Subgroups — C₄×#2247 #496100

Order 512.

792 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2247	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₃₂	C₂	1	4	yes	maximal
3	C₄×#494	C₂	1	2	yes	maximal
4	#13236	C₂	1	4	yes	maximal
5	#13203	C₂	1	2	yes	maximal
6	#12905	C₂	1	1	yes	maximal, characteristic
7	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₃₂	C₂²	1	4	yes	—
9	C₂×#494	C₂²	1	2	yes	—
10	C₂×C₄×Q₁₆	—	2	2	no	—
11	C₂×C₄×Q₁₆	C₂²	1	4	yes	—
12	C₄×C₄⋊Q₈	—	2	2	no	—
13	#2247	C₄	1	8	yes	direct product factor
14	C₄×Q₃₂	—	2	16	no	—
15	C₂×#366	C₂²	1	4	yes	—
16	C₂×#358	C₂²	1	8	yes	—
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	2	yes	—
21	#1969	—	2	2	no	—
22	#1969	C₂²	1	4	yes	—
23	#1956	—	2	2	no	—
24	#1845	C₂²	1	2	yes	—
25	C₄²×C₈	C₂²	1	1	yes	characteristic
26	C₂²×Q₁₆	—	2	2	no	—
27	C₂²×Q₁₆	C₂³	1	4	yes	—
28	C₂×C₄⋊Q₈	—	2	2	no	—
29	C₂×C₄×Q₈	—	2	4	no	—
30	C₂×Q₃₂	—	2	8	no	—
31	C₂×Q₃₂	—	2	8	no	—
32	C₂×Q₃₂	C₂×C₄	1	16	yes	—
33	C₂²×C₁₆	C₂³	1	2	yes	—
34	C₂²×C₁₆	C₂³	1	2	yes	—
35	#494	—	2	4	no	—
36	#494	C₂×C₄	1	8	yes	—
37	C₄×Q₁₆	—	2	16	no	—
38	C₄×Q₁₆	—	4	8	no	—
39	C₂×#107	—	2	2	no	—
40	C₂×#107	C₂³	1	4	yes	—
41	C₂×#103	—	2	4	no	—
42	C₂×#103	—	2	4	no	—
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	D₄	1	1	yes	characteristic
45	C₂×C₄×C₈	D₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	#391	—	2	4	no	—
50	#389	—	4	4	no	—
51	#384	—	2	2	no	—
52	C₄×C₄⋊C₄	—	4	4	no	—
53	C₄³	D₄	1	1	yes	characteristic
54	#366	—	2	8	no	—
55	#358	—	2	16	no	—
56	C₄×C₁₆	—	2	4	no	—
57	C₄×C₁₆	—	2	2	no	—
58	C₄×C₁₆	C₂×C₄	1	4	yes	—
59	C₄×C₁₆	D₄	1	8	yes	—
60	C₂²×Q₈	—	2	4	no	—
61	C₂×Q₁₆	—	2	8	no	—
62	C₂×Q₁₆	—	2	8	no	—
63	C₂×Q₁₆	—	2	8	no	—
64	C₂×Q₁₆	—	2	4	no	—
65	C₂×Q₁₆	—	4	4	no	—
66	C₂×Q₁₆	—	4	4	no	—
67	C₂×Q₁₆	C₂²×C₄	1	8	yes	—
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₈	C₂⁴	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	8	no	—
73	C₄⋊Q₈	—	4	4	no	—
74	C₄×Q₈	—	4	8	no	—
75	C₄×Q₈	—	4	8	no	—
76	C₂×C₄⋊C₄	—	2	4	no	—
77	C₂×C₄⋊C₄	—	4	4	no	—
78	C₂×C₄⋊C₄	—	4	4	no	—
79	C₂×C₄⋊C₄	—	4	4	no	—
80	C₂×C₄²	—	4	4	no	—
81	C₂×C₄²	D₈	1	2	yes	—
82	C₂×C₄²	D₈	1	2	yes	—
83	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
86	Q₃₂	—	2	32	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	C₂²×C₄	1	4	yes	—
90	C₂×C₁₆	C₂×D₄	1	4	yes	—
91	C₂×C₁₆	C₂×D₄	1	4	yes	—
92	C₂×C₁₆	#41	1	4	yes	—
93	#107	—	2	8	no	—
94	#107	—	4	4	no	—
95	#103	—	4	8	no	—
96	#103	—	4	8	no	—
97	C₄×C₈	—	2	2	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	D₈	1	4	yes	—
101	C₄×C₈	C₂²×C₄	1	2	yes	—
102	C₄×C₈	C₂×D₄	1	4	yes	—
103	C₄×C₈	#41	1	2	yes	—
104	#95	—	4	8	no	—
105	C₂×Q₈	—	2	8	no	—
106	C₂×Q₈	—	4	4	no	—
107	C₂×Q₈	—	4	4	no	—
108	C₂×Q₈	—	4	4	no	—
109	C₂×Q₈	—	4	4	no	—
110	C₂×Q₈	—	4	4	no	—
111	C₂²×C₄	—	4	4	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₈	1	2	yes	—
