Subgroups — C₂×#69394 #10795566

Order 768.

1470 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	6	yes	maximal
3	#69394	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1842	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄²	C₄	1	3	yes	—
6	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄²	C₂²	1	3	yes	—
8	C₂³×C₃⋊C₈	C₂²	1	4	yes	—
9	C₃×C₄³	C₄	1	4	yes	—
10	C₃×C₄³	C₂²	1	4	yes	—
11	C₂×C₄⋊(C₃⋊C₈)	—	2	24	no	—
12	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	24	yes	—
13	C₂²×C₄⋊C₈	—	3	6	no	—
14	C₂×C₄³	D₃	1	1	yes	characteristic
15	#1842	—	3	8	no	—
16	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
17	C₂³×C₃×C₄	C₂×C₄	1	3	yes	—
18	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
19	C₂×C₃×C₄²	—	2	12	no	—
20	C₂×C₃×C₄²	C₈	1	8	yes	complemented
21	C₂×C₃×C₄²	C₂×C₄	1	12	yes	—
22	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
23	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₄²	D₄	1	12	yes	—
25	C₂×C₃×C₄²	Q₈	1	12	yes	—
26	C₂×C₃×C₄²	C₂³	1	6	yes	—
27	C₂²×C₃⋊C₈	—	2	24	no	—
28	C₂²×C₃⋊C₈	C₂³	1	8	yes	—
29	C₄⋊(C₃⋊C₈)	—	2	48	no	—
30	C₂³×C₈	—	3	4	no	—
31	C₂²×C₄²	Dic₃	1	3	yes	—
32	C₂²×C₄²	Dic₃	1	1	yes	characteristic
33	C₂²×C₄²	C₂×D₃	1	3	yes	—
34	C₂×C₄⋊C₈	—	3	24	no	—
35	C₂×C₄⋊C₈	—	6	24	no	—
36	C₄³	Dic₃	1	4	yes	—
37	C₄³	C₂×D₃	1	4	yes	—
38	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₄	—	2	12	no	—
40	C₂²×C₃×C₄	C₄⋊C₄	1	12	yes	—
41	C₂²×C₃×C₄	C₄⋊C₄	1	12	yes	—
42	C₂²×C₃×C₄	C₂×C₈	1	12	yes	—
43	C₂²×C₃×C₄	M₁₆	1	12	yes	—
44	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
45	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₃×C₄	C₂²×C₄	1	6	yes	—
47	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
48	C₂²×C₃×C₄	C₂×Q₈	1	6	yes	—
49	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
50	C₃×C₄²	—	2	24	no	—
51	C₃×C₄²	C₄⋊C₄	1	24	yes	—
52	C₃×C₄²	C₂×C₈	1	16	yes	complemented
53	C₃×C₄²	C₂×D₄	1	12	yes	—
54	C₃×C₄²	C₂×Q₈	1	12	yes	—
55	C₂×C₃⋊C₈	—	2	24	no	—
56	C₂×C₃⋊C₈	—	4	16	no	—
57	C₂³×C₄	C₂×Dic₃	1	3	yes	—
58	C₂³×C₄	C₂×Dic₃	1	3	yes	—
59	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
60	C₂²×C₈	—	3	8	no	—
61	C₂²×C₈	—	6	24	no	—
62	C₂×C₄²	—	2	12	no	—
63	C₂×C₄²	C₃⋊C₈	1	8	yes	complemented
64	C₂×C₄²	Dic₆	1	12	yes	—
65	C₂×C₄²	D₁₂	1	12	yes	—
66	C₂×C₄²	C₂×Dic₃	1	12	yes	—
67	C₂×C₄²	C₂×Dic₃	1	4	yes	—
68	C₂×C₄²	C₂×Dic₃	1	6	yes	—
69	C₂×C₄²	C₂²×D₃	1	6	yes	—
70	C₄⋊C₈	—	6	48	no	—
71	C₂³×C₃	C₂×C₄⋊C₄	1	6	yes	—
72	C₂³×C₃	C₂²×C₈	1	2	yes	—
73	C₂³×C₃	C₂×M₁₆	1	6	yes	—
74	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	12	no	—
76	C₂×C₃×C₄	—	2	24	no	—
77	C₂×C₃×C₄	C₄⋊C₈	1	48	yes	complemented
78	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
79	C₂×C₃×C₄	C₂×C₄⋊C₄	1	6	yes	—
80	C₂×C₃×C₄	C₂×C₄⋊C₄	1	12	yes	—
81	C₂×C₃×C₄	#127	1	4	yes	—
82	C₂×C₃×C₄	C₄⋊Q₈	1	12	yes	—
83	C₂×C₃×C₄	C₂²×C₈	1	12	yes	—
84	C₂×C₃×C₄	C₂×M₁₆	1	12	yes	—
85	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
86	C₂×C₃×C₄	C₂²×Q₈	1	3	yes	—
87	C₃⋊C₈	—	4	8	no	—
88	C₂⁴	C₂²×Dic₃	1	1	yes	characteristic
