Subgroups — #18067

Order 256.

327 conjugacy classes of subgroups.

The subgroup lattice has 214 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	#2962	C₂	1	1	yes	maximal, characteristic, complemented
3	#2935	C₂	1	1	yes	maximal, characteristic
4	#1590	C₂	1	2	yes	maximal, complemented
5	#1583	C₂	1	2	yes	maximal, complemented
6	#1582	C₂	1	2	yes	maximal, complemented
7	C₈×SD₁₆	C₂	1	2	yes	maximal, complemented
8	C₂×#334	C₂	1	1	yes	maximal, characteristic, complemented
9	#1481	C₂	1	1	yes	maximal, characteristic, complemented
10	#1480	C₂	1	1	yes	maximal, characteristic
11	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#131	—	2	1	no	—
13	#533	C₂²	1	1	yes	characteristic
14	#532	C₂²	1	1	yes	characteristic
15	#498	C₄	1	1	yes	characteristic
16	#492	C₄	1	1	yes	characteristic
17	#487	C₄	1	1	yes	characteristic
18	#486	C₄	1	1	yes	characteristic
19	#465	C₄	1	2	yes	—
20	#465	C₄	1	2	yes	—
21	#446	—	2	2	no	—
22	#446	C₂²	1	2	yes	—
23	C₈×Q₈	C₂²	1	2	yes	—
24	C₄×SD₁₆	C₂²	1	2	yes	—
25	C₄×SD₁₆	C₂²	1	2	yes	—
26	#436	C₂²	1	2	yes	—
27	#435	—	2	2	no	—
28	C₈×D₄	C₂²	1	2	yes	—
29	#431	C₂²	1	1	yes	characteristic
30	#424	—	2	1	no	—
31	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	#408	—	2	1	no	—
34	#405	—	2	1	no	—
35	C₄×M₁₆	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
37	#334	C₂²	1	1	yes	characteristic
38	#334	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	C₈²	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	#131	—	4	4	no	—
51	C₂×M₁₆	—	2	2	no	—
52	C₂×M₁₆	D₄	1	1	yes	characteristic
53	C₂×M₁₆	D₄	1	1	yes	characteristic
54	C₂²×C₈	—	2	1	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	—	2	2	no	—
57	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
60	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
61	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
62	#127	C₂×C₄	1	1	yes	characteristic
63	#125	—	2	1	no	—
64	#124	—	2	1	no	—
65	C₂²⋊Q₈	—	2	1	no	—
66	C₂²⋊Q₈	C₂×C₄	1	2	yes	—
67	#121	—	2	1	no	—
68	#121	C₂×C₄	1	2	yes	—
69	C₄×Q₈	—	2	1	no	—
70	C₄×Q₈	C₂³	1	2	yes	—
71	C₄×D₄	—	2	1	no	—
72	C₄×D₄	C₂³	1	2	yes	—
73	#117	—	2	1	no	—
74	C₂×C₄²	C₂³	1	1	yes	characteristic
75	#106	C₂×C₄	1	2	yes	—
76	#106	C₂×C₄	1	2	yes	—
77	C₄⋊C₈	—	2	2	no	—
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	2	1	no	—
80	C₄⋊C₈	—	2	1	no	—
81	C₄⋊C₈	—	2	1	no	—
82	C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₄⋊C₈	C₂³	1	1	yes	characteristic
86	#103	C₂×C₄	1	2	yes	—
87	#103	C₂×C₄	1	2	yes	—
88	#102	C₂×C₄	1	2	yes	—
89	#102	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₂²⋊C₈	—	2	2	no	—
93	#97	—	2	2	no	—
94	#97	D₄	1	2	yes	—
95	C₄×C₈	—	2	2	no	—
96	C₄×C₈	C₂×C₄	1	1	yes	characteristic
97	C₄×C₈	C₂×C₄	1	1	yes	characteristic
98	C₄×C₈	C₂³	1	2	yes	—
99	C₄×C₈	C₂³	1	1	yes	characteristic
100	C₄×C₈	C₂³	1	1	yes	characteristic
101	C₄×C₈	C₂³	1	2	yes	—
102	#41	—	4	2	no	—
103	#41	—	4	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	C₂²×C₄	1	2	yes	—
107	C₂×D₄	—	2	2	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂²×C₄	1	2	yes	—
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄	#41	1	1	yes	characteristic
115	SD₁₆	—	2	4	no	—
116	SD₁₆	—	4	2	no	—
117	M₁₆	—	2	2	no	—
118	M₁₆	—	4	4	no	—
119	M₁₆	SD₁₆	1	4	yes	complemented
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈	C₂×D₄	1	2	yes	—
135	C₂×C₈	C₂×D₄	1	2	yes	—
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	1	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	C₂²×C₄	1	2	yes	—
143	C₄⋊C₄	C₂²×C₄	1	2	yes	—
144	C₂²⋊C₄	—	2	2	no	—
145	C₂²⋊C₄	—	2	2	no	—
146	C₂²⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	C₂²×C₄	1	1	yes	characteristic
149	C₄²	C₂×D₄	1	1	yes	characteristic
150	C₄²	#41	1	1	yes	characteristic
151	C₄²	C₂⁴	1	1	yes	characteristic
152	C₂³	—	2	2	no	—
153	C₂³	C₄×D₄	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	Q₈	—	2	2	no	—
156	Q₈	—	4	2	no	—
157	Q₈	—	8	1	no	—
158	D₄	—	2	2	no	—
159	D₄	—	2	2	no	—
160	D₄	—	4	4	no	—
161	D₄	—	4	2	no	—
162	D₄	—	8	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	C₄×D₄	1	1	yes	characteristic
172	C₂×C₄	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄	C₂×SD₁₆	1	2	yes	—
175	C₂×C₄	#135	1	2	yes	—
176	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄	C₂×#41	1	1	yes	characteristic
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	2	no	—
183	C₈	—	4	2	no	—
184	C₈	—	4	1	no	—
185	C₈	—	4	1	no	—
186	C₈	—	4	1	no	—
187	C₈	—	4	1	no	—
188	C₈	#131	1	2	yes	complemented
189	C₈	C₂×SD₁₆	1	4	yes	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	2	no	—
195	C₂²	C₄×SD₁₆	1	2	yes	—
196	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	C₄×SD₁₆	1	2	yes	—
204	C₄	C₂×#131	1	1	yes	characteristic
205	C₄	#568	1	1	yes	characteristic
206	C₄	C₂²×SD₁₆	1	1	yes	characteristic
207	C₄	C₂×#135	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	—	8	2	no	—
210	C₂	#2935	1	1	yes	characteristic
211	C₂	C₂×C₄×SD₁₆	1	1	yes	characteristic
212	C₂	#2957	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 ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2962	C₂	semidirect
4. #1590	C₂	semidirect
5. #1583	C₂	semidirect
6. #1582	C₂	semidirect
7. C₈×SD₁₆	C₂	semidirect
8. C₂×#334	C₂	semidirect
9. #1481	C₂	semidirect
11. C₈×M₁₆	C₂	semidirect
119. M₁₆	SD₁₆	semidirect
188. C₈	#131	semidirect

Maximal subgroups

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

subgroup	index
#3167	2
#2962	2
#2935	2
#1590	2
#1583	2
#1582	2
C₈×SD₁₆	2
C₂×#334	2
#1481	2
#1480	2
C₈×M₁₆	2