Subgroups — #18089

Order 256.

302 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3167	C₂	1	1	yes	maximal, characteristic
2	#2986	C₂	1	1	yes	maximal, characteristic, complemented
3	#2979	C₂	1	1	yes	maximal, characteristic
4	#1592	C₂	1	2	yes	maximal, complemented
5	#1583	C₂	1	2	yes	maximal, complemented
6	#1582	C₂	1	2	yes	maximal, complemented
7	SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	#1485	C₂	1	1	yes	maximal, characteristic, complemented
10	#1484	C₂	1	1	yes	maximal, characteristic
11	#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	#533	C₂²	1	1	yes	characteristic
13	#532	C₂²	1	1	yes	characteristic
14	#498	C₄	1	1	yes	characteristic
15	#492	C₄	1	1	yes	characteristic
16	#487	C₄	1	1	yes	characteristic
17	#486	C₄	1	1	yes	characteristic
18	#465	C₄	1	2	yes	—
19	#465	C₄	1	2	yes	—
20	#446	—	2	1	no	—
21	#446	C₂²	1	2	yes	—
22	C₈×Q₈	—	2	1	no	—
23	C₈×Q₈	C₂²	1	2	yes	—
24	C₄×SD₁₆	C₂²	1	2	yes	—
25	C₄×SD₁₆	C₂²	1	2	yes	—
26	#436	—	2	1	no	—
27	#436	C₂²	1	2	yes	—
28	C₈×D₄	—	2	1	no	—
29	C₈×D₄	C₂²	1	2	yes	—
30	#433	—	2	2	no	—
31	#431	C₂²	1	1	yes	characteristic
32	#431	C₂²	1	1	yes	characteristic
33	#431	C₂²	1	1	yes	characteristic
34	#408	—	2	2	no	—
35	C₄×M₁₆	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#335	C₂²	1	1	yes	characteristic
38	#335	C₂²	1	1	yes	characteristic
39	#334	C₂²	1	1	yes	characteristic
40	#334	C₂²	1	1	yes	characteristic
41	Q₈⋊C₈	C₂²	1	2	yes	—
42	Q₈⋊C₈	C₂²	1	2	yes	—
43	D₄⋊C₈	C₂²	1	2	yes	—
44	D₄⋊C₈	C₂²	1	2	yes	—
45	#322	C₂²	1	2	yes	—
46	#322	C₂²	1	2	yes	—
47	C₂×#41	—	2	1	no	—
48	C₂×SD₁₆	C₂×C₄	1	2	yes	—
49	C₂×SD₁₆	C₂×C₄	1	2	yes	—
50	C₂×M₁₆	—	2	1	no	—
51	C₂×M₁₆	—	2	2	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
58	#127	C₂×C₄	1	1	yes	characteristic
59	#125	—	2	1	no	—
60	#124	—	2	1	no	—
61	C₂²⋊Q₈	—	2	1	no	—
62	C₂²⋊Q₈	C₂×C₄	1	2	yes	—
63	#121	—	2	1	no	—
64	#121	C₂×C₄	1	2	yes	—
65	C₄×Q₈	—	2	1	no	—
66	C₄×Q₈	C₂³	1	2	yes	—
67	C₄×D₄	—	2	1	no	—
68	C₄×D₄	C₂³	1	2	yes	—
69	#117	—	2	1	no	—
70	C₂×C₄²	C₂³	1	1	yes	characteristic
71	#106	C₂×C₄	1	2	yes	—
72	#106	C₂×C₄	1	2	yes	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₄⋊C₈	D₄	1	1	yes	characteristic
77	C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊C₈	C₂³	1	1	yes	characteristic
81	#103	C₂×C₄	1	2	yes	—
82	#103	C₂×C₄	1	2	yes	—
83	#102	C₂×C₄	1	2	yes	—
84	#102	C₂×C₄	1	2	yes	—
85	C₂²⋊C₈	—	2	1	no	—
86	C₂²⋊C₈	—	2	1	no	—
87	C₂²⋊C₈	—	2	2	no	—
88	C₂²⋊C₈	—	2	2	no	—
89	C₂²⋊C₈	D₄	1	2	yes	—
90	#97	—	2	2	no	—
91	#97	—	2	1	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	C₂×C₄	1	1	yes	characteristic
