Subgroups — #33509

Order 256.

242 conjugacy classes of subgroups.

The subgroup lattice has 163 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2966	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	2	yes	maximal, complemented
3	#2169	C₂	1	2	yes	maximal, complemented
4	#2168	C₂	1	2	yes	maximal, complemented
5	#2168	C₂	1	2	yes	maximal, complemented
6	#2164	C₂	1	2	yes	maximal, complemented
7	#2163	C₂	1	1	yes	maximal, characteristic, complemented
8	#2151	C₂	1	2	yes	maximal, complemented
9	#2150	C₂	1	1	yes	maximal, characteristic, complemented
10	#529	C₄	1	1	yes	characteristic
11	C₂×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×M₃₂	C₂²	1	2	yes	—
14	C₂²×C₁₆	C₂²	1	2	yes	—
15	#446	C₄	1	1	yes	characteristic
16	C₈×Q₈	C₂²	1	1	yes	characteristic
17	#436	C₄	1	1	yes	characteristic
18	#435	C₄	1	2	yes	—
19	C₈×D₄	C₂²	1	1	yes	characteristic
20	C₈×D₄	C₂²	1	2	yes	—
21	#433	C₄	1	2	yes	—
22	#432	C₄	1	1	yes	characteristic
23	#405	C₂²	1	2	yes	—
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₄⋊C₁₆	C₂²	1	2	yes	—
27	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
31	C₂²⋊C₁₆	C₂²	1	2	yes	—
32	C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₂²⋊C₁₆	C₂²	1	2	yes	—
34	#346	C₂²	1	2	yes	—
35	#346	C₂²	1	2	yes	—
36	C₄×C₁₆	C₂²	1	2	yes	—
37	C₂×M₁₆	C₂×C₄	1	2	yes	—
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	2	yes	—
41	#126	C₈	1	2	yes	—
42	#125	C₈	1	1	yes	characteristic
43	#124	C₈	1	1	yes	characteristic
44	#123	C₈	1	2	yes	—
45	C₂²⋊Q₈	C₈	1	1	yes	characteristic
46	#121	C₈	1	1	yes	characteristic
47	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄	C₂×C₄	1	2	yes	—
49	C₄×D₄	C₂×C₄	1	1	yes	characteristic
50	#117	C₂×C₄	1	2	yes	—
51	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
52	M₃₂	—	4	1	no	—
53	M₃₂	—	4	2	no	—
54	M₃₂	—	4	1	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₆	C₂³	1	2	yes	—
59	C₂×C₁₆	C₂³	1	2	yes	—
60	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈	C₂×C₄	1	2	yes	—
63	C₄⋊C₈	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈	C₂×C₄	1	2	yes	—
67	#97	C₂×C₄	1	2	yes	—
68	#97	C₂×C₄	1	2	yes	—
69	C₄×C₈	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂³	1	2	yes	—
72	C₄×C₈	C₂³	1	1	yes	characteristic
73	C₄×C₈	C₂³	1	1	yes	characteristic
74	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄	C₂×C₈	1	2	yes	—
76	C₂×D₄	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂²×C₄	1	2	yes	—
81	M₁₆	—	4	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₈	C₂²×C₄	1	2	yes	—
92	C₂×C₈	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄	C₂×C₈	1	2	yes	—
94	C₄⋊C₄	C₂×C₈	1	2	yes	—
95	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
96	C₄⋊C₄	C₂×C₈	1	2	yes	—
97	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
98	C₄⋊C₄	C₂×C₈	1	2	yes	—
99	C₂²⋊C₄	C₂×C₈	1	2	yes	—
100	C₂²⋊C₄	C₂×C₈	1	2	yes	—
101	C₂²⋊C₄	C₂×C₈	1	2	yes	—
102	C₂²⋊C₄	C₂×C₈	1	2	yes	—
103	C₂²⋊C₄	C₂×C₈	1	2	yes	—
104	C₄²	C₂×C₈	1	1	yes	characteristic
105	C₄²	C₂×C₈	1	1	yes	characteristic
106	C₄²	C₂²×C₄	1	1	yes	characteristic
107	C₄²	C₂²×C₄	1	1	yes	characteristic
108	C₄²	C₂²×C₄	1	2	yes	—
109	C₁₆	—	2	2	no	—
110	C₁₆	—	2	2	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	2	no	—
113	C₂³	C₂²×C₈	1	2	yes	—
114	C₂³	C₂²×C₈	1	1	yes	characteristic
115	Q₈	—	4	1	no	—
116	D₄	—	4	1	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄	C₂²×C₈	1	2	yes	—
130	C₂×C₄	C₂²×C₈	1	2	yes	—
131	C₂×C₄	C₂²×C₈	1	2	yes	—
132	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	2	2	no	—
138	C₈	H(5,𝔽₂)	1	1	yes	characteristic
139	C₈	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	2	2	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂³×C₈	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	2	no	—
155	C₄	#520	1	1	yes	characteristic
156	C₄	#568	1	1	yes	characteristic
157	C₂	—	4	1	no	—
158	C₂	—	4	2	no	—
159	C₂	#2988	1	1	yes	characteristic
160	C₂	#3407	1	1	yes	characteristic
161	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	#121	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. #2169	C₂	semidirect
4. #2168	C₂	semidirect
5. #2168	C₂	semidirect
6. #2164	C₂	semidirect
7. #2163	C₂	semidirect
8. #2151	C₂	semidirect
9. #2150	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2966	2
#2183	2
#2169	2
#2168	2
#2164	2
#2163	2
#2151	2
#2150	2