Subgroups — #33438

Order 256.

341 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	2	yes	maximal, complemented
2	#2931	C₂	1	1	yes	maximal, characteristic
3	#2169	C₂	1	2	yes	maximal, complemented
4	#2168	C₂	1	4	yes	maximal, complemented
5	D₄×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#2113	C₂	1	2	yes	maximal, complemented
8	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131	C₂²	1	2	yes	—
10	#532	C₄	1	1	yes	characteristic
11	#504	—	2	8	no	—
12	C₂×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×M₃₂	C₂²	1	4	yes	—
14	C₂×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆	C₂²	1	2	yes	—
16	C₂²×C₁₆	C₂²	1	4	yes	—
17	#436	C₄	1	2	yes	—
18	#436	C₄	1	2	yes	—
19	C₈×D₄	C₂²	1	2	yes	—
20	C₈×D₄	C₂²	1	2	yes	—
21	#423	C₄	1	1	yes	characteristic
22	#408	C₄	1	2	yes	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₂²⋊C₁₆	C₂²	1	4	yes	—
29	#346	C₂²	1	2	yes	—
30	C₄×C₁₆	C₂²	1	2	yes	—
31	C₂×#41	C₂×C₄	1	2	yes	—
32	#131	—	2	4	no	—
33	#131	—	2	4	no	—
34	C₂×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×M₁₆	C₂×C₄	1	4	yes	—
36	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂³	1	4	yes	—
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈	C₈	1	1	yes	characteristic
40	#127	C₈	1	1	yes	characteristic
41	#124	C₈	1	2	yes	—
42	#121	C₈	1	2	yes	—
43	#121	C₈	1	2	yes	—
44	C₄×D₄	C₂×C₄	1	2	yes	—
45	C₄×D₄	C₂×C₄	1	2	yes	—
46	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
47	M₃₂	—	2	2	no	—
48	M₃₂	—	2	8	no	—
49	M₃₂	D₄	1	4	yes	complemented
50	C₂×C₁₆	—	2	4	no	—
51	C₂×C₁₆	—	2	8	no	—
52	C₂×C₁₆	C₂³	1	2	yes	—
53	C₂×C₁₆	C₂³	1	2	yes	—
54	C₂×C₁₆	C₂³	1	2	yes	—
55	C₂×C₁₆	C₂³	1	2	yes	—
56	C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₄⋊C₈	C₂×C₄	1	2	yes	—
58	C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈	C₂×C₄	1	4	yes	—
60	C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₈	C₂³	1	1	yes	characteristic
64	#41	—	2	4	no	—
65	#41	—	2	4	no	—
66	C₂×Q₈	C₂×C₈	1	2	yes	—
67	C₂×D₄	C₂×C₈	1	2	yes	—
68	C₂×D₄	C₂×C₈	1	4	yes	—
69	C₂×D₄	C₂×C₈	1	2	yes	—
70	C₂×D₄	C₂×C₈	1	2	yes	—
71	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄	C₂²×C₄	1	4	yes	—
74	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	M₁₆	—	2	8	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈	C₂²×C₄	1	4	yes	—
83	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈	C₂×D₄	1	2	yes	—
85	C₂×C₈	#41	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	2	yes	—
89	C₂²⋊C₄	C₂×C₈	1	4	yes	—
90	C₂²⋊C₄	C₂×C₈	1	4	yes	—
91	C₄²	C₂×C₈	1	1	yes	characteristic
92	C₄²	C₂×C₈	1	1	yes	characteristic
93	C₄²	C₂²×C₄	1	1	yes	characteristic
94	C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₁₆	—	2	2	no	—
97	C₁₆	—	2	2	no	—
98	C₁₆	C₂×D₄	1	4	yes	—
99	C₂³	C₂²×C₈	1	4	yes	—
100	C₂³	C₂²×C₈	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	D₄	—	2	8	no	—
103	D₄	—	2	4	no	—
104	D₄	—	2	4	no	—
105	D₄	—	4	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	C₄×D₄	1	2	yes	—
111	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄	C₂²×C₈	1	4	yes	—
115	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄	#131	1	2	yes	—
118	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	C₄×D₄	1	2	yes	—
123	C₈	C₂²×D₄	1	1	yes	characteristic
124	C₈	C₂×#41	1	1	yes	characteristic
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	4	no	—
130	C₂²	C₈×D₄	1	2	yes	—
131	C₂²	C₂³×C₈	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	4	no	—
136	C₄	C₈×D₄	1	2	yes	—
137	C₄	#504	1	2	yes	complemented
138	C₄	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄	C₂×#131	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	—	4	4	no	—
142	C₂	C₂×C₈×D₄	1	1	yes	characteristic
143	C₂	C₂×#504	1	1	yes	characteristic
144	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#532	C₄
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
1. C₂×#504	C₂	semidirect
3. #2169	C₂	semidirect
4. #2168	C₂	semidirect
5. D₄×C₁₆	C₂	semidirect
6. C₂×C₄⋊C₁₆	C₂	semidirect
7. #2113	C₂	semidirect
8. C₄×M₃₂	C₂	semidirect
49. M₃₂	D₄	semidirect
137. C₄	#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504	2
#2931	2
#2169	2
#2168	2
D₄×C₁₆	2
C₂×C₄⋊C₁₆	2
#2113	2
C₄×M₃₂	2