Subgroups — #22821

Order 256.

392 conjugacy classes of subgroups.

The subgroup lattice has 185 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
2	#1984	C₂	1	2	yes	maximal
3	#1916	C₂	1	4	yes	maximal
4	#1842	C₂	1	1	yes	maximal, characteristic
5	#1831	C₂	1	2	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	2	yes	maximal
9	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₄	1	2	yes	—
21	C₄×C₄⋊C₄	C₄	1	2	yes	—
22	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
23	C₄×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄³	C₂²	1	2	yes	—
26	C₄³	C₂²	1	1	yes	characteristic
27	#336	C₂²	1	4	yes	—
28	#336	C₂²	1	4	yes	—
29	#336	C₂²	1	2	yes	—
30	#336	C₂²	1	4	yes	—
31	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	2	yes	—
33	C₂²×C₈	C₂³	1	4	yes	—
34	C₂²×C₈	C₂³	1	2	yes	—
35	C₄×Q₈	—	2	4	no	—
36	C₄×Q₈	—	2	2	no	—
37	C₄×Q₈	—	2	2	no	—
38	C₄×Q₈	C₈	1	8	yes	complemented
39	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
44	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²	C₂×C₄	1	2	yes	—
47	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²	C₂×C₄	1	2	yes	—
49	C₂×C₄²	C₂×C₄	1	2	yes	—
50	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₄	1	2	yes	—
52	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²	C₂³	1	2	yes	—
54	C₂×C₄²	C₂³	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	C₄⋊C₈	—	2	4	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	2	4	no	—
61	C₄⋊C₈	—	4	2	no	—
62	C₄⋊C₈	Q₈	1	4	yes	complemented
63	C₄×C₈	—	2	4	no	—
64	C₄×C₈	—	4	4	no	—
65	#95	C₂×C₄	1	2	yes	—
66	#95	C₂×C₄	1	2	yes	—
67	#95	C₂×C₄	1	2	yes	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	C₂×C₈	1	4	yes	—
71	C₂²×C₄	C₂²×C₄	1	2	yes	—
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²×C₄	1	2	yes	—
76	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂²×C₄	1	2	yes	—
78	C₂²×C₄	C₂²×C₄	1	2	yes	—
79	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂⁴	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂×Q₈	1	4	yes	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	C₂×C₈	1	8	yes	—
95	C₄⋊C₄	C₂×C₈	1	4	yes	—
96	C₄⋊C₄	M₁₆	1	4	yes	—
97	C₄²	—	2	4	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	2	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	2	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	4	2	no	—
109	C₄²	C₂×C₈	1	8	yes	—
110	C₄²	C₂×C₈	1	4	yes	—
111	C₄²	C₂×Q₈	1	2	yes	—
112	C₄²	#41	1	2	yes	—
113	C₄²	#41	1	2	yes	—
114	C₄²	#41	1	2	yes	—
115	C₂³	C₂³×C₄	1	1	yes	characteristic
116	Q₈	—	2	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	#117	1	2	yes	—
136	C₂×C₄	#117	1	2	yes	—
137	C₂×C₄	C₄×Q₈	1	2	yes	—
138	C₂×C₄	C₄×Q₈	1	2	yes	—
139	C₂×C₄	C₂²×C₈	1	4	yes	—
140	C₂×C₄	C₂²×C₈	1	2	yes	—
141	C₂×C₄	C₂²×C₈	1	4	yes	—
142	C₂×C₄	C₂²×C₈	1	2	yes	—
143	C₂×C₄	C₂²×C₈	1	2	yes	—
144	C₂×C₄	C₂×M₁₆	1	4	yes	—
145	C₂×C₄	C₂×M₁₆	1	2	yes	—
146	C₂×C₄	#131	1	2	yes	—
147	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄	C₂×#41	1	1	yes	characteristic
149	C₂×C₄	C₂×#41	1	1	yes	characteristic
150	C₂×C₄	C₂×#41	1	1	yes	characteristic
151	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
153	C₈	—	2	4	no	—
154	C₈	—	4	2	no	—
155	C₈	—	4	4	no	—
156	C₂²	C₂×#117	1	1	yes	characteristic
157	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²	#520	1	1	yes	characteristic
159	C₂²	C₂³×C₈	1	1	yes	characteristic
160	C₂²	C₂²×M₁₆	1	1	yes	characteristic
161	C₂²	C₂×#131	1	1	yes	characteristic
162	C₂²	#568	1	1	yes	characteristic
163	C₄	—	2	4	no	—
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	1	no	—
171	C₄	#431	1	4	yes	—
172	C₄	C₈×Q₈	1	4	yes	—
173	C₄	#554	1	1	yes	characteristic
174	C₄	#556	1	1	yes	characteristic
175	C₄	#557	1	1	yes	characteristic
176	C₄	#559	1	1	yes	characteristic
177	C₂	#2355	1	1	yes	characteristic
178	C₂	C₂×#431	1	1	yes	characteristic
179	C₂	#2925	1	1	yes	characteristic
180	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
181	C₂	Q₈×M₁₆	1	1	yes	characteristic
182	C₂	#2988	1	1	yes	characteristic
183	C₂	#2990	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
38. C₄×Q₈	C₈	semidirect
62. 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
C₄²×Q₈	2
#1984	2
#1916	2
#1842	2
#1831	2
C₈×C₄⋊C₄	2
C₄×C₄⋊C₈	2