Subgroups — #23149

Order 256.

418 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	#1984	C₂	1	2	yes	maximal
4	#1939	C₂	1	2	yes	maximal
5	#1916	C₂	1	4	yes	maximal
6	C₈×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈	C₂²	1	2	yes	—
11	C₈×Q₈	—	2	8	no	—
12	C₂×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	4	yes	—
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	#391	C₄	1	2	yes	—
21	#389	C₄	1	2	yes	—
22	#389	C₄	1	2	yes	—
23	#384	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄³	C₂²	1	1	yes	characteristic
27	#336	C₂²	1	2	yes	—
28	#336	C₂²	1	2	yes	—
29	#336	C₂²	1	4	yes	—
30	#336	C₂²	1	4	yes	—
31	C₂²×Q₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈	C₂³	1	2	yes	—
33	C₂²×C₈	C₂³	1	2	yes	—
34	C₂²×C₈	C₂³	1	2	yes	—
35	C₂²×C₈	C₂³	1	2	yes	—
36	C₄⋊Q₈	C₈	1	8	yes	complemented
37	C₄×Q₈	—	2	4	no	—
38	C₄×Q₈	—	2	4	no	—
39	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²	C₂³	1	4	yes	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	C₄⋊C₈	—	2	8	no	—
54	C₄⋊C₈	—	2	2	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	D₄	1	4	yes	—
57	C₄⋊C₈	Q₈	1	4	yes	complemented
58	C₄×C₈	—	2	8	no	—
59	C₄×C₈	—	2	4	no	—
60	C₄×C₈	—	2	4	no	—
61	#95	C₂×C₄	1	4	yes	—
62	#95	C₂×C₄	1	4	yes	—
63	C₂×Q₈	—	2	4	no	—
64	C₂×Q₈	—	2	4	no	—
65	C₂×Q₈	C₂×C₈	1	8	yes	—
66	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄	C₂²×C₄	1	4	yes	—
68	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄	C₂²×C₄	1	4	yes	—
71	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂⁴	1	1	yes	characteristic
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	C₂×D₄	1	4	yes	—
86	C₂×C₈	C₂×Q₈	1	4	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	2	4	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	C₂×C₈	1	8	yes	—
92	C₄⋊C₄	C₂×C₈	1	8	yes	—
93	C₄²	—	2	2	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₄²	—	4	1	no	—
100	C₄²	C₂×C₈	1	4	yes	—
101	C₄²	C₂×D₄	1	2	yes	—
102	C₄²	C₂×Q₈	1	2	yes	—
103	C₄²	#41	1	2	yes	—
104	C₄²	#41	1	2	yes	—
105	C₂³	C₂³×C₄	1	1	yes	characteristic
106	Q₈	—	2	16	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	C₄×D₄	1	2	yes	—
122	C₂×C₄	C₄×D₄	1	2	yes	—
123	C₂×C₄	C₄×Q₈	1	2	yes	—
124	C₂×C₄	C₄×Q₈	1	2	yes	—
125	C₂×C₄	C₂²×C₈	1	2	yes	—
126	C₂×C₄	C₂²×C₈	1	8	yes	—
127	C₂×C₄	C₂²×C₈	1	2	yes	—
128	C₂×C₄	C₂²×C₈	1	2	yes	—
129	C₂×C₄	#131	1	2	yes	—
130	C₂×C₄	#131	1	2	yes	—
131	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄	C₂×#41	1	1	yes	characteristic
134	C₂×C₄	C₂×#41	1	1	yes	characteristic
135	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
136	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	2	4	no	—
139	C₈	—	4	2	no	—
140	C₈	—	4	2	no	—
141	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
143	C₂²	#519	1	1	yes	characteristic
144	C₂²	C₂³×C₈	1	1	yes	characteristic
145	C₂²	C₂×#131	1	1	yes	characteristic
146	C₂²	C₂×#131	1	1	yes	characteristic
147	C₂²	#568	1	1	yes	characteristic
148	C₄	—	2	8	no	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	4	1	no	—
154	C₄	C₈×D₄	1	4	yes	—
155	C₄	C₈×Q₈	1	4	yes	complemented
156	C₄	#547	1	1	yes	characteristic
157	C₄	D₄×Q₈	1	1	yes	characteristic
158	C₄	#556	1	1	yes	characteristic
159	C₄	#557	1	1	yes	characteristic
160	C₂	#2365	1	1	yes	characteristic
161	C₂	C₂×C₈×D₄	1	1	yes	characteristic
162	C₂	#2935	1	1	yes	characteristic
163	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
164	C₂	#2962	1	1	yes	characteristic
165	C₂	#2982	1	1	yes	characteristic
166	C₂	#2986	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator	C₂²	C₂³×C₈
socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
36. C₄⋊Q₈	C₈	semidirect
57. C₄⋊C₈	Q₈	semidirect
155. 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×C₈×Q₈	2
C₄×C₄⋊Q₈	2
#1984	2
#1939	2
#1916	2
C₈×C₄⋊C₄	2
C₄×C₄⋊C₈	2