96	C₄×C₈	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈	C₂³	1	2	yes	—
98	C₄×C₈	C₂³	1	1	yes	characteristic
99	C₄×C₈	C₂³	1	1	yes	characteristic
100	C₄×C₈	C₂³	1	2	yes	—
101	#41	—	8	1	no	—
102	#41	—	8	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	C₂²×C₄	1	2	yes	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	C₂²×C₄	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄	#41	1	1	yes	characteristic
114	SD₁₆	—	2	4	no	—
115	SD₁₆	—	4	2	no	—
116	M₁₆	—	4	2	no	—
117	M₁₆	—	8	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	1	no	—
125	C₂×C₈	—	8	2	no	—
126	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈	C₂×D₄	1	2	yes	—
131	C₂×C₈	C₂×D₄	1	2	yes	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	2	1	no	—
134	C₄⋊C₄	—	2	1	no	—
135	C₄⋊C₄	—	2	1	no	—
136	C₄⋊C₄	—	2	2	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄	C₂²×C₄	1	2	yes	—
140	C₂²⋊C₄	—	2	2	no	—
141	C₂²⋊C₄	—	2	2	no	—
142	C₂²⋊C₄	—	2	2	no	—
143	C₄²	—	2	2	no	—
144	C₄²	C₂²×C₄	1	1	yes	characteristic
145	C₄²	C₂×D₄	1	1	yes	characteristic
146	C₄²	#41	1	1	yes	characteristic
147	C₄²	C₂⁴	1	1	yes	characteristic
148	C₂³	—	2	2	no	—
149	C₂³	C₄×D₄	1	1	yes	characteristic
150	Q₈	—	2	2	no	—
151	Q₈	—	2	2	no	—
152	Q₈	—	4	2	no	—
153	Q₈	—	8	1	no	—
154	D₄	—	2	2	no	—
155	D₄	—	2	2	no	—
156	D₄	—	4	2	no	—
157	D₄	—	8	1	no	—
158	D₄	—	8	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	C₄×D₄	1	1	yes	characteristic
170	C₂×C₄	C₄×D₄	1	1	yes	characteristic
171	C₂×C₄	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄	C₂×#41	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₈	—	4	2	no	—
181	C₈	—	4	1	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	1	no	—
184	C₈	—	4	1	no	—
185	C₈	#131	1	2	yes	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	4	2	no	—
197	C₄	—	4	2	no	—
198	C₄	—	4	2	no	—
199	C₄	#444	1	2	yes	—
200	C₄	C₂×#131	1	1	yes	characteristic
201	C₄	#568	1	1	yes	characteristic
202	C₄	#577	1	1	yes	characteristic
203	C₄	#577	1	1	yes	characteristic
204	C₂	—	4	1	no	—
205	C₂	—	8	2	no	—
206	C₂	#2935	1	1	yes	characteristic
207	C₂	#2949	1	1	yes	characteristic
208	C₂	C₂×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2986	C₂	semidirect
4. #1592	C₂	semidirect
5. #1583	C₂	semidirect
6. #1582	C₂	semidirect
7. SD₁₆⋊C₈	C₂	semidirect
8. #1564	C₂	semidirect
9. #1485	C₂	semidirect
11. #1451	C₂	semidirect

Maximal subgroups

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

subgroup	index
#3167	2
#2986	2
#2979	2
#1592	2
#1583	2
#1582	2
SD₁₆⋊C₈	2
#1564	2
#1485	2
#1484	2
#1451	2