Subgroups — C₂×#12000 #483321

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1564	C₂	1	1	yes	maximal, characteristic
3	C₂²×#334	C₂	1	1	yes	maximal, characteristic
4	#12000	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1368	C₂	1	4	yes	maximal, complemented
6	C₂²×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×M₃₂	C₄	1	1	yes	characteristic
8	C₂×#431	—	2	1	no	—
9	C₂²×C₄⋊C₈	—	2	1	no	—
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₄×M₃₂	C₄	1	4	yes	—
12	C₄×M₃₂	C₂²	1	4	yes	—
13	C₂×#346	C₂²	1	2	yes	—
14	C₂×C₄×C₁₆	C₄	1	2	yes	—
15	#1564	—	2	2	no	—
16	#1564	C₂²	1	4	yes	—
17	C₂×#335	—	2	1	no	—
18	C₂×#334	—	2	1	no	—
19	C₂×#334	—	2	2	no	—
20	C₂×#334	C₂²	1	2	yes	—
21	C₂×#334	C₂²	1	4	yes	—
22	C₂×#334	C₂²	1	2	yes	—
23	#1368	C₂²	1	16	yes	complemented
24	C₂³×C₈	—	4	1	no	—
25	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₈	Q₈	1	1	yes	characteristic
27	C₂²×C₄²	D₄	1	1	yes	characteristic
28	C₂×M₃₂	—	2	4	no	—
29	C₂×M₃₂	C₈	1	8	yes	complemented
30	C₂×M₃₂	C₂×C₄	1	4	yes	—
31	C₂×M₃₂	C₂×C₄	1	4	yes	—
32	C₂²×C₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₁₆	C₂×C₄	1	2	yes	—
34	#431	—	2	4	no	—
35	#431	—	4	2	no	—
36	C₂×C₄⋊C₈	—	2	2	no	—
37	C₂×C₄⋊C₈	—	2	4	no	—
38	C₂×C₄⋊C₈	—	2	2	no	—
39	C₂×C₄⋊C₈	—	4	1	no	—
40	C₂×C₄⋊C₈	—	4	2	no	—
41	C₂×C₂²⋊C₈	—	4	1	no	—
42	C₂×C₄×C₈	—	4	1	no	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	D₄	1	2	yes	—
47	C₂×C₄×C₈	Q₈	1	2	yes	—
48	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	2	yes	—
51	#346	C₂×C₄	1	4	yes	—
52	#346	C₂³	1	4	yes	—
53	C₄×C₁₆	C₂×C₄	1	8	yes	—
54	#335	—	2	4	no	—
55	#334	—	2	2	no	—
56	#334	—	2	4	no	—
57	#334	—	2	2	no	—
58	#334	C₂³	1	4	yes	—
59	#334	C₂³	1	4	yes	—
60	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₄	SD₁₆	1	1	yes	characteristic
62	C₂³×C₄	SD₁₆	1	1	yes	characteristic
63	C₂²×C₈	—	2	4	no	—
64	C₂²×C₈	—	4	2	no	—
65	C₂²×C₈	—	4	4	no	—
66	C₂²×C₈	—	4	2	no	—
67	C₂²×C₈	—	8	4	no	—
68	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₈	C₄⋊C₄	1	2	yes	—
72	C₂²×C₈	C₂×C₈	1	4	yes	—
73	C₂²×C₈	M₁₆	1	4	yes	—
74	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂²×C₄	1	2	yes	—
76	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×Q₈	1	2	yes	—
78	C₂×C₄²	C₄⋊C₄	1	2	yes	—
79	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄²	SD₁₆	1	1	yes	characteristic
81	C₂×C₄²	SD₁₆	1	2	yes	—
82	C₂×C₄²	SD₁₆	1	2	yes	—
83	C₂×C₄²	SD₁₆	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₄	1	2	yes	—
86	M₃₂	—	2	8	no	—
87	M₃₂	C₂×C₈	1	16	yes	complemented
88	C₂×C₁₆	—	2	4	no	—
89	C₂×C₁₆	C₂×C₈	1	8	yes	complemented
90	C₂×C₁₆	C₂²×C₄	1	4	yes	—
91	C₂×C₁₆	C₂²×C₄	1	4	yes	—
92	C₄⋊C₈	—	2	4	no	—
93	C₄⋊C₈	—	2	4	no	—
94	C₄⋊C₈	—	4	2	no	—
95	C₄⋊C₈	—	4	4	no	—
96	C₄⋊C₈	—	4	2	no	—
97	C₂²⋊C₈	—	4	4	no	—
98	C₄×C₈	—	4	4	no	—
99	C₄×C₈	C₄⋊C₄	1	2	yes	—
100	C₄×C₈	C₄⋊C₄	1	2	yes	—
101	C₄×C₈	C₄⋊C₄	1	2	yes	—
102	C₄×C₈	C₂²×C₄	1	2	yes	—
103	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
104	C₄×C₈	C₂×D₄	1	1	yes	characteristic
105	C₄×C₈	C₂×D₄	1	2	yes	—
106	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
107	C₄×C₈	C₂×Q₈	1	2	yes	—
108	C₄×C₈	C₂⁴	1	1	yes	characteristic
109	C₂⁴	#106	1	1	yes	characteristic
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	4	2	no	—
112	C₂²×C₄	C₄⋊C₈	1	4	yes	—
113	C₂²×C₄	#106	1	1	yes	characteristic
