Subgroups — C₂×#12026 #483410

Order 512.

408 conjugacy classes of subgroups.

The subgroup lattice has 203 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2166	C₂	1	2	yes	maximal
2	C₂×#2108	C₂	1	1	yes	maximal, characteristic
3	#12026	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1373	C₂	1	2	yes	maximal, complemented
5	C₂×#1369	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂	C₄	1	1	yes	characteristic
7	C₂³×C₁₆	C₄	1	1	yes	characteristic
8	C₂×#405	C₂²	1	1	yes	characteristic
9	#2166	—	2	4	no	—
10	#2166	C₂²	1	8	yes	—
11	C₂×#364	C₂²	1	4	yes	—
12	C₂×#349	—	2	2	no	—
13	C₂×#347	—	2	2	no	—
14	#2108	C₄	1	4	yes	—
15	#2108	C₂²	1	4	yes	—
16	C₂×#346	C₄	1	1	yes	characteristic
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	#1373	C₂²	1	8	yes	complemented
21	#1369	C₂²	1	8	yes	complemented
22	C₂²×M₁₆	D₄	1	1	yes	characteristic
23	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×#117	Q₈	1	1	yes	characteristic
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	C₂×M₃₂	—	2	4	no	—
28	C₂×M₃₂	C₂×C₄	1	4	yes	—
29	C₂²×C₁₆	—	2	2	no	—
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	C₂×C₄	1	4	yes	—
34	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
36	#405	C₂×C₄	1	2	yes	—
37	#405	D₄	1	2	yes	—
38	#405	Q₈	1	2	yes	—
39	#405	C₂³	1	2	yes	—
40	C₂×#97	D₄	1	1	yes	characteristic
41	C₂×#97	Q₈	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#364	—	2	4	no	—
45	#364	C₂³	1	8	yes	—
46	#349	—	2	8	no	—
47	#347	—	2	8	no	—
48	#346	C₂×C₄	1	4	yes	—
49	#346	C₂×C₄	1	2	yes	—
50	#346	C₂³	1	2	yes	—
51	C₄×C₁₆	C₂×C₄	1	4	yes	—
52	C₄×C₁₆	C₂×C₄	1	2	yes	—
53	C₄×C₁₆	C₂³	1	2	yes	—
54	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
55	C₂×M₁₆	—	2	2	no	—
56	C₂×M₁₆	—	2	2	no	—
57	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
58	C₂×M₁₆	C₂×D₄	1	2	yes	—
59	C₂²×C₈	—	2	1	no	—
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₈	C₄⋊C₄	1	2	yes	—
65	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₈	C₂²×C₄	1	2	yes	—
67	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	#117	—	2	2	no	—
70	#117	C₄⋊C₄	1	2	yes	—
71	#117	C₂×Q₈	1	2	yes	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₂²⋊C₄	—	2	1	no	—
74	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
76	M₃₂	—	2	8	no	—
77	M₃₂	—	2	8	no	—
78	M₃₂	—	4	4	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	—	2	1	no	—
83	C₂×C₁₆	—	2	8	no	—
84	C₂×C₁₆	—	4	4	no	—
85	C₂×C₁₆	M₁₆	1	2	yes	—
86	C₂×C₁₆	M₁₆	1	2	yes	—
87	C₂×C₁₆	M₁₆	1	2	yes	—
88	C₂×C₁₆	M₁₆	1	2	yes	—
89	C₂×C₁₆	C₂²×C₄	1	2	yes	—
90	C₂×C₁₆	C₂²×C₄	1	2	yes	—
91	C₂×C₁₆	C₂²×C₄	1	2	yes	—
92	C₂×C₁₆	C₂²×C₄	1	2	yes	—
93	#97	C₄⋊C₄	1	2	yes	—
94	#97	C₄⋊C₄	1	2	yes	—
95	#97	C₂×D₄	1	2	yes	—
96	#97	C₂×Q₈	1	2	yes	—
97	C₄×C₈	C₄⋊C₄	1	2	yes	—
98	C₄×C₈	C₂²×C₄	1	2	yes	—
99	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
102	C₄×C₈	C₂⁴	1	1	yes	characteristic
103	C₂⁴	#108	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	1	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	#108	1	1	yes	characteristic
111	C₂²×C₄	#108	1	2	yes	—
112	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
113	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
114	M₁₆	—	2	4	no	—
115	M₁₆	—	2	4	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₈	#135	1	2	yes	—
133	C₂×C₈	#135	1	2	yes	—
134	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₂²⋊C₄	—	2	2	no	—
139	C₂²⋊C₄	—	2	2	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂×C₄⋊C₄	1	2	yes	—
143	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₄²	C₂²×Q₈	1	1	yes	characteristic
145	C₁₆	—	2	2	no	—
146	C₁₆	—	2	2	no	—
147	C₁₆	—	4	8	no	—
148	C₁₆	C₂×M₁₆	1	4	yes	—
149	C₁₆	C₂×M₁₆	1	4	yes	—
150	C₂³	—	2	1	no	—
151	C₂³	—	2	1	no	—
152	C₂³	—	4	2	no	—
153	C₂³	C₂×#108	1	1	yes	characteristic
154	C₂³	C₂×#108	1	2	yes	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	#423	1	1	yes	characteristic
167	C₂×C₄	#423	1	1	yes	characteristic
168	C₂×C₄	#427	1	1	yes	characteristic
169	C₂×C₄	#427	1	1	yes	characteristic
170	C₂×C₄	C₂×#108	1	2	yes	—
171	C₂×C₄	C₂×#108	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
173	C₈	—	2	2	no	—
174	C₈	—	2	2	no	—
175	C₈	#423	1	1	yes	characteristic
176	C₈	#423	1	1	yes	characteristic
177	C₈	#427	1	2	yes	—
178	C₈	C₂²×M₁₆	1	1	yes	characteristic
179	C₈	C₂²×M₁₆	1	1	yes	characteristic
180	C₈	C₂×#135	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	#1568	1	1	yes	characteristic
187	C₂²	#1568	1	1	yes	characteristic
188	C₂²	C₂²×#108	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	4	2	no	—
191	C₄	—	4	1	no	—
192	C₄	—	4	1	no	—
193	C₄	#1568	1	1	yes	characteristic
194	C₄	#1568	1	1	yes	characteristic
195	C₄	C₂×#423	1	1	yes	characteristic
196	C₄	C₂×#427	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	—	4	2	no	—
200	C₂	#12026	1	2	yes	direct product factor
201	C₂	C₂×#1568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	#1568
prime radical	C₂⁴	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12026	C₂	direct (N × H)
4. C₂×#1373	C₂	semidirect
5. C₂×#1369	C₂	semidirect
20. #1373	C₂²	semidirect
21. #1369	C₂²	semidirect
200. C₂	#12026	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×#2166	2
C₂×#2108	2
#12026	2
C₂×#1373	2
C₂×#1369	2

Subgroups — C2×#12026 #483410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12026 #483410