Subgroups — C₄×#2252 #496147

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2252	C₂	1	1	yes	maximal, characteristic
2	C₄×#500	C₂	1	2	yes	maximal
3	#13282	C₂	1	2	yes	maximal
4	#13165	C₂	1	4	yes	maximal
5	#12905	C₂	1	1	yes	maximal, characteristic
6	C₄×#366	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	#2252	C₄	1	8	yes	direct product factor
11	C₂×#366	C₂²	1	4	yes	—
12	C₂×#366	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	#366	—	2	8	no	—
49	#366	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	1	yes	characteristic
72	C₂×C₄²	D₈	1	1	yes	characteristic
73	C₂×C₄²	D₈	1	1	yes	characteristic
74	C₂×C₄²	D₈	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₁₆	—	2	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₂²×C₄	1	4	yes	—
81	C₂×C₁₆	C₂×Q₈	1	4	yes	—
82	C₂×C₁₆	C₂×Q₈	1	4	yes	—
83	C₂×C₁₆	#41	1	4	yes	—
84	#107	—	2	8	no	—
85	#107	—	2	4	no	—
86	#107	—	2	8	no	—
87	#107	—	4	4	no	—
88	#107	C₂²×C₄	1	8	yes	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	Q₁₆	1	4	yes	—
93	C₄×C₈	C₂²×C₄	1	2	yes	—
94	C₄×C₈	C₂×Q₈	1	4	yes	—
95	C₄×C₈	#41	1	2	yes	—
96	#95	—	4	4	no	—
97	#95	—	4	4	no	—
98	C₂×Q₈	—	4	8	no	—
99	C₂×Q₈	—	8	4	no	—
100	C₂×Q₈	—	8	4	no	—
101	C₂²×C₄	—	4	4	no	—
102	C₂²×C₄	—	4	4	no	—
103	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
106	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
108	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	C₄×D₄	1	2	yes	—
115	C₂×C₈	C₄⋊Q₈	1	2	yes	—
116	C₂×C₈	C₂×Q₁₆	1	2	yes	—
117	C₂×C₈	C₂×Q₁₆	1	2	yes	—
118	C₂×C₈	#135	1	2	yes	—
119	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₈	C₂²×Q₈	1	2	yes	—
121	C₂×C₈	C₂×#41	1	2	yes	—
122	C₂×C₈	C₂×#41	1	1	yes	characteristic
123	C₄⋊C₄	—	2	8	no	—
124	C₄⋊C₄	—	4	4	no	—
125	C₄⋊C₄	—	4	8	no	—
126	C₄⋊C₄	—	4	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₄⋊C₄	—	8	8	no	—
129	C₄⋊C₄	—	8	4	no	—
130	C₄⋊C₄	—	8	4	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	1	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	8	8	no	—
136	C₄²	D₁₆	1	4	yes	—
137	C₄²	Q₃₂	1	4	yes	—
138	C₄²	C₄×D₄	1	2	yes	—
139	C₄²	C₄⋊Q₈	1	2	yes	—
140	C₄²	C₂×Q₁₆	1	2	yes	—
141	C₄²	#135	1	2	yes	—
142	C₁₆	—	2	4	no	—
143	C₁₆	C₄×Q₈	1	8	yes	complemented
144	C₂³	C₂²×D₈	1	1	yes	characteristic
145	Q₈	—	8	16	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	4	8	no	—
153	C₂×C₄	—	8	4	no	—
154	C₂×C₄	—	8	4	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	8	4	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	C₄×D₈	1	2	yes	—
159	C₂×C₄	C₄×D₈	1	2	yes	—
160	C₂×C₄	#500	1	2	yes	—
161	C₂×C₄	#500	1	2	yes	—
162	C₂×C₄	C₂×D₁₆	1	2	yes	—
163	C₂×C₄	C₂×D₁₆	1	2	yes	—
164	C₂×C₄	C₂×D₁₆	1	2	yes	—
165	C₂×C₄	C₂×Q₃₂	1	2	yes	—
166	C₂×C₄	C₂×Q₃₂	1	2	yes	—
167	C₂×C₄	C₂×Q₃₂	1	2	yes	—
168	C₂×C₄	#508	1	2	yes	—
169	C₂×C₄	#508	1	2	yes	—
170	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
172	C₂×C₄	#532	1	1	yes	characteristic
173	C₂×C₄	C₂²×Q₁₆	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×Q₁₆	1	4	yes	—
179	C₈	C₂×C₄×Q₈	1	2	yes	—
180	C₈	#533	1	2	yes	—
181	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
182	C₂²	C₂×#500	1	1	yes	characteristic
183	C₂²	#3169	1	1	yes	characteristic
184	C₂²	C₂²×D₁₆	1	1	yes	characteristic
185	C₂²	C₂²×Q₃₂	1	1	yes	characteristic
186	C₂²	C₂×#508	1	1	yes	characteristic
187	C₂²	C₂×#508	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	8	8	no	—
192	C₄	—	16	4	no	—
193	C₄	C₄×D₁₆	1	4	yes	—
194	C₄	C₄×Q₃₂	1	4	yes	—
195	C₄	#2252	1	4	yes	direct product factor
196	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
197	C₄	C₂×C₄×Q₁₆	1	1	yes	characteristic
198	C₄	#3160	1	1	yes	characteristic
199	C₄	#3170	1	1	yes	characteristic
200	C₂	C₄×#500	1	1	yes	characteristic
201	C₂	C₂×C₄×D₁₆	1	1	yes	characteristic
202	C₂	C₂×C₄×Q₃₂	1	1	yes	characteristic
203	C₂	C₂×#2252	1	1	yes	characteristic
204	C₂	#33726	1	1	yes	characteristic
205	C₂	#33733	1	1	yes	characteristic
206	C₂	#33734	1	1	yes	characteristic

	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. #2252	C₄	direct (N × H)
53. C₄×C₁₆	Q₈	semidirect
143. C₁₆	C₄×Q₈	semidirect
195. C₄	#2252	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×#2252	2
C₄×#500	2
#13282	2
#13165	2
#12905	2
C₄×#366	2
C₄²×C₁₆	2

Subgroups — C4×#2252 #496147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#2252 #496147