Subgroups — C₂×#12025 #483404

Order 512.

518 conjugacy classes of subgroups.

The subgroup lattice has 230 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×#1568	C₂	1	1	yes	maximal, characteristic
3	C₂×#1567	C₂	1	1	yes	maximal, characteristic
4	#12025	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1372	C₂	1	2	yes	maximal, complemented
6	C₂×#1371	C₂	1	2	yes	maximal, complemented
7	C₂²×M₃₂	C₄	1	2	yes	—
8	C₂×#432	—	2	1	no	—
9	C₂×#423	—	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	C₄	1	1	yes	characteristic
14	C₂×#346	C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	#1568	—	2	2	no	—
17	#1568	C₂²	1	4	yes	—
18	#1567	—	2	2	no	—
19	#1567	C₂²	1	4	yes	—
20	C₂×#335	—	2	1	no	—
21	C₂×#335	C₂²	1	2	yes	—
22	C₂×#335	C₂²	1	1	yes	characteristic
23	C₂×#335	C₂²	1	1	yes	characteristic
24	C₂×#334	—	2	1	no	—
25	#1372	C₂²	1	8	yes	complemented
26	#1371	C₂²	1	8	yes	complemented
27	C₂²×M₁₆	—	4	1	no	—
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₈	Q₈	1	1	yes	characteristic
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	8	no	—
32	C₂×M₃₂	C₂×C₄	1	8	yes	—
33	C₂²×C₁₆	C₂×C₄	1	2	yes	—
34	C₂²×C₁₆	C₂×C₄	1	2	yes	—
35	#432	—	2	4	no	—
36	#432	—	4	2	no	—
37	#423	—	2	4	no	—
38	#423	—	4	2	no	—
39	C₂×C₄⋊C₈	—	2	1	no	—
40	C₂×C₄⋊C₈	—	2	1	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	C₂×C₄⋊C₈	—	4	1	no	—
43	C₂×C₂²⋊C₈	—	4	1	no	—
44	C₂×#97	—	4	1	no	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	2	yes	—
49	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
50	C₂×C₄×C₈	Q₈	1	2	yes	—
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	2	yes	—
53	#346	C₂×C₄	1	4	yes	—
54	#346	C₂×C₄	1	4	yes	—
55	C₄×C₁₆	C₂×C₄	1	4	yes	—
56	C₄×C₁₆	C₂³	1	4	yes	—
57	#335	—	2	4	no	—
58	#335	—	2	2	no	—
59	#335	—	2	1	no	—
60	#335	—	2	1	no	—
61	#335	C₂³	1	4	yes	—
62	#335	C₂³	1	2	yes	—
63	#335	C₂³	1	2	yes	—
64	#334	—	2	4	no	—
65	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
66	C₂³×C₄	D₈	1	1	yes	characteristic
67	C₂³×C₄	Q₁₆	1	1	yes	characteristic
68	C₂×M₁₆	—	4	4	no	—
69	C₂×M₁₆	—	8	4	no	—
70	C₂²×C₈	—	2	4	no	—
71	C₂²×C₈	—	2	4	no	—
72	C₂²×C₈	—	4	2	no	—
73	C₂²×C₈	—	4	1	no	—
74	C₂²×C₈	—	4	1	no	—
75	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₈	C₄⋊C₄	1	2	yes	—
77	C₂²×C₈	C₄⋊C₄	1	2	yes	—
78	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²×C₄	1	2	yes	—
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₈	C₂×Q₈	1	2	yes	—
83	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄²	C₄⋊C₄	1	2	yes	—
85	C₂×C₄²	D₈	1	2	yes	—
86	C₂×C₄²	D₈	1	1	yes	characteristic
87	C₂×C₄²	Q₁₆	1	1	yes	characteristic
88	C₂×C₄²	Q₁₆	1	2	yes	—
89	C₂×C₄²	C₂×D₄	1	2	yes	—
90	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
91	M₃₂	—	2	16	no	—
92	C₂×C₁₆	—	2	4	no	—
93	C₂×C₁₆	M₁₆	1	8	yes	complemented
94	C₂×C₁₆	C₂²×C₄	1	4	yes	—
95	C₂×C₁₆	C₂²×C₄	1	4	yes	—
96	C₄⋊C₈	—	2	2	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	2	4	no	—
99	C₄⋊C₈	—	4	2	no	—
100	C₄⋊C₈	—	4	1	no	—
101	C₄⋊C₈	—	4	1	no	—
102	C₄⋊C₈	—	4	4	no	—
103	C₂²⋊C₈	—	4	4	no	—
104	#97	—	4	4	no	—
105	C₄×C₈	C₄⋊C₄	1	2	yes	—
106	C₄×C₈	C₄⋊C₄	1	2	yes	—
107	C₄×C₈	C₄⋊C₄	1	2	yes	—
108	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂²×C₄	1	2	yes	—
110	C₄×C₈	C₂×D₄	1	2	yes	—
111	C₄×C₈	C₂×D₄	1	1	yes	characteristic
112	C₄×C₈	C₂×Q₈	1	2	yes	—
