Subgroups — C₂×#69434 #10792514

Order 768.

1614 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#376	C₂	1	1	yes	maximal, characteristic
2	C₂²×#803	C₂	1	3	yes	maximal
3	C₂²×C₄×C₃⋊C₈	C₂	1	3	yes	maximal
4	#69434	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1726	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄²	C₂²	1	3	yes	—
7	C₂³×C₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₃⋊C₈	C₂²	1	3	yes	—
9	C₃×#376	C₄	1	4	yes	—
10	C₃×#376	C₂²	1	4	yes	—
11	C₂×C₃×#95	C₄	1	4	yes	—
12	C₂×#803	C₄	1	24	yes	complemented
13	C₂×#803	C₂²	1	12	yes	—
14	C₂×C₄×C₃⋊C₈	C₄	1	24	yes	complemented
15	C₂×C₄×C₃⋊C₈	C₂²	1	12	yes	—
16	C₂²×#97	—	3	3	no	—
17	C₂²×C₄×C₈	—	3	3	no	—
18	C₂×#376	D₃	1	1	yes	characteristic
19	#1726	—	3	8	no	—
20	C₂³×C₃×C₄	C₂×C₄	1	6	yes	—
21	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
23	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
24	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
25	C₂×C₃×C₄²	C₂³	1	6	yes	—
26	C₂²×C₃⋊C₈	C₂×C₄	1	12	yes	—
27	C₂²×C₃⋊C₈	C₂×C₄	1	12	yes	—
28	C₂²×C₃⋊C₈	C₂×C₄	1	24	yes	—
29	C₂²×C₃⋊C₈	C₂³	1	2	yes	—
30	C₂²×C₃⋊C₈	C₂³	1	6	yes	—
31	C₃×#95	C₂×C₄	1	16	yes	—
32	#803	C₂×C₄	1	48	yes	complemented
33	C₄×C₃⋊C₈	C₂×C₄	1	48	yes	complemented
34	C₂³×C₈	—	3	1	no	—
35	C₂³×C₈	—	3	3	no	—
36	C₂²×C₄²	C₂×D₃	1	3	yes	—
37	C₂×#97	—	3	24	no	—
38	C₂×#97	—	3	12	no	—
39	C₂×C₄×C₈	—	3	24	no	—
40	C₂×C₄×C₈	—	3	12	no	—
41	#376	Dic₃	1	4	yes	—
42	#376	C₂×D₃	1	4	yes	—
43	C₂×#95	Dic₃	1	4	yes	—
44	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₃×C₄	—	2	24	no	—
46	C₂²×C₃×C₄	C₄²	1	24	yes	—
47	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
48	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
49	C₂²×C₃×C₄	C₂²×C₄	1	12	yes	—
50	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
52	C₃×C₄²	C₄²	1	24	yes	—
53	C₃×C₄²	C₂²×C₄	1	12	yes	—
54	C₃×C₄²	C₂²×C₄	1	12	yes	—
55	C₂×C₃⋊C₈	C₄²	1	16	yes	—
56	C₂×C₃⋊C₈	C₄²	1	48	yes	—
57	C₂×C₃⋊C₈	C₂²×C₄	1	12	yes	—
58	C₂×C₃⋊C₈	C₂²×C₄	1	12	yes	—
59	C₂×C₃⋊C₈	C₂²×C₄	1	24	yes	—
60	C₂³×C₄	C₂×Dic₃	1	6	yes	—
61	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
62	C₂²×C₈	—	3	12	no	—
63	C₂²×C₈	—	3	12	no	—
64	C₂²×C₈	—	3	24	no	—
65	C₂²×C₈	—	3	2	no	—
66	C₂²×C₈	—	3	6	no	—
67	C₂×C₄²	C₄×D₃	1	12	yes	—
68	C₂×C₄²	C₄×D₃	1	12	yes	—
69	C₂×C₄²	C₂×Dic₃	1	6	yes	—
70	C₂×C₄²	C₂²×D₃	1	6	yes	—
71	#97	—	3	48	no	—
72	C₄×C₈	—	3	48	no	—
73	#95	C₂×Dic₃	1	16	yes	—
74	C₂³×C₃	C₂×C₄²	1	6	yes	—
75	C₂³×C₃	#131	1	8	yes	—
76	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄	—	2	24	no	—
78	C₂×C₃×C₄	—	2	48	no	—
79	C₂×C₃×C₄	C₂×C₄²	1	4	yes	—
