Subgroups — #33504

Order 256.

268 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#41	C₂	1	1	yes	maximal, characteristic
2	#2183	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	2	yes	maximal, complemented
6	#2162	C₂	1	1	yes	maximal, characteristic, complemented
7	#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₁₆	C₂	1	2	yes	maximal, complemented
9	C₄×#41	C₄	1	1	yes	characteristic
10	C₂×M₃₂	C₂²	1	2	yes	—
11	C₂²×C₁₆	C₂²	1	4	yes	—
12	C₈×Q₈	C₄	1	1	yes	characteristic
13	C₈×Q₈	C₂²	1	1	yes	characteristic
14	C₈×D₄	C₄	1	2	yes	—
15	C₈×D₄	C₄	1	1	yes	characteristic
16	C₈×D₄	C₂²	1	1	yes	characteristic
17	C₈×D₄	C₂²	1	2	yes	—
18	#431	C₄	1	1	yes	characteristic
19	#431	C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆	C₂²	1	2	yes	—
24	C₄⋊C₁₆	C₂²	1	4	yes	—
25	C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₂²⋊C₁₆	C₂²	1	4	yes	—
29	#346	C₂²	1	4	yes	—
30	C₄×C₁₆	C₂²	1	2	yes	—
31	C₂×#41	C₈	1	1	yes	characteristic
32	C₂²×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂³	1	2	yes	—
35	C₂²×C₈	C₂³	1	1	yes	characteristic
36	C₄×Q₈	C₈	1	1	yes	characteristic
37	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₄×D₄	C₈	1	2	yes	—
39	C₄×D₄	C₈	1	1	yes	characteristic
40	C₄×D₄	C₂×C₄	1	2	yes	—
41	C₄×D₄	C₂×C₄	1	1	yes	characteristic
42	#117	C₈	1	1	yes	characteristic
43	#117	C₈	1	2	yes	—
44	C₂×C₄²	C₂×C₄	1	2	yes	—
45	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
46	M₃₂	—	4	2	no	—
47	C₂×C₁₆	—	4	4	no	—
48	C₂×C₁₆	C₂³	1	2	yes	—
49	C₂×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₆	C₂³	1	4	yes	—
51	C₄⋊C₈	C₂×C₄	1	2	yes	—
52	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₂²⋊C₈	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈	C₂×C₄	1	2	yes	—
57	C₂²⋊C₈	C₂×C₄	1	2	yes	—
58	C₄×C₈	C₂×C₄	1	2	yes	—
59	C₄×C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈	C₂×C₄	1	2	yes	—
61	C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₈	C₂³	1	2	yes	—
64	C₄×C₈	C₂³	1	1	yes	characteristic
65	#41	—	2	4	no	—
66	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
67	C₂×D₄	C₂×C₈	1	2	yes	—
68	C₂×D₄	C₂×C₈	1	1	yes	characteristic
69	C₂²×C₄	C₂×C₈	1	2	yes	—
70	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×C₈	1	2	yes	—
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×C₄	1	2	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈	C₂²×C₄	1	2	yes	—
83	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₈	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄	C₂×C₈	1	2	yes	—
88	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
89	C₄⋊C₄	C₂×C₈	1	2	yes	—
90	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
91	C₂²⋊C₄	C₂×C₈	1	2	yes	—
92	C₂²⋊C₄	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄	C₂×C₈	1	2	yes	—
94	C₄²	C₂×C₈	1	2	yes	—
95	C₄²	C₂×C₈	1	1	yes	characteristic
96	C₄²	C₂×C₈	1	2	yes	—
97	C₄²	C₂×C₈	1	1	yes	characteristic
98	C₄²	C₂²×C₄	1	1	yes	characteristic
99	C₄²	C₂²×C₄	1	2	yes	—
100	C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₁₆	—	2	2	no	—
102	C₁₆	—	2	4	no	—
103	C₁₆	—	2	2	no	—
104	C₂³	C₂²×C₈	1	1	yes	characteristic
105	C₂³	C₂²×C₈	1	2	yes	—
106	Q₈	—	2	2	no	—
107	D₄	—	2	4	no	—
108	D₄	M₃₂	1	4	yes	complemented
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₈	1	2	yes	—
119	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄	C₂²×C₈	1	2	yes	—
121	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄	C₂²×C₈	1	2	yes	—
123	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	2	2	no	—
131	C₈	H(5,𝔽₂)	1	1	yes	characteristic
132	C₈	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	C₂×M₃₂	1	4	yes	—
139	C₂²	C₂³×C₈	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	2	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	1	no	—
146	C₄	C₂×M₃₂	1	2	yes	—
147	C₄	#504	1	2	yes	—
148	C₄	#520	1	1	yes	characteristic
149	C₄	#568	1	1	yes	characteristic
150	C₂	—	2	4	no	—
151	C₂	—	4	1	no	—
152	C₂	#2988	1	1	yes	characteristic
153	C₂	C₂²×M₃₂	1	1	yes	characteristic
154	C₂	C₂×#504	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂×#41	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. #2183	C₂	semidirect
3. #2168	C₂	semidirect
4. D₄×C₁₆	C₂	semidirect
5. #2163	C₂	semidirect
6. #2162	C₂	semidirect
7. #2150	C₂	semidirect
8. C₂×C₄⋊C₁₆	C₂	semidirect
108. D₄	M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41	2
#2183	2
#2168	2
D₄×C₁₆	2
#2163	2
#2162	2
#2150	2
C₂×C₄⋊C₁₆	2