Subgroups — C₄×#2254 #496148

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2254	C₂	1	1	yes	maximal, characteristic
2	C₄×#500	C₂	1	2	yes	maximal
3	#13282	C₂	1	2	yes	maximal
4	#13166	C₂	1	4	yes	maximal
5	#12906	C₂	1	1	yes	maximal, characteristic
6	C₄×#367	C₂	1	4	yes	maximal
7	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	2	yes	—
9	C₄×C₄⋊Q₈	—	2	2	no	—
10	#2254	C₄	1	8	yes	direct product factor
11	C₂×#367	C₂²	1	4	yes	—
12	C₂×#367	C₂²	1	4	yes	—
13	C₂×C₄×C₁₆	C₂²	1	2	yes	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₆	C₂²	1	2	yes	—
17	#1990	—	2	2	no	—
18	#1942	—	2	2	no	—
19	#1942	C₂²	1	4	yes	—
20	#1845	C₂²	1	2	yes	—
21	C₄×#107	—	2	2	no	—
22	C₄×#107	C₂²	1	4	yes	—
23	C₄²×C₈	C₂²	1	1	yes	characteristic
24	#1380	C₂²	1	8	yes	—
25	C₂×C₄⋊Q₈	—	2	2	no	—
26	C₂×C₄×Q₈	—	4	2	no	—
27	C₂²×C₁₆	C₂³	1	2	yes	—
28	C₂²×C₁₆	C₂³	1	2	yes	—
29	#500	—	2	4	no	—
30	#500	C₂×C₄	1	8	yes	—
31	C₂×#107	—	2	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	C₂³	1	4	yes	—
34	C₂×#107	C₂³	1	4	yes	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	#391	—	4	2	no	—
42	#389	—	2	4	no	—
43	#389	—	4	2	no	—
44	#384	—	2	2	no	—
45	C₄×C₄⋊C₄	—	2	4	no	—
46	C₄×C₄⋊C₄	—	4	2	no	—
47	C₄³	D₄	1	1	yes	characteristic
48	#367	—	2	8	no	—
49	#367	C₂×C₄	1	16	yes	—
50	C₄×C₁₆	—	2	4	no	—
51	C₄×C₁₆	—	2	2	no	—
52	C₄×C₁₆	C₂×C₄	1	4	yes	—
53	C₄×C₁₆	Q₈	1	8	yes	complemented
54	#340	—	2	4	no	—
55	#340	—	2	4	no	—
56	C₂²×Q₈	—	4	2	no	—
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂⁴	1	1	yes	characteristic
61	C₄⋊Q₈	—	2	8	no	—
62	C₄⋊Q₈	—	4	4	no	—
63	C₄×Q₈	—	8	8	no	—
64	C₂×C₄⋊C₄	—	2	4	no	—
65	C₂×C₄⋊C₄	—	2	4	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄⋊C₄	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	4	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄²	—	4	4	no	—
71	C₂×C₄²	D₈	1	2	yes	—
72	C₂×C₄²	D₈	1	2	yes	—
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	C₂²×C₄	1	4	yes	—
79	C₂×C₁₆	C₂×Q₈	1	4	yes	—
80	C₂×C₁₆	C₂×Q₈	1	4	yes	—
81	C₂×C₁₆	#41	1	4	yes	—
82	#107	—	2	8	no	—
83	#107	—	2	4	no	—
84	#107	—	2	8	no	—
85	#107	—	4	4	no	—
86	#107	C₂²×C₄	1	8	yes	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	Q₁₆	1	4	yes	—
91	C₄×C₈	C₂²×C₄	1	2	yes	—
92	C₄×C₈	C₂×Q₈	1	4	yes	—
93	C₄×C₈	#41	1	2	yes	—
94	#95	—	4	4	no	—
95	#95	—	4	4	no	—
96	C₂×Q₈	—	4	8	no	—
97	C₂×Q₈	—	8	4	no	—
98	C₂×Q₈	—	8	4	no	—
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	—	4	4	no	—
101	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
102	C₂²×C₄	C₂×D₈	1	2	yes	—
103	C₂²×C₄	C₂×D₈	1	2	yes	—
104	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
