Subgroups — C₂×#33414 #7622540

Order 512.

1856 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#504	C₂	1	1	yes	maximal, characteristic
2	C₂×#2916	C₂	1	2	yes	maximal
3	#33414	C₂	1	16	yes	maximal, direct product factor
4	C₂×#2161	C₂	1	6	yes	maximal, complemented
5	C₂²×#368	C₂	1	6	yes	maximal, complemented
6	C₂×#568	—	2	2	no	—
7	C₂²×#131	C₂²	1	1	yes	characteristic
8	C₂×#504	C₄	1	8	yes	—
9	C₂×#504	C₄	1	8	yes	—
10	C₂×#504	C₂²	1	8	yes	—
11	C₂²×M₃₂	C₂²	1	3	yes	—
12	C₂³×C₁₆	C₂²	1	3	yes	—
13	#2916	—	2	8	no	—
14	#2916	C₂²	1	16	yes	—
15	C₂×#430	—	2	6	no	—
16	C₂²×#108	C₂²	1	12	yes	—
17	#2161	C₂²	1	48	yes	complemented
18	C₂×#368	C₂²	1	4	yes	—
19	C₂×#368	C₂²	1	12	yes	—
20	C₂×#368	C₂²	1	48	yes	complemented
21	C₂²×#41	D₄	1	1	yes	characteristic
22	#568	—	2	16	no	—
23	#568	—	4	8	no	—
24	C₂×#131	—	4	8	no	—
25	C₂×#131	C₂×C₄	1	4	yes	—
26	C₂×#131	C₂×C₄	1	4	yes	—
27	C₂×#131	D₄	1	4	yes	—
28	C₂×#131	Q₈	1	4	yes	—
29	C₂×#131	Q₈	1	4	yes	—
30	C₂×#131	C₂³	1	4	yes	—
31	C₂²×M₁₆	—	2	12	no	—
32	C₂²×M₁₆	D₄	1	3	yes	—
33	C₂³×C₈	C₂³	1	3	yes	—
34	#504	C₂×C₄	1	64	yes	—
35	C₂×M₃₂	C₂×C₄	1	12	yes	—
36	C₂×M₃₂	C₂×C₄	1	12	yes	—
37	C₂×M₃₂	C₂³	1	12	yes	—
38	C₂²×C₁₆	C₂×C₄	1	12	yes	—
39	C₂²×C₁₆	C₂×C₄	1	12	yes	—
40	C₂²×C₁₆	C₂³	1	1	yes	characteristic
41	C₂²×C₁₆	C₂³	1	3	yes	—
42	C₂²×C₁₆	C₂³	1	12	yes	—
43	#430	—	2	48	no	—
44	C₂×#108	—	2	12	no	—
45	C₂×#108	—	2	24	no	—
46	C₂×#108	C₂³	1	8	yes	—
47	C₂×#108	C₂³	1	48	yes	—
48	#368	C₂³	1	16	yes	—
49	#368	C₂³	1	48	yes	—
50	C₂×#41	C₄⋊C₄	1	4	yes	—
51	C₂×#41	C₄⋊C₄	1	4	yes	—
52	C₂×#41	D₈	1	4	yes	—
53	C₂×#41	D₈	1	4	yes	—
54	C₂×#41	Q₁₆	1	4	yes	—
55	C₂×#41	C₂×D₄	1	4	yes	—
56	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
57	C₂²×D₄	Q₁₆	1	3	yes	—
58	C₂³×C₄	C₂×D₄	1	3	yes	—
59	#131	—	4	64	no	—
60	#131	C₄⋊C₄	1	16	yes	—
61	#131	C₄⋊C₄	1	16	yes	—
62	#131	C₂²×C₄	1	16	yes	—
63	#131	C₂×Q₈	1	16	yes	—
64	C₂×M₁₆	—	2	8	no	—
65	C₂×M₁₆	—	2	48	no	—
66	C₂×M₁₆	—	4	12	no	—
67	C₂×M₁₆	—	4	24	no	—
68	C₂×M₁₆	C₄⋊C₄	1	12	yes	—
69	C₂×M₁₆	C₄⋊C₄	1	12	yes	—
70	C₂×M₁₆	C₂×D₄	1	12	yes	—
71	C₂²×C₈	—	4	12	no	—
72	C₂²×C₈	C₂²×C₄	1	6	yes	—
73	C₂²×C₈	C₂²×C₄	1	6	yes	—
74	C₂²×C₈	C₂×D₄	1	3	yes	—
75	C₂²×C₈	C₂×D₄	1	6	yes	—
76	C₂²×C₈	C₂×Q₈	1	6	yes	—
77	C₂²×C₈	C₂×Q₈	1	6	yes	—
78	C₂²×C₈	C₂⁴	1	1	yes	characteristic
79	C₂²×C₈	C₂⁴	1	6	yes	—
80	M₃₂	C₂²×C₄	1	48	yes	—
81	C₂×C₁₆	C₂²×C₄	1	2	yes	—
82	C₂×C₁₆	C₂²×C₄	1	2	yes	—
83	C₂×C₁₆	C₂²×C₄	1	6	yes	—
84	C₂×C₁₆	C₂²×C₄	1	6	yes	—
85	C₂×C₁₆	C₂²×C₄	1	48	yes	—
86	C₂×C₁₆	C₂⁴	1	2	yes	—
87	C₂×C₁₆	C₂⁴	1	6	yes	—
88	#108	—	2	8	no	—
89	#108	—	2	48	no	—
90	#108	C₂⁴	1	16	yes	—
91	C₂⁴	C₂×Q₁₆	1	3	yes	—
92	#41	#107	1	16	yes	—
93	#41	#107	1	16	yes	—
94	#41	C₂×C₄⋊C₄	1	16	yes	—
95	#41	C₂×D₈	1	16	yes	—
96	C₂×Q₈	#107	1	4	yes	—
97	C₂×Q₈	#107	1	4	yes	—
98	C₂×Q₈	C₂×Q₁₆	1	4	yes	—
99	C₂×D₄	#107	1	12	yes	—
100	C₂×D₄	#107	1	12	yes	—
101	C₂×D₄	C₂×Q₁₆	1	12	yes	—
102	C₂²×C₄	C₂×C₄⋊C₄	1	6	yes	—
