Subgroups — #8095

Order 256.

337 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×#423	C₂	1	1	yes	maximal, characteristic
3	#2285	C₂	1	1	yes	maximal, characteristic
4	#1303	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆	—	2	1	no	—
6	C₂×#117	—	2	1	no	—
7	C₂²×C₄²	C₂²	1	1	yes	characteristic
8	#431	—	2	1	no	—
9	#431	—	2	1	no	—
10	#431	C₄	1	2	yes	complemented
11	#431	C₂²	1	2	yes	—
12	#423	—	2	1	no	—
13	#423	—	2	1	no	—
14	#423	C₄	1	2	yes	complemented
15	#423	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈	—	2	1	no	—
17	C₂×C₄⋊C₈	C₄	1	2	yes	complemented
18	C₂×C₄⋊C₈	C₄	1	2	yes	complemented
19	C₂×C₂²⋊C₈	—	2	1	no	—
20	C₂×C₄×C₈	—	2	1	no	—
21	#384	—	2	1	no	—
22	#382	C₄	1	2	yes	—
23	#380	—	2	1	no	—
24	#380	C₄	1	2	yes	—
25	C₄×C₄⋊C₄	—	2	1	no	—
26	C₄×C₄⋊C₄	C₂²	1	2	yes	—
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄	D₄	1	1	yes	characteristic
29	C₂³×C₄	D₄	1	1	yes	characteristic
30	C₂×M₁₆	—	2	2	no	—
31	C₂×M₁₆	—	2	2	no	—
32	C₂×M₁₆	—	4	2	no	—
33	C₂×M₁₆	—	4	2	no	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	—	2	2	no	—
36	#117	—	4	2	no	—
37	#117	—	4	2	no	—
38	C₂×C₄⋊C₄	—	2	2	no	—
39	C₂×C₄⋊C₄	—	2	2	no	—
40	C₂×C₄⋊C₄	—	2	2	no	—
41	C₂×C₂²⋊C₄	—	2	2	no	—
42	C₂×C₄²	—	2	2	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	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²	D₄	1	1	yes	characteristic
49	C₂×C₄²	D₄	1	1	yes	characteristic
50	C₂×C₄²	D₄	1	1	yes	characteristic
51	C₂×C₄²	D₄	1	1	yes	characteristic
52	C₂×C₄²	Q₈	1	1	yes	characteristic
53	C₂×C₄²	Q₈	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	2	no	—
59	C₄⋊C₈	C₂×C₄	1	4	yes	—
60	C₄⋊C₈	C₂×C₄	1	4	yes	—
61	C₂²⋊C₈	—	2	2	no	—
62	C₂²⋊C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	#95	—	2	2	no	—
66	#95	C₂×C₄	1	4	yes	—
67	C₂⁴	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄	—	2	4	no	—
69	C₂²×C₄	—	4	2	no	—
70	C₂²×C₄	—	4	2	no	—
71	C₂²×C₄	C₄²	1	1	yes	characteristic
72	C₂²×C₄	C₄²	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊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	D₈	1	2	yes	—
87	C₂²×C₄	SD₁₆	1	2	yes	—
88	C₂²×C₄	SD₁₆	1	2	yes	—
89	C₂²×C₄	Q₁₆	1	2	yes	—
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
93	M₁₆	—	4	4	no	—
94	M₁₆	—	4	2	no	—
95	M₁₆	—	4	2	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	4	4	no	—
99	C₂×C₈	—	4	4	no	—
100	C₄⋊C₄	—	4	4	no	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	4	4	no	—
103	C₂²⋊C₄	—	4	4	no	—
104	C₄²	—	4	2	no	—
105	C₄²	—	4	2	no	—
106	C₄²	C₄²	1	1	yes	characteristic
107	C₄²	C₄²	1	1	yes	characteristic
108	C₄²	C₂²⋊C₄	1	1	yes	characteristic
109	C₄²	C₂²⋊C₄	1	1	yes	characteristic
110	C₄²	C₂²⋊C₄	1	1	yes	characteristic
111	C₄²	C₂²⋊C₄	1	1	yes	characteristic
112	C₄²	C₄⋊C₄	1	1	yes	characteristic
113	C₄²	C₄⋊C₄	1	1	yes	characteristic
114	C₄²	C₄⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
119	C₄²	C₂²×C₄	1	1	yes	characteristic
120	C₄²	C₂×D₄	1	1	yes	characteristic
121	C₄²	C₂×Q₈	1	1	yes	characteristic
122	C₂³	—	4	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	#95	1	1	yes	characteristic
125	C₂³	#95	1	1	yes	characteristic
126	C₂³	#102	1	2	yes	—
127	C₂³	#103	1	2	yes	—
128	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	#95	1	1	yes	characteristic
145	C₂×C₄	#95	1	1	yes	characteristic
146	C₂×C₄	#95	1	1	yes	characteristic
147	C₂×C₄	#95	1	1	yes	characteristic
148	C₂×C₄	#95	1	1	yes	characteristic
149	C₂×C₄	#95	1	1	yes	characteristic
150	C₂×C₄	#102	1	2	yes	—
151	C₂×C₄	#102	1	2	yes	—
152	C₂×C₄	#102	1	2	yes	—
153	C₂×C₄	#103	1	2	yes	—
154	C₂×C₄	#103	1	2	yes	—
155	C₂×C₄	#103	1	2	yes	—
156	C₂×C₄	#106	1	2	yes	—
157	C₂×C₄	#106	1	2	yes	—
158	C₂×C₄	#107	1	2	yes	—
159	C₂×C₄	#107	1	2	yes	—
160	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
161	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₄	C₂×D₈	1	1	yes	characteristic
167	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
168	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
169	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	4	4	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	—	2	1	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	#340	1	2	yes	—
181	C₂²	#340	1	2	yes	—
182	C₂²	C₂×#95	1	1	yes	characteristic
183	C₂²	C₂×#102	1	1	yes	characteristic
184	C₂²	C₂×#103	1	1	yes	characteristic
185	C₄	—	2	1	no	—
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	—	8	4	no	—
192	C₄	#340	1	2	yes	—
193	C₄	#340	1	2	yes	—
194	C₄	#344	1	2	yes	—
195	C₄	C₂×#102	1	1	yes	characteristic
196	C₄	C₂×#103	1	1	yes	characteristic
197	C₄	#421	1	2	yes	—
198	C₄	C₂×#106	1	1	yes	characteristic
199	C₄	C₂×#107	1	1	yes	characteristic
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	—	2	2	no	—
203	C₂	—	2	1	no	—
204	C₂	#1733	1	1	yes	characteristic
205	C₂	C₂×#340	1	1	yes	characteristic
206	C₂	C₂×#344	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₂×#95
prime radical	C₂⁴	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1303	C₂	semidirect
10. #431	C₄	semidirect
14. #423	C₄	semidirect
17. C₂×C₄⋊C₈	C₄	semidirect
18. C₂×C₄⋊C₈	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431	2
C₂×#423	2
#2285	2
#1303	2