Subgroups — #21339

Order 256.

407 conjugacy classes of subgroups.

The subgroup lattice has 176 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	#1939	C₂	1	2	yes	maximal
4	#1916	C₂	1	4	yes	maximal
5	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊C₈	C₂	1	2	yes	maximal
7	#1744	C₂	1	2	yes	maximal
8	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
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	4	yes	—
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	1	yes	characteristic
19	C₄×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
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	1	yes	characteristic
26	C₄³	C₂²	1	2	yes	—
27	#336	C₂²	1	2	yes	—
28	#336	C₂²	1	4	yes	—
29	#336	C₂²	1	4	yes	—
30	#336	C₂²	1	4	yes	—
31	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂³	1	4	yes	—
33	C₂²×C₈	C₂³	1	2	yes	—
34	C₂²×C₈	C₂³	1	2	yes	—
35	C₄×Q₈	—	2	4	no	—
36	C₄×Q₈	—	2	4	no	—
37	C₄×Q₈	C₈	1	8	yes	complemented
38	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
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×C₄²	C₂×C₄	1	2	yes	—
45	C₂×C₄²	C₂×C₄	1	2	yes	—
46	C₂×C₄²	C₂×C₄	1	2	yes	—
47	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²	C₂×C₄	1	2	yes	—
50	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²	C₂³	1	2	yes	—
53	C₂×C₄²	C₂³	1	1	yes	characteristic
54	C₂×C₄²	C₂³	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	1	yes	characteristic
57	C₄⋊C₈	—	2	8	no	—
58	C₄⋊C₈	—	2	4	no	—
59	C₄×C₈	—	2	4	no	—
60	C₄×C₈	—	2	4	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	Q₈	1	4	yes	complemented
64	#95	C₂×C₄	1	2	yes	—
65	#95	C₂×C₄	1	2	yes	—
66	#95	C₂×C₄	1	2	yes	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	C₂×C₈	1	4	yes	—
70	C₂²×C₄	C₂²×C₄	1	2	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	2	yes	—
75	C₂²×C₄	C₂²×C₄	1	2	yes	—
76	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄	C₂⁴	1	1	yes	characteristic
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	—	2	4	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	C₂×Q₈	1	4	yes	—
89	C₂×C₈	#41	1	4	yes	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	C₂×C₈	1	8	yes	—
95	C₄⋊C₄	C₂×C₈	1	4	yes	—
96	C₄²	—	2	4	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	4	no	—
103	C₄²	—	2	4	no	—
104	C₄²	C₂×C₈	1	8	yes	—
105	C₄²	C₂×C₈	1	4	yes	—
106	C₄²	M₁₆	1	4	yes	—
107	C₄²	C₂×Q₈	1	2	yes	—
108	C₄²	#41	1	4	yes	—
109	C₄²	#41	1	2	yes	—
110	C₂³	C₂³×C₄	1	1	yes	characteristic
111	Q₈	—	2	8	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	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	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	#117	1	2	yes	—
129	C₂×C₄	#117	1	2	yes	—
130	C₂×C₄	C₄×Q₈	1	2	yes	—
131	C₂×C₄	C₄×Q₈	1	2	yes	—
132	C₂×C₄	C₂²×C₈	1	4	yes	—
133	C₂×C₄	C₂²×C₈	1	2	yes	—
134	C₂×C₄	C₂²×C₈	1	4	yes	—
135	C₂×C₄	C₂²×C₈	1	2	yes	—
136	C₂×C₄	C₂²×C₈	1	2	yes	—
137	C₂×C₄	C₂×M₁₆	1	2	yes	—
138	C₂×C₄	C₂×M₁₆	1	2	yes	—
139	C₂×C₄	C₂×M₁₆	1	2	yes	—
140	C₂×C₄	#131	1	2	yes	—
141	C₂×C₄	#131	1	2	yes	—
142	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄	C₂×#41	1	1	yes	characteristic
144	C₂×C₄	C₂×#41	1	2	yes	—
145	C₂×C₄	C₂×#41	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	—	2	4	no	—
148	C₈	—	4	4	no	—
149	C₂²	C₂×#117	1	1	yes	characteristic
150	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
151	C₂²	C₄×#41	1	1	yes	characteristic
152	C₂²	C₂³×C₈	1	1	yes	characteristic
153	C₂²	C₂²×M₁₆	1	1	yes	characteristic
154	C₂²	C₂×#131	1	1	yes	characteristic
155	C₂²	C₂×#131	1	1	yes	characteristic
156	C₄	—	2	4	no	—
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	4	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	2	no	—
162	C₄	—	4	2	no	—
163	C₄	#431	1	4	yes	—
164	C₄	C₈×Q₈	1	4	yes	—
165	C₄	#446	1	4	yes	—
166	C₄	#529	1	2	yes	—
167	C₄	#533	1	2	yes	—
168	C₂	#2295	1	1	yes	characteristic
169	C₂	C₂×#431	1	1	yes	characteristic
170	C₂	#2922	1	1	yes	characteristic
171	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
172	C₂	C₂×#446	1	1	yes	characteristic
173	C₂	C₈×#41	1	1	yes	characteristic
174	C₂	#2965	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂³
commutator	C₂²	C₂³×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴
socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
37. C₄×Q₈	C₈	semidirect
63. 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
#1939	2
#1916	2
C₄×C₄⋊C₈	2
#1744	2
C₄²×C₈	2