89	C₂²×C₄	—	2	12	no	—
90	C₂²×C₄	C₂×C₃⋊C₈	1	12	yes	—
91	C₂²×C₄	#208	1	12	yes	—
92	C₂²×C₄	C₄⋊Dic₃	1	12	yes	—
93	C₂²×C₄	C₄⋊Dic₃	1	12	yes	—
94	C₂²×C₄	C₂×Dic₆	1	6	yes	—
95	C₂²×C₄	C₂×D₁₂	1	6	yes	—
96	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
97	C₂²×C₄	C₂²×Dic₃	1	6	yes	—
98	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
99	C₂²×C₄	C₂³×D₃	1	1	yes	characteristic
100	C₂×C₈	—	6	24	no	—
101	C₂×C₈	—	12	16	no	—
102	C₄²	—	2	24	no	—
103	C₄²	C₂×C₃⋊C₈	1	16	yes	complemented
104	C₄²	C₄⋊Dic₃	1	24	yes	—
105	C₄²	C₂×Dic₆	1	12	yes	—
106	C₄²	C₂×D₁₂	1	12	yes	—
107	C₂²×C₃	#384	1	4	yes	—
108	C₂²×C₃	C₂×C₄⋊C₈	1	12	yes	—
109	C₂²×C₃	#423	1	12	yes	—
110	C₂²×C₃	C₂²×C₄⋊C₄	1	3	yes	—
111	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
112	C₂²×C₃	C₂²×M₁₆	1	3	yes	—
113	C₃×C₄	—	2	12	no	—
114	C₃×C₄	#384	1	4	yes	—
115	C₃×C₄	C₂×C₄⋊C₈	1	24	yes	complemented
116	C₃×C₄	C₂×#127	1	1	yes	characteristic
117	C₃×C₄	C₂×C₄⋊Q₈	1	3	yes	—
118	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
119	C₂³	C₂×#208	1	6	yes	—
120	C₂³	C₂×C₄⋊Dic₃	1	6	yes	—
121	C₂³	C₂³×Dic₃	1	1	yes	characteristic
122	C₂×C₄	—	2	24	no	—
123	C₂×C₄	—	2	12	no	—
124	C₂×C₄	C₄⋊(C₃⋊C₈)	1	48	yes	complemented
125	C₂×C₄	#869	1	12	yes	—
126	C₂×C₄	#874	1	4	yes	—
127	C₂×C₄	C₂²×C₃⋊C₈	1	12	yes	—
128	C₂×C₄	C₂×#208	1	12	yes	—
129	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
130	C₂×C₄	C₂×C₄⋊Dic₃	1	12	yes	—
131	C₂×C₄	C₂×C₄⋊Dic₃	1	6	yes	—
132	C₂×C₄	C₂²×Dic₆	1	3	yes	—
133	C₂×C₄	C₂²×D₁₂	1	3	yes	—
134	C₈	—	12	8	no	—
135	C₂×C₃	#1842	1	8	yes	complemented
136	C₂×C₃	C₂×#384	1	1	yes	characteristic
137	C₂×C₃	C₂²×C₄⋊C₈	1	3	yes	—
138	C₂×C₃	C₂×#423	1	3	yes	—
139	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	12	yes	—
140	C₂²	#4924	1	12	yes	—
141	C₂²	#4935	1	4	yes	—
142	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
143	C₂²	C₂²×#208	1	3	yes	—
144	C₂²	C₂²×C₄⋊Dic₃	1	3	yes	—
145	C₄	—	2	12	no	—
146	C₄	C₂×C₄⋊(C₃⋊C₈)	1	24	yes	complemented
147	C₄	#4935	1	4	yes	—
148	C₄	C₂×#869	1	3	yes	—
149	C₄	C₂×#874	1	1	yes	characteristic
150	C₃	C₂×#1842	1	1	yes	characteristic, complemented, Sylow
151	C₂	#69394	1	8	yes	direct product factor
152	C₂	C₂²×C₄⋊(C₃⋊C₈)	1	3	yes	—
153	C₂	C₂×#4924	1	3	yes	—
154	C₂	C₂×#4935	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄³	C₂
socle, prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂⁴
hypercenter	C₂×C₄³	D₃
hypocenter	C₃	C₂×#1842
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. #69394	C₂	direct (N × H)
20. C₂×C₃×C₄²	C₈	semidirect
52. C₃×C₄²	C₂×C₈	semidirect
63. C₂×C₄²	C₃⋊C₈	semidirect
77. C₂×C₃×C₄	C₄⋊C₈	semidirect
103. C₄²	C₂×C₃⋊C₈	semidirect
115. C₃×C₄	C₂×C₄⋊C₈	semidirect
124. C₂×C₄	C₄⋊(C₃⋊C₈)	semidirect
135. C₂×C₃	#1842	semidirect
146. C₄	C₂×C₄⋊(C₃⋊C₈)	semidirect
150. C₃	C₂×#1842	semidirect
151. C₂	#69394	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×C₄³	2
C₂²×C₄⋊(C₃⋊C₈)	2
#69394	2
C₂×#1842	3

Subgroups — C2×#69394 #10795566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69394 #10795566