Subgroups — #17816

Order 256.

320 conjugacy classes of subgroups.

The subgroup lattice has 187 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2920	C₂	1	1	yes	maximal, characteristic
2	C₂×#432	C₂	1	1	yes	maximal, characteristic
3	C₂×#431	C₂	1	1	yes	maximal, characteristic
4	#1572	C₂	1	4	yes	maximal, complemented
5	#1571	C₂	1	2	yes	maximal, complemented
6	#1570	C₂	1	2	yes	maximal, complemented
7	#1568	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	#432	—	2	1	no	—
10	#432	—	2	1	no	—
11	#432	—	2	1	no	—
12	#432	C₂²	1	2	yes	—
13	#432	C₂²	1	2	yes	—
14	#432	C₂²	1	2	yes	—
15	#431	—	2	1	no	—
16	#431	C₄	1	2	yes	—
17	#431	C₄	1	2	yes	—
18	#431	C₂²	1	2	yes	—
19	#431	C₂²	1	1	yes	characteristic
20	#431	C₂²	1	1	yes	characteristic
21	#431	C₂²	1	2	yes	—
22	#423	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈	C₄	1	2	yes	—
24	C₂×C₄⋊C₈	C₂²	1	2	yes	—
25	#407	—	2	1	no	—
26	C₂×C₂²⋊C₈	—	2	1	no	—
27	C₂×C₂²⋊C₈	C₄	1	2	yes	—
28	C₄×M₁₆	—	2	1	no	—
29	C₂×#97	—	2	1	no	—
30	C₂×C₄×C₈	C₂²	1	2	yes	—
31	#335	C₂²	1	8	yes	complemented
32	#334	C₂²	1	8	yes	complemented
33	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄	D₄	1	1	yes	characteristic
35	C₂³×C₄	Q₈	1	1	yes	characteristic
36	C₂×M₁₆	—	2	2	no	—
37	C₂²×C₈	—	2	2	no	—
38	C₂²×C₈	C₂×C₄	1	4	yes	—
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²	D₄	1	1	yes	characteristic
43	C₂×C₄²	D₄	1	1	yes	characteristic
44	C₂×C₄²	D₄	1	1	yes	characteristic
45	C₂×C₄²	Q₈	1	1	yes	characteristic
46	C₂×C₄²	Q₈	1	1	yes	characteristic
47	C₂×C₄²	Q₈	1	1	yes	characteristic
48	C₂×C₄²	C₂³	1	1	yes	characteristic
49	C₂×C₄²	C₂³	1	1	yes	characteristic
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₄⋊C₈	—	2	2	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₄⋊C₈	C₂³	1	4	yes	—
57	C₄⋊C₈	C₂³	1	4	yes	—
58	C₂²⋊C₈	—	2	2	no	—
59	C₂²⋊C₈	—	2	2	no	—
60	C₂²⋊C₈	—	2	2	no	—
61	C₂²⋊C₈	—	2	2	no	—
62	C₂²⋊C₈	C₂×C₄	1	2	yes	—
63	C₂²⋊C₈	C₂×C₄	1	2	yes	—
64	C₂²⋊C₈	C₂×C₄	1	4	yes	—
65	#97	—	2	2	no	—
66	#97	—	2	2	no	—
67	#97	—	2	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	C₂×C₄	1	4	yes	—
70	C₄×C₈	C₂³	1	2	yes	—
71	C₄×C₈	C₂³	1	2	yes	—
72	C₂⁴	C₄⋊C₄	1	1	yes	characteristic
73	C₂²×C₄	—	2	4	no	—
74	C₂²×C₄	—	4	2	no	—
75	C₂²×C₄	—	4	2	no	—
76	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
94	M₁₆	—	4	4	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	4	4	no	—
98	C₂×C₈	—	4	4	no	—
99	C₂×C₈	C₂²×C₄	1	4	yes	—
100	C₂×C₈	C₂²×C₄	1	4	yes	—
101	C₄²	C₄⋊C₄	1	1	yes	characteristic
102	C₄²	C₄⋊C₄	1	1	yes	characteristic
103	C₄²	C₄⋊C₄	1	1	yes	characteristic
104	C₄²	C₄⋊C₄	1	1	yes	characteristic
105	C₄²	C₄⋊C₄	1	1	yes	characteristic
106	C₄²	C₄⋊C₄	1	1	yes	characteristic
107	C₄²	C₂²×C₄	1	1	yes	characteristic
108	C₄²	C₂²×C₄	1	1	yes	characteristic
109	C₄²	C₂²×C₄	1	1	yes	characteristic
110	C₄²	C₂×D₄	1	1	yes	characteristic
111	C₄²	C₂×D₄	1	1	yes	characteristic
112	C₄²	C₂×D₄	1	1	yes	characteristic
113	C₄²	C₂×Q₈	1	1	yes	characteristic
114	C₄²	C₂×Q₈	1	1	yes	characteristic
115	C₄²	C₂×Q₈	1	1	yes	characteristic
116	C₄²	C₂⁴	1	1	yes	characteristic
117	C₂³	—	2	2	no	—
118	C₂³	—	2	2	no	—
119	C₂³	—	4	1	no	—
120	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
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	—	4	1	no	—
134	C₂×C₄	—	4	1	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	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄	#135	1	4	yes	—
150	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	—	2	4	no	—
155	C₈	—	4	4	no	—
156	C₈	—	4	4	no	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	—	2	1	no	—
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	4	4	no	—
165	C₂²	#427	1	2	yes	—
166	C₂²	#430	1	2	yes	—
167	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	#427	1	2	yes	—
178	C₄	#568	1	2	yes	—
179	C₄	C₂×#135	1	2	yes	—
180	C₂	—	2	2	no	—
181	C₂	—	2	2	no	—
182	C₂	—	4	1	no	—
183	C₂	#2905	1	1	yes	characteristic
184	C₂	C₂×#427	1	1	yes	characteristic
185	C₂	C₂×#430	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #1572	C₂	semidirect
5. #1571	C₂	semidirect
6. #1570	C₂	semidirect
7. #1568	C₂	semidirect
31. #335	C₂²	semidirect
32. #334	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#2920	2
C₂×#432	2
C₂×#431	2
#1572	2
#1571	2
#1570	2
#1568	2