Subgroups — #17796

Order 256.

378 conjugacy classes of subgroups.

The subgroup lattice has 206 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×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#1569	C₂	1	4	yes	maximal, complemented
5	#1567	C₂	1	2	yes	maximal, complemented
6	#1566	C₂	1	2	yes	maximal, complemented
7	#1565	C₂	1	4	yes	maximal, complemented
8	C₂²×M₁₆	—	2	1	no	—
9	C₂²×M₁₆	C₄	1	2	yes	—
10	C₂²×C₄²	C₂²	1	1	yes	characteristic
11	#432	—	2	1	no	—
12	#432	C₂²	1	2	yes	—
13	#431	—	2	1	no	—
14	#431	C₂²	1	2	yes	—
15	#423	—	2	1	no	—
16	#423	—	2	1	no	—
17	#423	C₂²	1	1	yes	characteristic
18	#423	C₂²	1	2	yes	—
19	#423	C₂²	1	1	yes	characteristic
20	#423	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈	—	2	1	no	—
22	C₂×C₄⋊C₈	C₂²	1	2	yes	—
23	#407	—	2	1	no	—
24	C₄×M₁₆	—	2	1	no	—
25	C₄×M₁₆	C₄	1	1	yes	characteristic
26	C₄×M₁₆	C₄	1	2	yes	—
27	C₄×M₁₆	C₄	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	#335	C₂²	1	8	yes	complemented
33	#334	C₂²	1	8	yes	complemented
34	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄	D₄	1	1	yes	characteristic
36	C₂³×C₄	Q₈	1	1	yes	characteristic
37	C₂×M₁₆	—	2	8	no	—
38	C₂×M₁₆	—	2	2	no	—
39	C₂×M₁₆	—	2	2	no	—
40	C₂×M₁₆	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	4	4	no	—
43	C₂×M₁₆	C₂×C₄	1	2	yes	—
44	C₂×M₁₆	C₂×C₄	1	4	yes	—
45	C₂×M₁₆	C₂×C₄	1	2	yes	—
46	C₂²×C₈	—	2	2	no	—
47	C₂²×C₈	C₂×C₄	1	4	yes	—
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	D₄	1	1	yes	characteristic
52	C₂×C₄²	D₄	1	1	yes	characteristic
53	C₂×C₄²	D₄	1	1	yes	characteristic
54	C₂×C₄²	Q₈	1	1	yes	characteristic
55	C₂×C₄²	Q₈	1	1	yes	characteristic
56	C₂×C₄²	Q₈	1	1	yes	characteristic
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	C₂³	1	2	yes	—
66	C₄⋊C₈	C₂³	1	4	yes	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₂²⋊C₈	—	2	2	no	—
69	C₂²⋊C₈	—	2	2	no	—
70	#97	—	2	2	no	—
71	#97	C₂×C₄	1	2	yes	—
72	#97	C₂×C₄	1	4	yes	—
73	#97	C₂×C₄	1	2	yes	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	C₂×C₄	1	2	yes	—
76	C₄×C₈	C₂×C₄	1	2	yes	—
77	C₄×C₈	C₂³	1	4	yes	—
78	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄	—	2	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	4	2	no	—
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	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
99	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
101	M₁₆	—	2	4	no	—
102	M₁₆	—	2	8	no	—
103	M₁₆	—	4	4	no	—
104	M₁₆	—	4	2	no	—
105	M₁₆	—	4	4	no	—
106	M₁₆	—	4	2	no	—
107	M₁₆	M₁₆	1	8	yes	complemented
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	4	4	no	—
113	C₂×C₈	C₂²×C₄	1	4	yes	—
114	C₂×C₈	C₂²×C₄	1	4	yes	—
115	C₄²	C₄⋊C₄	1	1	yes	characteristic
116	C₄²	C₄⋊C₄	1	1	yes	characteristic
117	C₄²	C₄⋊C₄	1	1	yes	characteristic
118	C₄²	C₄⋊C₄	1	1	yes	characteristic
119	C₄²	C₄⋊C₄	1	1	yes	characteristic
120	C₄²	C₄⋊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	C₂×D₄	1	1	yes	characteristic
125	C₄²	C₂×D₄	1	1	yes	characteristic
126	C₄²	C₂×D₄	1	1	yes	characteristic
127	C₄²	C₂×Q₈	1	1	yes	characteristic
128	C₄²	C₂×Q₈	1	1	yes	characteristic
129	C₄²	C₂×Q₈	1	1	yes	characteristic
130	C₄²	C₂⁴	1	1	yes	characteristic
131	C₂³	—	2	2	no	—
132	C₂³	—	2	2	no	—
133	C₂³	—	4	1	no	—
134	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
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	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
161	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄	C₂×M₁₆	1	4	yes	—
165	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
166	C₂×C₄	#137	1	2	yes	—
167	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
170	C₈	—	2	4	no	—
171	C₈	—	4	4	no	—
172	C₈	—	4	2	no	—
173	C₈	—	4	2	no	—
174	C₈	C₂×M₁₆	1	8	yes	complemented
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	1	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	4	4	no	—
183	C₂²	#423	1	2	yes	—
184	C₂²	#428	1	2	yes	—
185	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	1	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	#423	1	2	yes	—
194	C₄	#428	1	2	yes	—
195	C₄	#430	1	2	yes	—
196	C₄	C₂²×M₁₆	1	2	yes	—
197	C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄	C₂×#137	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	—	2	2	no	—
201	C₂	—	4	1	no	—
202	C₂	C₂×#423	1	1	yes	characteristic
203	C₂	C₂×#428	1	1	yes	characteristic
204	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. #1569	C₂	semidirect
5. #1567	C₂	semidirect
6. #1566	C₂	semidirect
7. #1565	C₂	semidirect
32. #335	C₂²	semidirect
33. #334	C₂²	semidirect
107. M₁₆	M₁₆	semidirect
174. C₈	C₂×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×#423	2
C₂×C₄×M₁₆	2
#1569	2
#1567	2
#1566	2
#1565	2