Subgroups — #17822

Order 256.

332 conjugacy classes of subgroups.

The subgroup lattice has 196 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×#432	C₂	1	1	yes	maximal, characteristic
3	C₂×#423	C₂	1	1	yes	maximal, characteristic
4	#1573	C₂	1	4	yes	maximal, complemented
5	#1571	C₂	1	2	yes	maximal, complemented
6	#1570	C₂	1	2	yes	maximal, complemented
7	#1569	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆	—	2	1	no	—
9	C₂²×C₄²	C₂²	1	1	yes	characteristic
10	#432	—	2	1	no	—
11	#432	—	2	1	no	—
12	#432	C₄	1	2	yes	—
13	#432	C₄	1	2	yes	—
14	#432	C₂²	1	2	yes	—
15	#432	C₂²	1	2	yes	—
16	#431	C₂²	1	2	yes	—
17	#431	C₂²	1	2	yes	—
18	#423	—	2	1	no	—
19	#423	—	2	1	no	—
20	#423	C₄	1	2	yes	—
21	#423	C₂²	1	2	yes	—
22	#423	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈	—	2	1	no	—
24	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈	C₂²	1	2	yes	—
27	#407	C₄	1	2	yes	—
28	C₂×C₂²⋊C₈	—	2	1	no	—
29	C₄×M₁₆	C₂²	1	2	yes	—
30	C₂×#97	—	2	1	no	—
31	#335	C₂²	1	8	yes	complemented
32	#334	C₂²	1	8	yes	complemented
33	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄	D₄	1	1	yes	characteristic
35	C₂³×C₄	Q₈	1	1	yes	characteristic
36	C₂×M₁₆	—	2	2	no	—
37	C₂×M₁₆	—	2	2	no	—
38	C₂×M₁₆	—	4	4	no	—
39	C₂×M₁₆	C₂×C₄	1	4	yes	—
40	C₂²×C₈	—	2	2	no	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	2	1	no	—
43	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²	D₄	1	1	yes	characteristic
47	C₂×C₄²	D₄	1	1	yes	characteristic
48	C₂×C₄²	D₄	1	1	yes	characteristic
49	C₂×C₄²	Q₈	1	1	yes	characteristic
50	C₂×C₄²	Q₈	1	1	yes	characteristic
51	C₂×C₄²	Q₈	1	1	yes	characteristic
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	C₂×C₄²	C₂³	1	1	yes	characteristic
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	2	2	no	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	2	1	no	—
59	C₄⋊C₈	—	2	1	no	—
60	C₄⋊C₈	C₂×C₄	1	4	yes	—
61	C₄⋊C₈	C₂×C₄	1	4	yes	—
62	C₄⋊C₈	C₂³	1	2	yes	—
63	C₄⋊C₈	C₂³	1	4	yes	—
64	C₄⋊C₈	C₂³	1	2	yes	—
65	C₂²⋊C₈	—	2	2	no	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈	C₂×C₄	1	4	yes	—
69	#97	—	2	2	no	—
70	#97	—	2	2	no	—
71	#97	C₂×C₄	1	4	yes	—
72	C₄×C₈	C₂³	1	4	yes	—
73	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	4	2	no	—
77	C₂²×C₄	—	4	2	no	—
78	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
96	M₁₆	—	4	4	no	—
97	M₁₆	—	4	4	no	—
98	M₁₆	—	4	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	4	no	—
102	C₂×C₈	—	4	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	C₂²×C₄	1	8	yes	—
106	C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₄²	C₄⋊C₄	1	1	yes	characteristic
108	C₄²	C₄⋊C₄	1	1	yes	characteristic
109	C₄²	C₄⋊C₄	1	1	yes	characteristic
110	C₄²	C₄⋊C₄	1	1	yes	characteristic
111	C₄²	C₄⋊C₄	1	1	yes	characteristic
112	C₄²	C₂²×C₄	1	1	yes	characteristic
113	C₄²	C₂²×C₄	1	1	yes	characteristic
114	C₄²	C₂²×C₄	1	1	yes	characteristic
115	C₄²	C₂×D₄	1	1	yes	characteristic
116	C₄²	C₂×D₄	1	1	yes	characteristic
117	C₄²	C₂×D₄	1	1	yes	characteristic
118	C₄²	C₂×Q₈	1	1	yes	characteristic
119	C₄²	C₂×Q₈	1	1	yes	characteristic
120	C₄²	C₂×Q₈	1	1	yes	characteristic
121	C₄²	C₂⁴	1	1	yes	characteristic
122	C₂³	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	—	4	1	no	—
125	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	4	1	no	—
144	C₂×C₄	—	4	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
157	C₂×C₄	#137	1	2	yes	—
158	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
161	C₈	—	2	8	no	—
162	C₈	—	4	2	no	—
163	C₈	—	4	4	no	—
164	C₈	—	4	2	no	—
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	—	4	4	no	—
173	C₂²	#428	1	2	yes	—
174	C₂²	#430	1	2	yes	—
175	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	#428	1	2	yes	—
186	C₄	#568	1	2	yes	—
187	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
188	C₄	C₂×#137	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	—	2	2	no	—
191	C₂	—	4	1	no	—
192	C₂	#2905	1	1	yes	characteristic
193	C₂	C₂×#428	1	1	yes	characteristic
194	C₂	C₂×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1573	C₂	semidirect
5. #1571	C₂	semidirect
6. #1570	C₂	semidirect
7. #1569	C₂	semidirect
31. #335	C₂²	semidirect
32. #334	C₂²	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×#432	2
C₂×#423	2
#1573	2
#1571	2
#1570	2
#1569	2