Subgroups — #17327

Order 256.

418 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#1453	C₂	1	4	yes	maximal, complemented
5	#1451	C₂	1	2	yes	maximal, complemented
6	#1451	C₂	1	2	yes	maximal, complemented
7	C₈×M₁₆	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆	C₄	1	2	yes	—
10	C₂²×C₄²	C₂²	1	1	yes	characteristic
11	#432	C₄	1	2	yes	—
12	#432	C₄	1	2	yes	—
13	#431	C₄	1	2	yes	—
14	#431	C₄	1	2	yes	—
15	#431	C₂²	1	1	yes	characteristic
16	#431	C₂²	1	1	yes	characteristic
17	#431	C₂²	1	2	yes	—
18	#431	C₂²	1	2	yes	—
19	#431	C₂²	1	2	yes	—
20	#423	C₄	1	2	yes	—
21	C₂×C₄⋊C₈	C₄	1	2	yes	—
22	#407	C₄	1	2	yes	—
23	C₂×C₂²⋊C₈	C₄	1	2	yes	—
24	C₄×M₁₆	C₄	1	2	yes	—
25	C₄×M₁₆	C₄	1	2	yes	—
26	C₄×M₁₆	C₂²	1	1	yes	characteristic
27	C₄×M₁₆	C₂²	1	1	yes	characteristic
28	C₄×M₁₆	C₂²	1	2	yes	—
29	C₄×M₁₆	C₂²	1	2	yes	—
30	C₂×#97	C₄	1	2	yes	—
31	C₂×C₄×C₈	C₂²	1	2	yes	—
32	C₂×C₄×C₈	C₂²	1	2	yes	—
33	#322	C₂²	1	4	yes	complemented
34	#322	C₂²	1	4	yes	complemented
35	#322	C₂²	1	4	yes	complemented
36	C₈²	C₂²	1	4	yes	complemented
37	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆	—	2	8	no	—
41	C₂×M₁₆	C₂×C₄	1	2	yes	—
42	C₂×M₁₆	C₂×C₄	1	4	yes	—
43	C₂×M₁₆	C₂×C₄	1	4	yes	—
44	C₂×M₁₆	C₂×C₄	1	2	yes	—
45	C₂²×C₈	C₂×C₄	1	4	yes	—
46	C₂²×C₈	C₂×C₄	1	4	yes	—
47	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₄⋊C₈	C₂×C₄	1	4	yes	—
64	C₂²⋊C₈	C₂×C₄	1	2	yes	—
65	C₂²⋊C₈	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈	C₂×C₄	1	4	yes	—
67	C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	#97	C₂×C₄	1	2	yes	—
70	#97	C₂×C₄	1	4	yes	—
71	#97	C₂×C₄	1	2	yes	—
72	#97	C₂×C₄	1	4	yes	—
73	C₄×C₈	C₂×C₄	1	4	yes	—
74	C₄×C₈	C₂×C₄	1	4	yes	—
75	C₄×C₈	C₂³	1	2	yes	—
76	C₄×C₈	C₂³	1	2	yes	—
77	C₄×C₈	C₂³	1	4	yes	—
78	C₄×C₈	C₂³	1	2	yes	—
79	C₄×C₈	C₂³	1	2	yes	—
80	C₂⁴	C₄²	1	1	yes	characteristic
81	C₂²×C₄	—	2	4	no	—
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	—	4	2	no	—
84	C₂²×C₄	C₄²	1	1	yes	characteristic
85	C₂²×C₄	C₄²	1	1	yes	characteristic
86	C₂²×C₄	C₄²	1	1	yes	characteristic
87	C₂²×C₄	C₄²	1	1	yes	characteristic
88	C₂²×C₄	C₄²	1	1	yes	characteristic
89	C₂²×C₄	C₄²	1	1	yes	characteristic
90	C₂²×C₄	C₄²	1	1	yes	characteristic
91	C₂²×C₄	C₄²	1	1	yes	characteristic
92	C₂²×C₄	C₄²	1	1	yes	characteristic
93	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
102	M₁₆	—	2	4	no	—
103	M₁₆	—	2	8	no	—
104	M₁₆	—	2	8	no	—
105	M₁₆	M₁₆	1	8	yes	complemented
106	C₂×C₈	—	2	8	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	C₂²×C₄	1	4	yes	—
109	C₂×C₈	C₂²×C₄	1	4	yes	—
110	C₂×C₈	C₂²×C₄	1	4	yes	—
111	C₂×C₈	C₂²×C₄	1	8	yes	—
112	C₂×C₈	C₂²×C₄	1	4	yes	—
113	C₄²	C₄²	1	1	yes	characteristic
114	C₄²	C₄²	1	1	yes	characteristic
115	C₄²	C₄²	1	1	yes	characteristic
116	C₄²	C₄²	1	1	yes	characteristic
117	C₄²	C₄²	1	1	yes	characteristic
118	C₄²	C₄²	1	1	yes	characteristic
119	C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₄²	C₂²×C₄	1	1	yes	characteristic
121	C₄²	C₂²×C₄	1	1	yes	characteristic
122	C₄²	C₂²×C₄	1	1	yes	characteristic
123	C₄²	C₂²×C₄	1	1	yes	characteristic
124	C₄²	C₂²×C₄	1	1	yes	characteristic
125	C₄²	C₂²×C₄	1	1	yes	characteristic
126	C₄²	C₂²×C₄	1	1	yes	characteristic
127	C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₄²	C₂⁴	1	1	yes	characteristic
129	C₂³	—	2	2	no	—
130	C₂³	—	2	2	no	—
131	C₂³	—	4	1	no	—
132	C₂³	C₂×C₄²	1	1	yes	characteristic
133	C₂³	C₂×C₄²	1	1	yes	characteristic
134	C₂³	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	4	1	no	—
150	C₂×C₄	—	4	1	no	—
151	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
160	C₂×C₄	C₂×M₁₆	1	4	yes	—
161	C₂×C₄	#131	1	4	yes	—
162	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
163	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
164	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
165	C₈	—	2	4	no	—
166	C₈	—	2	8	no	—
167	C₈	—	2	4	no	—
168	C₈	—	2	4	no	—
169	C₈	C₂×M₁₆	1	8	yes	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	4	4	no	—
178	C₂²	C₄×M₁₆	1	2	yes	—
179	C₂²	#405	1	2	yes	—
180	C₂²	C₂²×C₄²	1	1	yes	characteristic
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	C₄×M₁₆	1	2	yes	—
189	C₄	#405	1	2	yes	—
190	C₄	C₂²×M₁₆	1	2	yes	—
191	C₄	C₂×#131	1	2	yes	—
192	C₄	#568	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	—	2	2	no	—
195	C₂	—	4	1	no	—
196	C₂	C₂×C₄×M₁₆	1	1	yes	characteristic
197	C₂	C₂×#405	1	1	yes	characteristic
198	C₂	#2873	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₄²	C₂⁴
commutator, socle	C₂²	C₂²×C₄²
prime radical	C₂⁴	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1453	C₂	semidirect
5. #1451	C₂	semidirect
6. #1451	C₂	semidirect
7. C₈×M₁₆	C₂	semidirect
8. C₈×M₁₆	C₂	semidirect
33. #322	C₂²	semidirect
34. #322	C₂²	semidirect
35. #322	C₂²	semidirect
36. C₈²	C₂²	semidirect
105. M₁₆	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#2920	2
C₂×#431	2
C₂×C₄×M₁₆	2
#1453	2
#1451	2
C₈×M₁₆	2