Subgroups — #33488

Order 256.

313 conjugacy classes of subgroups.

The subgroup lattice has 169 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	#2169	C₂	1	2	yes	maximal, complemented
3	#2168	C₂	1	4	yes	maximal, complemented
4	D₄×C₁₆	C₂	1	2	yes	maximal, complemented
5	#2163	C₂	1	1	yes	maximal, characteristic, complemented
6	#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	#2112	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈	C₂²	1	2	yes	—
10	C₂×C₄×D₄	C₄	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	4	yes	complemented
12	C₂²×C₁₆	C₂²	1	4	yes	complemented
13	C₈×D₄	C₄	1	2	yes	—
14	C₈×D₄	C₄	1	1	yes	characteristic
15	C₈×D₄	C₄	1	1	yes	characteristic
16	C₈×D₄	C₂²	1	1	yes	characteristic
17	C₈×D₄	C₂²	1	2	yes	—
18	C₈×D₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
20	C₂×C₂²⋊C₈	C₄	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆	C₂²	1	2	yes	complemented
23	C₄⋊C₁₆	C₂²	1	2	yes	complemented
24	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₂²⋊C₁₆	C₂²	1	2	yes	—
26	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
28	C₂²⋊C₁₆	C₂²	1	4	yes	complemented
29	#346	C₂²	1	2	yes	complemented
30	C₄×C₁₆	C₂²	1	2	yes	complemented
31	C₂²×D₄	C₈	1	1	yes	characteristic
32	C₂³×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₈	—	4	4	no	—
34	C₂²×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₈	C₂×C₄	1	2	yes	—
36	C₂²×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	2	yes	—
40	C₂²×C₈	C₂³	1	2	yes	—
41	C₄×D₄	C₈	1	2	yes	—
42	C₄×D₄	C₈	1	1	yes	characteristic
43	C₄×D₄	C₈	1	1	yes	characteristic
44	C₄×D₄	C₂×C₄	1	2	yes	—
45	C₄×D₄	C₂×C₄	1	1	yes	characteristic
46	C₄×D₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
48	C₂×C₂²⋊C₄	C₈	1	2	yes	—
49	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
50	M₃₂	—	4	4	no	—
51	C₂×C₁₆	—	4	4	no	—
52	C₂×C₁₆	C₂³	1	2	yes	—
53	C₂×C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₆	C₂³	1	4	yes	—
55	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂²⋊C₈	C₂×C₄	1	2	yes	—
59	C₂²⋊C₈	C₂×C₄	1	2	yes	—
60	C₂²⋊C₈	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈	C₂×C₄	1	2	yes	—
62	C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₈	C₂³	1	1	yes	characteristic
66	C₂⁴	C₂×C₈	1	2	yes	—
67	C₂×D₄	—	2	4	no	—
68	C₂×D₄	C₂×C₈	1	2	yes	—
69	C₂×D₄	C₂×C₈	1	1	yes	characteristic
70	C₂×D₄	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄	—	4	2	no	—
72	C₂²×C₄	—	4	2	no	—
73	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
74	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₈	1	2	yes	—
76	C₂²×C₄	C₂×C₈	1	2	yes	—
77	C₂²×C₄	C₂×C₈	1	2	yes	—
78	C₂²×C₄	C₂²×C₄	1	2	yes	—
79	C₂²×C₄	C₂²×C₄	1	2	yes	—
80	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₈	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
96	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
97	C₂²⋊C₄	C₂×C₈	1	2	yes	—
98	C₂²⋊C₄	C₂×C₈	1	2	yes	—
99	C₂²⋊C₄	C₂×C₈	1	2	yes	—
100	C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₄²	C₂×C₈	1	1	yes	characteristic
102	C₄²	C₂×C₈	1	1	yes	characteristic
103	C₄²	C₂²×C₄	1	1	yes	characteristic
104	C₄²	C₂²×C₄	1	1	yes	characteristic
105	C₁₆	—	2	2	no	—
106	C₁₆	—	2	4	no	—
107	C₁₆	—	2	2	no	—
108	C₂³	—	2	4	no	—
109	C₂³	—	2	4	no	—
110	C₂³	—	4	2	no	—
111	C₂³	C₂²×C₈	1	2	yes	—
112	C₂³	C₂²×C₈	1	2	yes	—
113	C₂³	C₂²×C₈	1	1	yes	characteristic
114	D₄	—	2	4	no	—
115	D₄	—	2	2	no	—
116	D₄	M₃₂	1	4	yes	complemented
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
131	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₈	1	2	yes	—
133	C₂×C₄	C₂²×C₈	1	2	yes	—
134	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	H(5,𝔽₂)	1	2	yes	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	4	8	no	—
150	C₂²	C₂×M₃₂	1	4	yes	—
151	C₂²	#504	1	2	yes	—
152	C₂²	C₂³×C₈	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	C₂×M₃₂	1	2	yes	—
160	C₄	#518	1	1	yes	characteristic
161	C₄	#568	1	1	yes	characteristic
162	C₂	—	2	2	no	—
163	C₂	—	2	4	no	—
164	C₂	—	4	2	no	—
165	C₂	#2972	1	1	yes	characteristic
166	C₂	C₂²×M₃₂	1	1	yes	characteristic
167	C₂	C₂×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂²×D₄	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2169	C₂	semidirect
3. #2168	C₂	semidirect
4. D₄×C₁₆	C₂	semidirect
5. #2163	C₂	semidirect
6. #2162	C₂	semidirect
7. #2112	C₂	semidirect
8. C₂×C₂²⋊C₁₆	C₂	semidirect
11. C₂×M₃₂	C₂²	semidirect
12. C₂²×C₁₆	C₂²	semidirect
22. C₄⋊C₁₆	C₂²	semidirect
23. C₄⋊C₁₆	C₂²	semidirect
27. C₂²⋊C₁₆	C₂²	semidirect
28. C₂²⋊C₁₆	C₂²	semidirect
29. #346	C₂²	semidirect
30. C₄×C₁₆	C₂²	semidirect
116. D₄	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄	2
#2169	2
#2168	2
D₄×C₁₆	2
#2163	2
#2162	2
#2112	2
C₂×C₂²⋊C₁₆	2