80	C₂×C₃×C₄	C₂×C₄²	1	12	yes	—
81	C₂×C₃×C₄	C₂×C₄²	1	24	yes	—
82	C₂×C₃×C₄	C₂×C₄²	1	6	yes	—
83	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
84	C₂×C₃×C₄	C₂³×C₄	1	3	yes	—
85	C₃⋊C₈	C₂×C₄²	1	8	yes	—
86	C₃⋊C₈	C₂×C₄²	1	24	yes	—
87	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
88	C₂²×C₄	—	2	24	no	—
89	C₂²×C₄	C₄×Dic₃	1	24	yes	—
90	C₂²×C₄	C₂×C₄×D₃	1	6	yes	—
91	C₂²×C₄	C₂×C₄×D₃	1	6	yes	—
92	C₂²×C₄	C₂²×Dic₃	1	12	yes	—
93	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
94	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
95	C₂×C₈	—	3	16	no	—
96	C₂×C₈	—	3	48	no	—
97	C₂×C₈	—	3	12	no	—
98	C₂×C₈	—	3	12	no	—
99	C₂×C₈	—	3	24	no	—
100	C₄²	C₄×Dic₃	1	24	yes	—
101	C₄²	C₂×C₄×D₃	1	12	yes	—
102	C₄²	C₂×C₄×D₃	1	12	yes	—
103	C₂²×C₃	C₄³	1	4	yes	—
104	C₂²×C₃	#405	1	24	yes	—
105	C₂²×C₃	C₂²×C₄²	1	3	yes	—
106	C₂²×C₃	C₂×#131	1	4	yes	—
107	C₃×C₄	—	2	24	no	—
108	C₃×C₄	C₄³	1	4	yes	—
109	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
110	C₃×C₄	C₂²×C₄²	1	3	yes	—
111	C₂³	C₂×C₄×Dic₃	1	6	yes	—
112	C₂³	#948	1	8	yes	—
113	C₂³	C₂³×Dic₃	1	1	yes	characteristic
114	C₂×C₄	—	2	48	no	—
115	C₂×C₄	—	2	24	no	—
116	C₂×C₄	C₄²×D₃	1	4	yes	—
117	C₂×C₄	C₄²×D₃	1	12	yes	—
118	C₂×C₄	C₂×C₄×Dic₃	1	24	yes	—
119	C₂×C₄	C₂×C₄×Dic₃	1	6	yes	—
120	C₂×C₄	C₂²×C₄×D₃	1	3	yes	—
121	C₂×C₄	C₂²×C₄×D₃	1	3	yes	—
122	C₈	—	3	8	no	—
123	C₈	—	3	24	no	—
124	C₂×C₃	#1726	1	8	yes	complemented
125	C₂×C₃	C₂×C₄³	1	1	yes	characteristic
126	C₂×C₃	C₂×#405	1	6	yes	—
127	C₂²	C₄²×Dic₃	1	4	yes	—
128	C₂²	#4997	1	24	yes	—
129	C₂²	C₂²×C₄×Dic₃	1	3	yes	—
130	C₂²	C₂×#948	1	4	yes	—
131	C₄	—	2	24	no	—
132	C₄	C₄²×Dic₃	1	4	yes	—
133	C₄	C₂×C₄²×D₃	1	1	yes	characteristic
134	C₄	C₂×C₄²×D₃	1	3	yes	—
135	C₃	C₂×#1726	1	1	yes	characteristic, complemented, Sylow
136	C₂	#69434	1	8	yes	direct product factor
137	C₂	C₂×C₄²×Dic₃	1	1	yes	characteristic
138	C₂	C₂×#4997	1	6	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₂×C₃×#376	C₂
socle, prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂×#376	D₃
hypocenter	C₃	C₂×#1726
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#376

Complements

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

N	H ≅ G/N	type
4. #69434	C₂	direct (N × H)
12. C₂×#803	C₄	semidirect
14. C₂×C₄×C₃⋊C₈	C₄	semidirect
32. #803	C₂×C₄	semidirect
33. C₄×C₃⋊C₈	C₂×C₄	semidirect
124. C₂×C₃	#1726	semidirect
135. C₃	C₂×#1726	semidirect
136. C₂	#69434	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×#376	2
C₂²×#803	2
C₂²×C₄×C₃⋊C₈	2
#69434	2
C₂×#1726	3

Subgroups — C2×#69434 #10792514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69434 #10792514