Subgroups — #23160

Order 256.

359 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499	C₂	1	1	yes	maximal, characteristic
2	#2370	C₂	1	2	yes	maximal
3	#1989	C₂	1	2	yes	maximal
4	#1942	C₂	1	1	yes	maximal, characteristic
5	#1942	C₂	1	1	yes	maximal, characteristic
6	#1920	C₂	1	2	yes	maximal
7	#1919	C₂	1	2	yes	maximal
8	C₄×#107	C₂	1	1	yes	maximal, characteristic
9	C₄×#107	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×#125	C₂²	1	2	yes	—
13	#499	C₄	1	8	yes	complemented
14	C₂×#107	C₂²	1	1	yes	characteristic
15	C₂×#107	C₂²	1	1	yes	characteristic
16	C₂×#107	C₂²	1	1	yes	characteristic
17	C₂×#107	C₂²	1	1	yes	characteristic
18	C₂×#106	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#389	—	2	2	no	—
24	#389	C₂²	1	2	yes	—
25	#389	C₂²	1	2	yes	—
26	#387	—	2	2	no	—
27	#387	—	2	2	no	—
28	C₄×C₄⋊C₄	—	2	2	no	—
29	C₄×C₄⋊C₄	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	#340	C₂²	1	2	yes	—
34	#340	C₂²	1	2	yes	—
35	#340	C₂²	1	2	yes	—
36	#340	C₂²	1	2	yes	—
37	#340	C₂²	1	2	yes	—
38	#336	C₂²	1	1	yes	characteristic
39	#336	C₂²	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	#125	—	2	4	no	—
45	#125	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄	—	2	2	no	—
47	C₂×C₄⋊C₄	—	2	2	no	—
48	C₂×C₄⋊C₄	—	2	2	no	—
49	C₂×C₄⋊C₄	—	2	2	no	—
50	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₄²	—	2	2	no	—
57	C₂×C₄²	—	2	2	no	—
58	C₂×C₄²	—	2	2	no	—
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#107	—	2	2	no	—
63	#107	C₂×C₄	1	4	yes	—
64	#107	C₂×C₄	1	4	yes	—
65	#107	Q₈	1	4	yes	—
66	#106	C₂×C₄	1	8	yes	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	C₂×C₄	1	4	yes	—
71	#95	—	2	2	no	—
72	#95	—	2	2	no	—
73	#95	—	2	2	no	—
74	#95	—	2	2	no	—
75	#95	—	2	2	no	—
76	#95	D₄	1	1	yes	characteristic
77	#95	D₄	1	1	yes	characteristic
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂⁴	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	C₂²×C₄	1	2	yes	—
98	C₂×C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₈	C₂×Q₈	1	2	yes	—
100	C₂×C₈	#41	1	2	yes	—
101	C₄⋊C₄	—	2	8	no	—
102	C₄⋊C₄	—	2	8	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	8	2	no	—
112	C₄⋊C₄	C₂²×C₄	1	4	yes	—
113	C₄⋊C₄	C₂²×C₄	1	4	yes	—
114	C₄⋊C₄	C₂×Q₈	1	4	yes	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	4	4	no	—
120	C₄²	—	8	2	no	—
121	C₄²	—	8	2	no	—
122	C₄²	C₂²×C₄	1	2	yes	—
123	C₄²	#41	1	2	yes	—
124	C₂³	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	4	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	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	C₄×D₄	1	2	yes	—
152	C₂×C₄	C₄×D₄	1	2	yes	—
153	C₂×C₄	#135	1	2	yes	—
154	C₂×C₄	#135	1	2	yes	—
155	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄	C₂×#41	1	1	yes	characteristic
158	C₂×C₄	C₂×#41	1	1	yes	characteristic
159	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
160	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
161	C₈	—	2	2	no	—
162	C₈	—	4	1	no	—
163	C₈	—	4	1	no	—
164	C₈	C₄×Q₈	1	4	yes	—
165	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²	#547	1	1	yes	characteristic
167	C₂²	D₄×Q₈	1	1	yes	characteristic
168	C₂²	C₂×#135	1	1	yes	characteristic
169	C₂²	C₂×#135	1	1	yes	characteristic
170	C₂²	#576	1	1	yes	characteristic
171	C₂²	#578	1	1	yes	characteristic
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	4	1	no	—
177	C₄	—	4	1	no	—
178	C₄	—	4	4	no	—
179	C₄	—	4	4	no	—
180	C₄	—	4	4	no	—
181	C₄	—	8	2	no	—
182	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
183	C₄	#519	1	1	yes	characteristic
184	C₄	#556	1	1	yes	characteristic
185	C₄	#557	1	1	yes	characteristic
186	C₂	#2365	1	1	yes	characteristic
187	C₂	C₄×#135	1	1	yes	characteristic
188	C₂	#2951	1	1	yes	characteristic
189	C₂	#3336	1	1	yes	characteristic
190	C₂	#3339	1	1	yes	characteristic
191	C₂	#3380	1	1	yes	characteristic
192	C₂	#3383	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂×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
13. #499	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#499	2
#2370	2
#1989	2
#1942	2
#1920	2
#1919	2
C₄×#107	2
C₈×C₄⋊C₄	2