. C₂×C₁₆	C₃×Q₈	semidirect
106. C₁₆	C₂×C₃×Q₈	semidirect
132. C₂×C₃	#2252	direct (N × H)
146. C₃	C₂×#2252	direct (N × H)
147. C₂	C₃×#2252	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×C₃×#500	2
C₃×#2252	2
C₂×C₃×#366	2
C₂×C₃×C₄×C₁₆	2
C₂×#2252	3

Subgroups — C2×C3×#2252 #10652330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#2252 #10652330