Finite Group Databasebeta

← C₂×#13694 #512931

Subgroups — C₂×#13694 #512931

Order 512.

498 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#2215	C₂	1	2	yes	maximal, complemented
4	#13694	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1423	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#1415	C₂	1	2	yes	maximal, complemented
7	C₂²×D₁₆	C₂²	1	2	yes	—
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#466	—	2	2	no	—
10	C₂×D₃₂	—	2	4	no	—
11	C₂×D₃₂	C₂²	1	4	yes	complemented
12	C₂²×C₃₂	—	2	4	no	—
13	C₂²×C₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₃₂	C₂²	1	4	yes	complemented
16	#2215	—	2	4	no	—
17	#2215	C₂²	1	8	yes	complemented
18	C₂×#366	C₂²	1	2	yes	—
19	C₂×#357	—	2	2	no	—
20	#1423	C₂²	1	4	yes	complemented
21	#1415	C₂²	1	8	yes	complemented
22	C₂²×D₈	—	2	2	no	—
23	C₂³×C₈	D₄	1	1	yes	characteristic
24	C₂×#121	—	4	2	no	—
25	C₂×D₁₆	—	2	4	no	—
26	C₂×D₁₆	—	2	2	no	—
27	C₂×D₁₆	—	2	4	no	—
28	C₂×D₁₆	C₂³	1	4	yes	—
29	C₂²×C₁₆	—	2	2	no	—
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₁₆	D₄	1	2	yes	—
33	C₂²×C₁₆	C₂³	1	2	yes	—
34	C₂²×C₁₆	C₂³	1	1	yes	characteristic
35	#466	—	2	8	no	—
36	#466	—	4	4	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#102	—	4	2	no	—
39	D₃₂	—	2	8	no	—
40	C₂×C₃₂	—	2	2	no	—
41	C₂×C₃₂	—	2	8	no	—
42	C₂×C₃₂	D₄	1	4	yes	complemented
43	C₂×C₃₂	C₂³	1	2	yes	—
44	C₂×C₃₂	C₂³	1	2	yes	—
45	#366	—	2	2	no	—
46	#366	C₂³	1	4	yes	—
47	#357	—	2	8	no	—
48	C₂²×D₄	—	4	2	no	—
49	C₂²×D₄	—	8	2	no	—
50	C₂³×C₄	D₈	1	1	yes	characteristic
51	C₂×D₈	—	2	4	no	—
52	C₂×D₈	—	4	4	no	—
53	C₂×D₈	—	4	2	no	—
54	C₂×D₈	—	4	4	no	—
55	C₂²×C₈	—	2	2	no	—
56	C₂²×C₈	—	2	2	no	—
57	C₂²×C₈	D₈	1	1	yes	characteristic
58	C₂²×C₈	D₈	1	2	yes	—
59	C₂²×C₈	C₂×D₄	1	2	yes	—
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	#121	—	4	8	no	—
62	#121	—	8	4	no	—
63	C₂×C₄⋊C₄	—	4	2	no	—
64	C₂×C₂²⋊C₄	—	8	2	no	—
65	D₁₆	—	2	4	no	—
66	D₁₆	—	2	4	no	—
67	D₁₆	—	4	4	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₂×D₄	1	2	yes	—
73	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
74	C₂×C₁₆	C₂×D₄	1	2	yes	—
75	C₂×C₁₆	#41	1	2	yes	—
76	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
77	#107	—	2	4	no	—
78	#107	—	4	2	no	—
79	#102	—	4	8	no	—
80	C₃₂	—	2	2	no	—
81	C₃₂	C₂×D₄	1	4	yes	complemented
82	C₂⁴	—	8	2	no	—
83	C₂⁴	D₁₆	1	1	yes	characteristic
84	C₂×D₄	—	4	4	no	—
85	C₂×D₄	—	8	8	no	—
86	C₂×D₄	—	8	4	no	—
87	C₂×D₄	—	8	2	no	—
88	C₂×D₄	—	8	4	no	—
89	C₂×D₄	—	16	8	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	8	2	no	—