114	C₂²×C₄	#106	1	2	yes	—
115	C₂²×C₄	#106	1	2	yes	—
116	C₂²×C₄	#106	1	2	yes	—
117	C₂²×C₄	#106	1	1	yes	characteristic
118	C₂²×C₄	#106	1	1	yes	characteristic
119	C₂²×C₄	#108	1	4	yes	—
120	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
121	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
123	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
124	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
125	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	4	4	no	—
132	C₂×C₈	—	8	4	no	—
133	C₂×C₈	—	8	4	no	—
134	C₂×C₈	—	8	8	no	—
135	C₂×C₈	C₄⋊C₈	1	8	yes	—
136	C₂×C₈	C₄⋊C₈	1	4	yes	—
137	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₈	C₂²×C₈	1	2	yes	—
142	C₂×C₈	C₂²×C₈	1	4	yes	—
143	C₂×C₈	C₂×M₁₆	1	2	yes	—
144	C₂×C₈	C₂×M₁₆	1	4	yes	—
145	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
147	C₄²	#106	1	2	yes	—
148	C₄²	#106	1	2	yes	—
149	C₄²	#106	1	2	yes	—
150	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₄²	C₂×C₄⋊C₄	1	2	yes	—
152	C₄²	C₂×SD₁₆	1	2	yes	—
153	C₄²	C₂×SD₁₆	1	1	yes	characteristic
154	C₄²	C₂×SD₁₆	1	2	yes	—
155	C₄²	C₂×SD₁₆	1	1	yes	characteristic
156	C₄²	C₂²×D₄	1	1	yes	characteristic
157	C₁₆	—	2	4	no	—
158	C₁₆	C₂²×C₈	1	8	yes	complemented
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	#334	1	4	yes	—
162	C₂³	C₂×#106	1	2	yes	—
163	C₂³	C₂×#106	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	#334	1	8	yes	—
172	C₂×C₄	#334	1	4	yes	—
173	C₂×C₄	#365	1	4	yes	—
174	C₂×C₄	C₂×C₄⋊C₈	1	4	yes	—
175	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
176	C₂×C₄	C₂×#106	1	1	yes	characteristic
177	C₂×C₄	C₂×#106	1	2	yes	—
178	C₂×C₄	C₂×#106	1	1	yes	characteristic
179	C₂×C₄	C₂×#106	1	2	yes	—
180	C₂×C₄	C₂×#106	1	1	yes	characteristic
181	C₂×C₄	C₂×#106	1	2	yes	—
182	C₂×C₄	C₂×#108	1	4	yes	—
183	C₂×C₄	C₂×#108	1	2	yes	—
184	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
186	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	8	4	no	—
191	C₈	—	8	4	no	—
192	C₈	C₂×C₄⋊C₈	1	2	yes	—
193	C₈	C₂×C₄⋊C₈	1	4	yes	—
194	C₈	C₂³×C₈	1	1	yes	characteristic
195	C₈	C₂²×M₁₆	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	8	no	—
199	C₂²	C₂×#334	1	4	yes	—
200	C₂²	C₂×#334	1	2	yes	—
201	C₂²	C₂×#365	1	2	yes	—
202	C₂²	#2155	1	2	yes	—
203	C₂²	C₂²×#106	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	2	2	no	—
209	C₄	C₂×#334	1	2	yes	—
210	C₄	C₂×#334	1	4	yes	—
211	C₄	C₂×#365	1	4	yes	—
212	C₄	C₂²×C₄⋊C₈	1	1	yes	characteristic
213	C₄	C₂²×#108	1	1	yes	characteristic
214	C₂	—	2	4	no	—
215	C₂	#12000	1	4	yes	direct product factor
216	C₂	C₂²×#334	1	1	yes	characteristic
217	C₂	C₂²×#365	1	1	yes	characteristic
218	C₂	C₂×#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#106
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×#106
prime radical	C₂⁴	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ 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
4. #12000	C₂	direct (N × H)
5. C₂×#1368	C₂	semidirect
23. #1368	C₂²	semidirect
29. C₂×M₃₂	C₈	semidirect
87. M₃₂	C₂×C₈	semidirect
89. C₂×C₁₆	C₂×C₈	semidirect
158. C₁₆	C₂²×C₈	semidirect
215. C₂	#12000	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×C₄×M₃₂	2
C₂×#1564	2
C₂²×#334	2
#12000	2
C₂×#1368	2

Subgroups — C2×#12000 #483321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12000 #483321