113	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
114	C₄×C₈	C₂⁴	1	1	yes	characteristic
115	C₂⁴	#107	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#107	1	1	yes	characteristic
122	C₂²×C₄	#107	1	1	yes	characteristic
123	C₂²×C₄	#107	1	2	yes	—
124	C₂²×C₄	#107	1	1	yes	characteristic
125	C₂²×C₄	#107	1	2	yes	—
126	C₂²×C₄	#107	1	2	yes	—
127	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
128	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₄	C₂×D₈	1	2	yes	—
130	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
131	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
132	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
133	M₁₆	—	8	8	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	4	no	—
143	C₂×C₈	—	8	2	no	—
144	C₂×C₈	—	8	4	no	—
145	C₂×C₈	—	8	2	no	—
146	C₂×C₈	#108	1	4	yes	—
147	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
151	C₂×C₈	C₂×M₁₆	1	4	yes	—
152	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
154	C₄²	#107	1	2	yes	—
155	C₄²	#107	1	2	yes	—
156	C₄²	#107	1	2	yes	—
157	C₄²	C₂×C₄⋊C₄	1	2	yes	—
158	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₄²	C₂×D₈	1	2	yes	—
160	C₄²	C₂×D₈	1	1	yes	characteristic
161	C₄²	C₂×Q₁₆	1	2	yes	—
162	C₄²	C₂×Q₁₆	1	1	yes	characteristic
163	C₄²	C₂²×D₄	1	1	yes	characteristic
164	C₁₆	—	2	4	no	—
165	C₁₆	C₂×M₁₆	1	8	yes	complemented
166	C₂³	—	2	2	no	—
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×#107	1	2	yes	—
170	C₂³	C₂×#107	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	#423	1	2	yes	—
184	C₂×C₄	C₂×#107	1	2	yes	—
185	C₂×C₄	C₂×#107	1	1	yes	characteristic
186	C₂×C₄	C₂×#107	1	2	yes	—
187	C₂×C₄	C₂×#107	1	2	yes	—
188	C₂×C₄	C₂×#107	1	1	yes	characteristic
189	C₂×C₄	C₂×#107	1	1	yes	characteristic
190	C₂×C₄	C₂×#108	1	2	yes	—
191	C₂×C₄	#509	1	2	yes	—
192	C₂×C₄	#510	1	2	yes	—
193	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
195	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	2	1	no	—
198	C₈	—	2	1	no	—
199	C₈	—	8	2	no	—
200	C₈	—	8	4	no	—
201	C₈	—	8	2	no	—
202	C₈	#423	1	2	yes	—
203	C₈	C₂×#108	1	4	yes	—
204	C₈	C₂²×M₁₆	1	2	yes	—
205	C₂²	—	2	2	no	—
206	C₂²	—	2	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	4	4	no	—
209	C₂²	#1567	1	2	yes	—
210	C₂²	#2159	1	2	yes	—
211	C₂²	#2161	1	2	yes	—
212	C₂²	C₂²×#107	1	1	yes	characteristic
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	2	1	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	1	no	—
218	C₄	#1567	1	2	yes	—
219	C₄	#2159	1	2	yes	—
220	C₄	C₂×#423	1	1	yes	characteristic
221	C₄	C₂²×#108	1	1	yes	characteristic
222	C₄	C₂×#509	1	1	yes	characteristic
223	C₄	C₂×#510	1	1	yes	characteristic
224	C₂	—	4	2	no	—
225	C₂	#12025	1	4	yes	direct product factor
226	C₂	C₂×#1567	1	1	yes	characteristic
227	C₂	C₂×#2159	1	1	yes	characteristic
228	C₂	C₂×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×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. #12025	C₂	direct (N × H)
5. C₂×#1372	C₂	semidirect
6. C₂×#1371	C₂	semidirect
25. #1372	C₂²	semidirect
26. #1371	C₂²	semidirect
93. C₂×C₁₆	M₁₆	semidirect
165. C₁₆	C₂×M₁₆	semidirect
225. C₂	#12025	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×#1568	2
C₂×#1567	2
#12025	2
C₂×#1372	2
C₂×#1371	2

Subgroups — C2×#12025 #483404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12025 #483404