93	C₂²×C₄	D₁₆	1	2	yes	—
94	C₂²×C₄	D₁₆	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₈	1	2	yes	—
96	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
97	D₈	—	4	4	no	—
98	D₈	—	4	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	#121	1	2	yes	—
105	C₂×C₈	C₂×D₈	1	2	yes	—
106	C₂×C₈	C₂×D₈	1	1	yes	characteristic
107	C₂×C₈	#135	1	2	yes	—
108	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
109	C₄⋊C₄	—	4	4	no	—
110	C₄⋊C₄	—	8	2	no	—
111	C₂²⋊C₄	—	8	8	no	—
112	C₁₆	—	2	2	no	—
113	C₁₆	#121	1	2	yes	—
114	C₁₆	C₂²×D₄	1	1	yes	characteristic
115	C₁₆	C₂×#41	1	1	yes	characteristic
116	C₂³	—	2	2	no	—
117	C₂³	—	2	2	no	—
118	C₂³	—	8	4	no	—
119	C₂³	—	16	4	no	—
120	C₂³	—	16	4	no	—
121	C₂³	—	16	4	no	—
122	C₂³	D₃₂	1	4	yes	complemented
123	C₂³	C₂×D₁₆	1	2	yes	—
124	C₂³	C₂×D₁₆	1	1	yes	characteristic
125	D₄	—	8	4	no	—
126	D₄	—	8	4	no	—
127	D₄	—	16	16	no	—
128	D₄	—	16	4	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	—	16	4	no	—
135	C₂×C₄	#466	1	2	yes	—
136	C₂×C₄	C₂×D₁₆	1	2	yes	—
137	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
138	C₂×C₄	#508	1	2	yes	—
139	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	#466	1	2	yes	—
142	C₈	C₂×#121	1	1	yes	characteristic
143	C₈	C₂×#135	1	1	yes	characteristic
144	C₂²	—	2	2	no	—
145	C₂²	—	2	8	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	16	4	no	—
148	C₂²	—	16	4	no	—
149	C₂²	—	16	4	no	—
150	C₂²	—	32	8	no	—
151	C₂²	#2215	1	2	yes	—
152	C₂²	C₂×D₃₂	1	2	yes	—
153	C₂²	C₂×D₃₂	1	4	yes	complemented
154	C₂²	#2262	1	2	yes	—
155	C₂²	C₂²×D₁₆	1	1	yes	characteristic
156	C₄	—	2	2	no	—
157	C₄	—	16	4	no	—
158	C₄	#2215	1	2	yes	—
159	C₄	C₂×#466	1	1	yes	characteristic
160	C₄	C₂×#508	1	1	yes	characteristic
161	C₂	—	2	4	no	—
162	C₂	—	32	4	no	—
163	C₂	#13694	1	4	yes	direct product factor
164	C₂	C₂×#2215	1	1	yes	characteristic
165	C₂	C₂²×D₃₂	1	1	yes	characteristic
166	C₂	C₂×#2262	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×D₃₂	C₂	semidirect
2. C₂³×C₃₂	C₂	semidirect
3. C₂×#2215	C₂	semidirect
4. #13694	C₂	direct (N × H)
5. C₂×#1423	C₂	semidirect
6. C₂×#1415	C₂	semidirect
11. C₂×D₃₂	C₂²	semidirect
15. C₂²×C₃₂	C₂²	semidirect
17. #2215	C₂²	semidirect
20. #1423	C₂²	semidirect
21. #1415	C₂²	semidirect
42. C₂×C₃₂	D₄	semidirect
81. C₃₂	C₂×D₄	semidirect
122. C₂³	D₃₂	semidirect
153. C₂²	C₂×D₃₂	semidirect
163. C₂	#13694	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₃₂	2
C₂³×C₃₂	2
C₂×#2215	2
#13694	2
C₂×#1423	2
C₂×#1415	2