Subgroups — #33508

Order 256.

251 conjugacy classes of subgroups.

The subgroup lattice has 170 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	1	yes	maximal, characteristic, complemented
3	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#2169	C₂	1	1	yes	maximal, characteristic, complemented
5	#2168	C₂	1	2	yes	maximal, complemented
6	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆	C₂	1	2	yes	maximal, complemented
8	#2164	C₂	1	2	yes	maximal, complemented
9	#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	#2151	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#529	C₄	1	1	yes	characteristic
13	C₂×M₃₂	C₂²	1	2	yes	—
14	C₂²×C₁₆	C₂²	1	2	yes	—
15	C₂²×C₁₆	C₂²	1	2	yes	—
16	#446	C₄	1	1	yes	characteristic
17	C₈×Q₈	C₂²	1	1	yes	characteristic
18	#436	C₄	1	1	yes	characteristic
19	#435	C₄	1	2	yes	—
20	C₈×D₄	C₂²	1	2	yes	—
21	C₈×D₄	C₂²	1	1	yes	characteristic
22	#433	C₄	1	2	yes	—
23	#432	C₄	1	1	yes	characteristic
24	#405	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₄⋊C₁₆	C₂²	1	2	yes	—
27	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₂²⋊C₁₆	C₂²	1	2	yes	—
36	C₂²⋊C₁₆	C₂²	1	2	yes	—
37	#346	C₂²	1	2	yes	—
38	C₄×C₁₆	C₂²	1	1	yes	characteristic
39	C₄×C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₁₆	C₂²	1	2	yes	—
41	C₂×M₁₆	C₂×C₄	1	2	yes	—
42	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	2	yes	—
45	#126	C₈	1	2	yes	—
46	#125	C₈	1	1	yes	characteristic
47	#124	C₈	1	1	yes	characteristic
48	#123	C₈	1	2	yes	—
49	C₂²⋊Q₈	C₈	1	1	yes	characteristic
50	#121	C₈	1	1	yes	characteristic
51	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄	C₂×C₄	1	2	yes	—
53	C₄×D₄	C₂×C₄	1	1	yes	characteristic
54	#117	C₂×C₄	1	2	yes	—
55	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
56	M₃₂	—	4	2	no	—
57	C₂×C₁₆	—	4	2	no	—
58	C₂×C₁₆	—	4	2	no	—
59	C₂×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₆	C₂³	1	2	yes	—
65	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈	C₂×C₄	1	2	yes	—
68	C₄⋊C₈	C₂×C₄	1	2	yes	—
69	C₂²⋊C₈	C₂×C₄	1	2	yes	—
70	C₂²⋊C₈	C₂×C₄	1	2	yes	—
71	C₂²⋊C₈	C₂×C₄	1	2	yes	—
72	#97	C₂×C₄	1	2	yes	—
73	#97	C₂×C₄	1	2	yes	—
74	C₄×C₈	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈	C₂³	1	2	yes	—
78	C₄×C₈	C₂³	1	1	yes	characteristic
79	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
80	C₂×D₄	C₂×C₈	1	2	yes	—
81	C₂×D₄	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄	1	2	yes	—
86	M₁₆	—	4	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	4	1	no	—
92	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₈	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄	C₂×C₈	1	2	yes	—
99	C₄⋊C₄	C₂×C₈	1	2	yes	—
100	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
101	C₄⋊C₄	C₂×C₈	1	2	yes	—
102	C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
103	C₄⋊C₄	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄	C₂×C₈	1	2	yes	—
105	C₂²⋊C₄	C₂×C₈	1	2	yes	—
106	C₂²⋊C₄	C₂×C₈	1	2	yes	—
107	C₂²⋊C₄	C₂×C₈	1	2	yes	—
108	C₂²⋊C₄	C₂×C₈	1	2	yes	—
109	C₄²	C₂×C₈	1	1	yes	characteristic
110	C₄²	C₂×C₈	1	1	yes	characteristic
111	C₄²	C₂²×C₄	1	1	yes	characteristic
112	C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₄²	C₂²×C₄	1	2	yes	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	2	2	no	—
116	C₁₆	—	2	2	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	2	1	no	—
119	C₁₆	—	2	1	no	—
120	C₂³	C₂²×C₈	1	2	yes	—
121	C₂³	C₂²×C₈	1	1	yes	characteristic
122	Q₈	—	2	2	no	—
123	D₄	—	2	4	no	—
124	D₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄	C₂²×C₈	1	2	yes	—
136	C₂×C₄	C₂²×C₈	1	2	yes	—
137	C₂×C₄	C₂²×C₈	1	2	yes	—
138	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	2	2	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	H(5,𝔽₂)	1	1	yes	characteristic
146	C₈	2₋¹⁺⁴	1	1	yes	characteristic
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	#504	1	2	yes	—
153	C₂²	C₂³×C₈	1	1	yes	characteristic
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	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	#504	1	2	yes	—
162	C₄	#520	1	1	yes	characteristic
163	C₄	#568	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	—	4	2	no	—
166	C₂	#2988	1	1	yes	characteristic
167	C₂	C₂×#504	1	1	yes	characteristic
168	C₂	C₂×#504	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. Q₈×C₁₆	C₂	semidirect
4. #2169	C₂	semidirect
5. #2168	C₂	semidirect
6. D₄×C₁₆	C₂	semidirect
7. D₄×C₁₆	C₂	semidirect
8. #2164	C₂	semidirect
9. #2162	C₂	semidirect
10. #2151	C₂	semidirect
11. C₂×C₄⋊C₁₆	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
Q₈×C₁₆	2
#2169	2
#2168	2
D₄×C₁₆	2
#2164	2
#2162	2
#2151	2
C₂×C₄⋊C₁₆	2