Subgroups — C₂×#7137 #126549

Order 512.

622 conjugacy classes of subgroups.

The subgroup lattice has 141 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#346	C₂	1	2	yes	maximal
2	C₂²×#322	C₂	1	1	yes	maximal, characteristic
3	#7137	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₁₆	C₄	1	4	yes	—
5	C₂²×C₄×C₈	C₄	1	2	yes	—
6	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×#346	—	2	4	no	—
8	C₂×#346	—	2	4	no	—
9	C₂×#346	C₄	1	4	yes	—
10	C₂×#346	C₂²	1	4	yes	—
11	C₂×#322	—	2	4	no	—
12	C₂×#322	C₄	1	2	yes	—
13	C₂×#322	C₂²	1	2	yes	—
14	C₂³×C₈	—	2	2	no	—
15	C₂³×C₈	C₂×C₄	1	2	yes	—
16	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	—	2	8	no	—
18	C₂²×C₁₆	—	2	8	no	—
19	C₂²×C₁₆	C₈	1	16	yes	complemented
20	C₂²×C₁₆	C₂×C₄	1	8	yes	—
21	C₂×C₄×C₈	—	2	4	no	—
22	C₂×C₄×C₈	—	2	2	no	—
23	C₂×C₄×C₈	C₈	1	8	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	4	yes	—
28	C₂×C₄×C₈	D₄	1	2	yes	—
29	C₂×C₄×C₈	Q₈	1	2	yes	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	#346	—	2	8	no	—
32	#346	—	2	4	no	—
33	#346	—	2	4	no	—
34	#322	—	2	4	no	—
35	#322	—	2	4	no	—
36	C₂³×C₄	C₄²	1	1	yes	characteristic
37	C₂³×C₄	M₁₆	1	2	yes	—
38	C₂²×C₈	—	2	8	no	—
39	C₂²×C₈	—	2	4	no	—
40	C₂²×C₈	—	4	8	no	—
41	C₂²×C₈	C₄²	1	2	yes	—
42	C₂²×C₈	C₂²⋊C₄	1	4	yes	—
43	C₂²×C₈	C₄⋊C₄	1	4	yes	—
44	C₂²×C₈	C₂×C₈	1	8	yes	—
45	C₂²×C₈	M₁₆	1	8	yes	—
46	C₂²×C₈	C₂²×C₄	1	2	yes	—
47	C₂×C₄²	C₄²	1	1	yes	characteristic
48	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄²	C₄⋊C₄	1	2	yes	—
50	C₂×C₄²	C₂×C₈	1	4	yes	—
51	C₂×C₄²	M₁₆	1	2	yes	—
52	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₁₆	—	2	32	no	—
54	C₂×C₁₆	—	2	8	no	—
55	C₂×C₁₆	—	2	8	no	—
56	C₂×C₁₆	C₂×C₈	1	16	yes	complemented
57	C₄×C₈	—	2	8	no	—
58	C₄×C₈	—	2	4	no	—
59	C₄×C₈	C₂²⋊C₄	1	4	yes	—
60	C₄×C₈	C₂²⋊C₄	1	2	yes	—
61	C₄×C₈	C₄⋊C₄	1	4	yes	—
62	C₄×C₈	C₄⋊C₄	1	2	yes	—
63	C₄×C₈	C₂×C₈	1	8	yes	—
64	C₄×C₈	C₂×D₄	1	2	yes	—
65	C₄×C₈	C₂×D₄	1	1	yes	characteristic
66	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂⁴	#97	1	1	yes	characteristic
68	C₂²×C₄	—	2	8	no	—
69	C₂²×C₄	#95	1	2	yes	—
70	C₂²×C₄	C₄×C₈	1	2	yes	—
71	C₂²×C₄	C₄×C₈	1	4	yes	—
72	C₂²×C₄	#97	1	2	yes	—
73	C₂²×C₄	#97	1	1	yes	characteristic
74	C₂²×C₄	#97	1	4	yes	—
75	C₂²×C₄	C₂²⋊C₈	1	4	yes	—
76	C₂²×C₄	C₄⋊C₈	1	4	yes	—
77	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
78	C₂²×C₄	C₂×M₁₆	1	2	yes	—
79	C₂×C₈	—	2	8	no	—
80	C₂×C₈	—	4	8	no	—
81	C₂×C₈	—	4	16	no	—
82	C₂×C₈	#95	1	4	yes	—
83	C₂×C₈	C₄×C₈	1	4	yes	—
84	C₂×C₈	#97	1	4	yes	—
85	C₂×C₈	C₂²⋊C₈	1	16	yes	—
86	C₂×C₈	C₄⋊C₈	1	16	yes	—
87	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₈	C₂²×C₈	1	4	yes	—
90	C₂×C₈	C₂×M₁₆	1	4	yes	—
91	C₄²	#95	1	2	yes	—
92	C₄²	C₄×C₈	1	2	yes	—
93	C₄²	C₂²⋊C₈	1	4	yes	—
94	C₄²	C₄⋊C₈	1	4	yes	—
95	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₄²	C₂²×C₈	1	2	yes	—
98	C₁₆	—	2	16	no	—
99	C₂³	#322	1	4	yes	—
100	C₂³	#336	1	2	yes	—
101	C₂³	#347	1	8	yes	—
102	C₂³	C₂×#97	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	8	no	—
106	C₂×C₄	—	2	8	no	—
107	C₂×C₄	C₈²	1	4	yes	—
108	C₂×C₄	#322	1	4	yes	—
109	C₂×C₄	#322	1	4	yes	—
110	C₂×C₄	#336	1	4	yes	—
111	C₂×C₄	#336	1	8	yes	—
112	C₂×C₄	#336	1	2	yes	—
113	C₂×C₄	C₂×#95	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
116	C₂×C₄	C₂×#97	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊C₈	1	2	yes	—
118	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
119	C₈	—	4	8	no	—
120	C₈	#336	1	8	yes	—
121	C₈	C₂×C₂²⋊C₈	1	4	yes	—
122	C₈	C₂×C₄⋊C₈	1	4	yes	—
123	C₂²	#1273	1	4	yes	—
124	C₂²	#1313	1	16	yes	—
125	C₂²	#1375	1	8	yes	—
126	C₂²	C₂×#322	1	2	yes	—
127	C₂²	C₂×#336	1	1	yes	characteristic
128	C₂²	C₂×#347	1	4	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	#1273	1	4	yes	—
133	C₄	C₂×C₈²	1	1	yes	characteristic
134	C₄	C₂×#322	1	1	yes	characteristic
135	C₄	C₂×#336	1	2	yes	—
136	C₂	#7137	1	8	yes	direct product factor
137	C₂	C₂×#1273	1	1	yes	characteristic
138	C₂	C₂×#1313	1	4	yes	—
139	C₂	C₂×#1375	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₄²
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂⁴	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #7137	C₂	direct (N × H)
19. C₂²×C₁₆	C₈	semidirect
56. C₂×C₁₆	C₂×C₈	semidirect
136. C₂	#7137	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×#346	2
C₂²×#322	2
#7137	2

Subgroups — C2×#7137 #126549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7137 #126549