Subgroups — #22722

Order 256.

416 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446	C₂	1	2	yes	maximal
2	C₄²×Q₈	C₂	1	1	yes	maximal, characteristic
3	#1940	C₂	1	4	yes	maximal
4	#1939	C₂	1	2	yes	maximal
5	#1842	C₂	1	1	yes	maximal, characteristic
6	#1842	C₂	1	2	yes	maximal
7	#1767	C₂	1	2	yes	maximal
8	C₄×C₄⋊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	#446	—	2	8	no	—
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈	C₂²	1	4	yes	—
18	C₂×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×#97	C₂²	1	4	yes	—
21	C₂×C₄×C₈	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄	C₄	1	2	yes	—
23	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄	C₄	1	2	yes	—
25	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂²	1	2	yes	—
28	C₄³	C₂²	1	1	yes	characteristic
29	C₄³	C₂²	1	2	yes	—
30	#336	C₂²	1	4	yes	—
31	#336	C₂²	1	2	yes	—
32	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂³	1	2	yes	—
34	C₂²×C₈	C₂³	1	4	yes	—
35	C₂²×C₈	C₂³	1	2	yes	—
36	C₄×Q₈	—	2	4	no	—
37	C₄×Q₈	—	2	4	no	—
38	C₄×Q₈	D₄	1	4	yes	—
39	C₄×Q₈	Q₈	1	4	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	2	yes	—
44	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄	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	1	yes	characteristic
50	C₂×C₄²	C₂×C₄	1	2	yes	—
51	C₂×C₄²	C₂×C₄	1	2	yes	—
52	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²	C₂×C₄	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	C₂×C₄²	C₂³	1	2	yes	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₄⋊C₈	—	2	4	no	—
60	C₄⋊C₈	—	2	8	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	2	4	no	—
63	C₄⋊C₈	—	4	4	no	—
64	C₄⋊C₈	—	4	2	no	—
65	C₄⋊C₈	Q₈	1	4	yes	complemented
66	#97	—	2	8	no	—
67	C₄×C₈	—	2	4	no	—
68	#95	C₂×C₄	1	2	yes	—
69	#95	C₂×C₄	1	2	yes	—
70	#95	C₂×C₄	1	2	yes	—
71	C₂×Q₈	—	2	2	no	—
72	C₂×Q₈	—	2	2	no	—
73	C₂×Q₈	C₄⋊C₄	1	4	yes	—
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	2	yes	—
81	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄	C₂⁴	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	C₄⋊C₄	1	8	yes	—
98	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
99	C₄⋊C₄	M₁₆	1	4	yes	—
100	C₄²	—	2	4	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	4	no	—
103	C₄²	—	2	4	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	2	no	—
111	C₄²	M₁₆	1	4	yes	—
112	C₄²	C₂×D₄	1	2	yes	—
113	C₄²	C₂×D₄	1	4	yes	—
114	C₄²	C₂×Q₈	1	4	yes	—
115	C₄²	C₂×Q₈	1	2	yes	—
116	C₄²	C₂×Q₈	1	2	yes	—
117	C₄²	#41	1	2	yes	—
118	C₂³	C₂³×C₄	1	1	yes	characteristic
119	Q₈	—	2	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	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	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄	C₄×Q₈	1	2	yes	—
143	C₂×C₄	C₄×Q₈	1	2	yes	—
144	C₂×C₄	C₂×M₁₆	1	4	yes	—
145	C₂×C₄	C₂×M₁₆	1	2	yes	—
146	C₂×C₄	C₂×M₁₆	1	2	yes	—
147	C₂×C₄	C₂×M₁₆	1	2	yes	—
148	C₂×C₄	C₂×M₁₆	1	2	yes	—
149	C₂×C₄	#131	1	2	yes	—
150	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄	C₂×#41	1	1	yes	characteristic
154	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
156	C₈	—	2	4	no	—
157	C₈	—	4	2	no	—
158	C₈	—	4	4	no	—
159	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₂²	#520	1	1	yes	characteristic
162	C₂²	C₂²×M₁₆	1	1	yes	characteristic
163	C₂²	C₂²×M₁₆	1	1	yes	characteristic
164	C₂²	C₂×#131	1	1	yes	characteristic
165	C₂²	#568	1	1	yes	characteristic
166	C₄	—	2	4	no	—
167	C₄	—	2	2	no	—
168	C₄	—	2	4	no	—
169	C₄	—	2	2	no	—
170	C₄	—	2	2	no	—
171	C₄	—	2	4	no	—
172	C₄	—	4	2	no	—
173	C₄	—	4	1	no	—
174	C₄	#423	1	4	yes	—
175	C₄	#446	1	4	yes	complemented
176	C₄	D₄×Q₈	1	1	yes	characteristic
177	C₄	#550	1	1	yes	characteristic
178	C₄	#557	1	1	yes	characteristic
179	C₄	Q₈²	1	1	yes	characteristic
180	C₂	Q₈×C₄⋊C₄	1	1	yes	characteristic
181	C₂	C₂×#423	1	1	yes	characteristic
182	C₂	#2906	1	1	yes	characteristic
183	C₂	C₂×#446	1	1	yes	characteristic
184	C₂	Q₈×M₁₆	1	1	yes	characteristic
185	C₂	#2989	1	1	yes	characteristic
186	C₂	#2990	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₄
LCS	G ▷ C₂³ ▷ 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
65. C₄⋊C₈	Q₈	semidirect
175. C₄	#446	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446	2
C₄²×Q₈	2
#1940	2
#1939	2
#1842	2
#1767	2
C₄×C₄⋊C₈	2