115	C₂²×C₄	C₂×D₈	1	2	yes	—
116	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
117	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
118	Q₁₆	—	2	16	no	—
119	Q₁₆	—	4	8	no	—
120	Q₁₆	—	4	16	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	C₄×D₄	1	2	yes	—
126	C₂×C₈	#127	1	2	yes	—
127	C₂×C₈	C₂×D₈	1	2	yes	—
128	C₂×C₈	C₂×D₈	1	2	yes	—
129	C₂×C₈	#135	1	2	yes	—
130	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₈	C₂²×D₄	1	2	yes	—
132	C₂×C₈	C₂×#41	1	2	yes	—
133	C₂×C₈	C₂×#41	1	1	yes	characteristic
134	C₄⋊C₄	—	4	16	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	8	8	no	—
138	C₄⋊C₄	—	8	8	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	2	1	no	—
141	C₄²	—	2	4	no	—
142	C₄²	—	8	8	no	—
143	C₄²	Q₃₂	1	8	yes	complemented
144	C₄²	C₄×D₄	1	2	yes	—
145	C₄²	#127	1	2	yes	—
146	C₄²	C₂×D₈	1	2	yes	—
147	C₄²	#135	1	2	yes	—
148	C₁₆	—	2	4	no	—
149	C₁₆	C₄×D₄	1	8	yes	—
150	C₂³	C₂²×D₈	1	1	yes	characteristic
151	Q₈	—	4	8	no	—
152	Q₈	—	4	8	no	—
153	Q₈	—	8	4	no	—
154	Q₈	—	8	4	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	4	8	no	—
159	C₂×C₄	—	8	4	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	8	8	no	—
163	C₂×C₄	C₄×D₈	1	4	yes	—
164	C₂×C₄	#493	1	2	yes	—
165	C₂×C₄	#493	1	2	yes	—
166	C₂×C₄	C₂×Q₃₂	1	4	yes	—
167	C₂×C₄	C₂×Q₃₂	1	4	yes	—
168	C₂×C₄	C₂×Q₃₂	1	4	yes	—
169	C₂×C₄	#508	1	4	yes	—
170	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×C₄	C₂×#127	1	1	yes	characteristic
172	C₂×C₄	#532	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
174	C₂×C₄	C₂×#135	1	1	yes	characteristic
175	C₂×C₄	C₂×#135	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	2	no	—
178	C₈	C₄×D₈	1	4	yes	—
179	C₈	C₂×C₄×D₄	1	2	yes	—
180	C₈	#532	1	2	yes	—
181	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
182	C₂²	C₂×#493	1	1	yes	characteristic
183	C₂²	#3169	1	1	yes	characteristic
184	C₂²	C₂²×Q₃₂	1	2	yes	—
185	C₂²	C₂×#508	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	2	4	no	—
188	C₄	—	8	8	no	—
189	C₄	—	16	4	no	—
190	C₄	C₄×Q₃₂	1	8	yes	complemented
191	C₄	#2247	1	4	yes	direct product factor
192	C₄	C₄×#127	1	1	yes	characteristic
193	C₄	C₂×C₄×D₈	1	1	yes	characteristic
194	C₄	#3147	1	1	yes	characteristic
195	C₄	#3169	1	1	yes	characteristic
196	C₂	C₄×#493	1	1	yes	characteristic
197	C₂	C₂×C₄×Q₃₂	1	2	yes	—
198	C₂	C₂×#2247	1	1	yes	characteristic
199	C₂	#33713	1	1	yes	characteristic
200	C₂	#33734	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle, prime radical	C₂³	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
13. #2247	C₄	direct (N × H)
143. C₄²	Q₃₂	semidirect
190. C₄	C₄×Q₃₂	semidirect
191. C₄	#2247	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×#2247	2
C₂×C₄×Q₃₂	2
C₄×#494	2
#13236	2
#13203	2
#12905	2
C₄²×C₁₆	2

Subgroups — C4×#2247 #496100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#2247 #496100