103	C₂²×C₄	C₂×C₄⋊C₄	1	6	yes	—
104	C₂²×C₄	C₂×D₈	1	6	yes	—
105	C₂²×C₄	C₂×D₈	1	6	yes	—
106	C₂²×C₄	C₂×Q₁₆	1	3	yes	—
107	C₂²×C₄	C₂×Q₁₆	1	6	yes	—
108	C₂²×C₄	C₂²×D₄	1	6	yes	—
109	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
110	M₁₆	—	2	16	no	—
111	M₁₆	—	4	8	no	—
112	M₁₆	—	4	48	no	—
113	M₁₆	C₂×C₄⋊C₄	1	48	yes	—
114	C₂×C₈	—	4	48	no	—
115	C₂×C₈	—	4	8	no	—
116	C₂×C₈	C₂×C₄⋊C₄	1	6	yes	—
117	C₂×C₈	C₂×C₄⋊C₄	1	6	yes	—
118	C₂×C₈	C₂×C₄⋊C₄	1	12	yes	—
119	C₂×C₈	C₂×C₄⋊C₄	1	12	yes	—
120	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₈	C₂³×C₄	1	12	yes	—
123	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₈	C₂²×D₄	1	6	yes	—
125	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
126	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₈	C₂²×Q₈	1	12	yes	—
128	C₂×C₈	C₂⁵	1	1	yes	characteristic
129	C₁₆	C₂³×C₄	1	4	yes	—
130	C₁₆	C₂³×C₄	1	12	yes	—
131	C₂³	—	4	6	no	—
132	C₂³	C₂×#107	1	6	yes	—
133	C₂³	C₂×#107	1	6	yes	—
134	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
135	C₂³	C₂²×Q₁₆	1	6	yes	—
136	Q₈	C₂×#107	1	16	yes	—
137	D₄	C₂×#107	1	48	yes	—
138	C₂×C₄	C₂×#107	1	6	yes	—
139	C₂×C₄	C₂×#107	1	6	yes	—
140	C₂×C₄	C₂×#107	1	12	yes	—
141	C₂×C₄	C₂×#107	1	12	yes	—
142	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₂²×C₄⋊C₄	1	12	yes	—
145	C₂×C₄	C₂²×D₈	1	12	yes	—
146	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
147	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
148	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
149	C₂×C₄	C₂²×Q₁₆	1	6	yes	—
150	C₂×C₄	C₂³×D₄	1	1	yes	characteristic
151	C₈	—	4	16	no	—
152	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
153	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₈	C₂²×C₄⋊C₄	1	12	yes	—
155	C₈	C₂⁴×C₄	1	1	yes	characteristic
156	C₈	C₂³×Q₈	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	—	2	12	no	—
159	C₂²	—	4	6	no	—
160	C₂²	—	4	6	no	—
161	C₂²	C₂²×#107	1	1	yes	characteristic
162	C₂²	C₂²×#107	1	12	yes	—
163	C₂²	C₂²×#107	1	1	yes	characteristic
164	C₂²	C₂³×Q₁₆	1	1	yes	characteristic
165	C₄	C₂²×#107	1	1	yes	characteristic
166	C₄	C₂²×#107	1	12	yes	—
167	C₄	C₂²×#107	1	1	yes	characteristic
168	C₄	C₂³×C₄⋊C₄	1	1	yes	characteristic
169	C₄	C₂³×D₈	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	—	2	12	no	—
173	C₂	#33414	1	2	yes	direct product factor
174	C₂	C₂³×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂⁴×C₄
Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂²×#107
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #33414	C₂	direct (N × H)
4. C₂×#2161	C₂	semidirect
5. C₂²×#368	C₂	semidirect
17. #2161	C₂²	semidirect
20. C₂×#368	C₂²	semidirect
173. C₂	#33414	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×#504	2
C₂×#2916	2
#33414	2
C₂×#2161	2
C₂²×#368	2

Subgroups — C2×#33414 #7622540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#33414 #7622540