105	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	C₄×D₄	1	2	yes	—
111	C₂×C₈	C₄⋊Q₈	1	2	yes	—
112	C₂×C₈	C₂×Q₁₆	1	2	yes	—
113	C₂×C₈	C₂×Q₁₆	1	2	yes	—
114	C₂×C₈	#135	1	2	yes	—
115	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₈	C₂²×Q₈	1	2	yes	—
117	C₂×C₈	C₂×#41	1	2	yes	—
118	C₂×C₈	C₂×#41	1	1	yes	characteristic
119	C₄⋊C₄	—	2	8	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	4	8	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	4	4	no	—
124	C₄⋊C₄	—	8	8	no	—
125	C₄⋊C₄	—	8	4	no	—
126	C₄⋊C₄	—	8	4	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	2	1	no	—
129	C₄²	—	2	4	no	—
130	C₄²	—	8	8	no	—
131	C₄²	SD₃₂	1	8	yes	—
132	C₄²	C₄×D₄	1	2	yes	—
133	C₄²	C₄⋊Q₈	1	2	yes	—
134	C₄²	C₂×Q₁₆	1	2	yes	—
135	C₄²	#135	1	2	yes	—
136	C₁₆	—	2	4	no	—
137	C₁₆	C₄×Q₈	1	8	yes	complemented
138	C₂³	C₂²×D₈	1	1	yes	characteristic
139	Q₈	—	8	16	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	4	8	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	—	8	4	no	—
146	C₂×C₄	—	8	4	no	—
147	C₂×C₄	—	8	4	no	—
148	C₂×C₄	—	8	4	no	—
149	C₂×C₄	C₄×D₈	1	4	yes	—
150	C₂×C₄	#500	1	2	yes	—
151	C₂×C₄	#500	1	2	yes	—
152	C₂×C₄	C₂×SD₃₂	1	4	yes	—
153	C₂×C₄	C₂×SD₃₂	1	4	yes	—
154	C₂×C₄	C₂×SD₃₂	1	4	yes	—
155	C₂×C₄	#508	1	4	yes	—
156	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
158	C₂×C₄	#532	1	1	yes	characteristic
159	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
160	C₂×C₄	C₂×#135	1	1	yes	characteristic
161	C₂×C₄	C₂×#135	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	2	2	no	—
164	C₈	C₄×Q₁₆	1	4	yes	—
165	C₈	C₂×C₄×Q₈	1	2	yes	—
166	C₈	#533	1	2	yes	—
167	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
168	C₂²	C₂×#500	1	1	yes	characteristic
169	C₂²	#3169	1	1	yes	characteristic
170	C₂²	C₂²×SD₃₂	1	2	yes	—
171	C₂²	C₂×#508	1	2	yes	—
172	C₄	—	2	2	no	—
173	C₄	—	2	4	no	—
174	C₄	—	8	8	no	—
175	C₄	—	16	4	no	—
176	C₄	C₄×SD₃₂	1	8	yes	—
177	C₄	#2254	1	4	yes	direct product factor
178	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
179	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
180	C₄	#3160	1	1	yes	characteristic
181	C₄	#3170	1	1	yes	characteristic
182	C₂	C₄×#500	1	1	yes	characteristic
183	C₂	C₂×C₄×SD₃₂	1	2	yes	—
184	C₂	C₂×#2254	1	1	yes	characteristic
185	C₂	#33726	1	1	yes	characteristic
186	C₂	#33736	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
10. #2254	C₄	direct (N × H)
53. C₄×C₁₆	Q₈	semidirect
137. C₁₆	C₄×Q₈	semidirect
177. C₄	#2254	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×#2254	2
C₄×#500	2
#13282	2
#13166	2
#12906	2
C₄×#367	2
C₄²×C₁₆	2

Subgroups — C4×#2254 #496148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#2254 #496148