Subgroups — #28812

Order 256.

367 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×#499	C₂	1	1	yes	maximal, characteristic
3	#2550	C₂	1	2	yes	maximal
4	#2059	C₂	1	2	yes	maximal
5	#2058	C₂	1	2	yes	maximal
6	#1949	C₂	1	2	yes	maximal
7	#1942	C₂	1	1	yes	maximal, characteristic
8	#1942	C₂	1	1	yes	maximal, characteristic
9	#1846	C₂	1	1	yes	maximal, characteristic
10	C₄×#107	C₂	1	1	yes	maximal, characteristic
11	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×#125	C₂²	1	2	yes	—
14	#500	—	2	4	no	—
15	#499	—	2	4	no	—
16	C₂×#107	C₂²	1	2	yes	—
17	C₂×#107	C₂²	1	1	yes	characteristic
18	C₂×#107	C₂²	1	1	yes	characteristic
19	C₂×#107	C₂²	1	1	yes	characteristic
20	C₂×#107	C₂²	1	1	yes	characteristic
21	C₂×#106	C₂²	1	2	yes	—
22	C₂×#106	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#398	—	2	2	no	—
28	#395	—	2	2	no	—
29	#391	—	2	2	no	—
30	#389	—	2	2	no	—
31	#389	C₂²	1	2	yes	—
32	#389	C₂²	1	2	yes	—
33	#382	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	#340	C₂²	1	2	yes	—
37	#340	C₂²	1	2	yes	—
38	#340	C₂²	1	2	yes	—
39	#336	C₂²	1	1	yes	characteristic
40	#336	C₂²	1	1	yes	characteristic
41	C₂²×Q₈	—	2	2	no	—
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈	—	2	4	no	—
47	C₄⋊Q₈	—	4	2	no	—
48	#125	—	2	4	no	—
49	#125	—	4	2	no	—
50	C₂×C₄⋊C₄	—	2	2	no	—
51	C₂×C₄⋊C₄	—	2	2	no	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄	C₂³	1	2	yes	—
61	C₂×C₄²	—	2	2	no	—
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	C₂×C₄²	C₂³	1	1	yes	characteristic
64	C₂×C₄²	C₂³	1	1	yes	characteristic
65	#107	—	2	2	no	—
66	#107	—	2	2	no	—
67	#107	—	2	4	no	—
68	#107	D₄	1	4	yes	—
69	#107	Q₈	1	4	yes	complemented
70	#106	—	2	4	no	—
71	#106	—	4	2	no	—
72	C₄⋊C₈	Q₈	1	4	yes	complemented
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	Q₈	1	4	yes	complemented
76	#95	—	2	2	no	—
77	#95	—	2	2	no	—
78	#95	—	2	2	no	—
79	#95	—	2	2	no	—
80	#95	—	2	2	no	—
81	#95	D₄	1	1	yes	characteristic
82	#95	D₄	1	1	yes	characteristic
83	C₂×Q₈	—	4	4	no	—
84	C₂×Q₈	—	4	4	no	—
85	C₂×Q₈	—	8	2	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	2	2	no	—
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×D₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄	C₂⁴	1	1	yes	characteristic
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₈	C₂×Q₈	1	2	yes	—
105	C₂×C₈	C₂×Q₈	1	2	yes	—
106	C₂×C₈	C₂×Q₈	1	2	yes	—
107	C₂×C₈	C₂×Q₈	1	2	yes	—
108	C₂×C₈	C₂×Q₈	1	2	yes	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	4	4	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄⋊C₄	—	4	4	no	—
120	C₄⋊C₄	—	4	4	no	—
121	C₄⋊C₄	—	8	2	no	—
122	C₄⋊C₄	C₂×D₄	1	4	yes	—
123	C₄⋊C₄	C₂×Q₈	1	4	yes	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	2	1	no	—
128	C₄²	—	8	2	no	—
129	C₄²	C₂×Q₈	1	2	yes	—
130	C₄²	C₂×Q₈	1	2	yes	—
131	C₂³	C₂²×D₄	1	1	yes	characteristic
132	Q₈	—	8	8	no	—
133	C₂×C₄	—	2	1	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	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	2	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	2	no	—
158	C₂×C₄	—	4	2	no	—
159	C₂×C₄	C₄⋊Q₈	1	2	yes	—
160	C₂×C₄	C₄⋊Q₈	1	2	yes	—
161	C₂×C₄	#135	1	2	yes	—
162	C₂×C₄	#135	1	2	yes	—
163	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
165	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
166	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	—	2	2	no	—
172	C₈	C₄⋊Q₈	1	4	yes	—
173	C₂²	C₂×C₄⋊Q₈	1	1	yes	characteristic
174	C₂²	D₄²	1	1	yes	characteristic
175	C₂²	D₄×Q₈	1	1	yes	characteristic
176	C₂²	C₂×#135	1	1	yes	characteristic
177	C₂²	C₂×#135	1	1	yes	characteristic
178	C₂²	#576	1	1	yes	characteristic
179	C₂²	#578	1	1	yes	characteristic
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	4	no	—
185	C₄	—	4	1	no	—
186	C₄	—	4	4	no	—
187	C₄	—	4	1	no	—
188	C₄	—	8	2	no	—
189	C₄	—	8	2	no	—
190	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
191	C₄	#544	1	1	yes	characteristic
192	C₄	D₄×Q₈	1	1	yes	characteristic
193	C₄	Q₈²	1	1	yes	characteristic
194	C₂	#2551	1	1	yes	characteristic
195	C₂	#3160	1	1	yes	characteristic
196	C₂	#3164	1	1	yes	characteristic
197	C₂	#3280	1	1	yes	characteristic
198	C₂	#3285	1	1	yes	characteristic
199	C₂	#3380	1	1	yes	characteristic
200	C₂	#3383	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
69. #107	Q₈	semidirect
72. C₄⋊C₈	Q₈	semidirect
75. C₄×C₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#500	2
C₂×#499	2
#2550	2
#2059	2
#2058	2
#1949	2
#1942	2
#1846	2
C₄×#107	2
C₈×C₄⋊C₄	2