Subgroups — #22310

Order 256.

315 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2399	C₂	1	1	yes	maximal, characteristic
2	#1985	C₂	1	2	yes	maximal
3	#1984	C₂	1	2	yes	maximal
4	#1940	C₂	1	1	yes	maximal, characteristic
5	#1940	C₂	1	1	yes	maximal, characteristic
6	#1917	C₂	1	2	yes	maximal
7	#1916	C₂	1	2	yes	maximal
8	#1832	C₂	1	1	yes	maximal, characteristic
9	#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#1746	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈	C₂²	1	2	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	2	yes	—
16	C₂×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×#97	C₂²	1	1	yes	characteristic
18	C₂×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#398	C₄	1	2	yes	—
22	#398	C₄	1	2	yes	—
23	#395	C₄	1	2	yes	—
24	#395	C₄	1	2	yes	—
25	C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	#376	C₂²	1	1	yes	characteristic
29	#336	C₂²	1	2	yes	—
30	#336	C₂²	1	2	yes	—
31	#336	C₂²	1	1	yes	characteristic
32	#336	C₂²	1	2	yes	—
33	#336	C₂²	1	2	yes	—
34	#336	C₂²	1	1	yes	characteristic
35	#336	C₂²	1	2	yes	—
36	#336	C₂²	1	2	yes	—
37	#336	C₂²	1	1	yes	characteristic
38	#336	C₂²	1	1	yes	characteristic
39	C₂²×Q₈	C₂×C₄	1	2	yes	—
40	C₂²×C₈	C₂³	1	2	yes	—
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₈	C₂³	1	2	yes	—
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₄×Q₈	—	4	2	no	—
47	C₄×Q₈	—	4	2	no	—
48	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
55	C₂×C₄²	C₂³	1	1	yes	characteristic
56	C₂×C₄²	C₂³	1	2	yes	—
57	C₂×C₄²	C₂³	1	2	yes	—
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₂×C₄²	C₂³	1	1	yes	characteristic
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	4	2	no	—
63	C₄⋊C₈	—	4	2	no	—
64	#97	—	2	2	no	—
65	#97	—	4	2	no	—
66	C₄×C₈	—	4	2	no	—
67	C₄×C₈	Q₈	1	4	yes	complemented
68	#95	C₂×C₄	1	2	yes	—
69	#95	C₂×C₄	1	2	yes	—
70	#95	C₂×C₄	1	2	yes	—
71	#95	C₂×C₄	1	1	yes	characteristic
72	#95	C₂×C₄	1	1	yes	characteristic
73	#95	C₂×C₄	1	2	yes	—
74	#95	C₂×C₄	1	1	yes	characteristic
75	#95	C₂×C₄	1	1	yes	characteristic
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	—	4	2	no	—
78	C₂×Q₈	—	4	2	no	—
79	C₂²×C₄	C₂²×C₄	1	2	yes	—
80	C₂²×C₄	C₂²×C₄	1	2	yes	—
81	C₂²×C₄	C₂²×C₄	1	2	yes	—
82	C₂²×C₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₄	C₂²×C₄	1	2	yes	—
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₄	1	2	yes	—
87	C₂²×C₄	C₂⁴	1	1	yes	characteristic
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	C₂×Q₈	1	2	yes	—
105	C₂×C₈	C₂×Q₈	1	2	yes	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	2	4	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	2	2	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	C₂×Q₈	1	2	yes	—
122	C₄²	#41	1	2	yes	—
123	C₂³	C₂³×C₄	1	1	yes	characteristic
124	Q₈	—	4	8	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	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	C₄×Q₈	1	2	yes	—
145	C₂×C₄	C₄×Q₈	1	2	yes	—
146	C₂×C₄	#131	1	4	yes	—
147	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄	C₂×#41	1	1	yes	characteristic
149	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
153	C₈	—	2	2	no	—
154	C₈	—	2	2	no	—
155	C₈	—	4	1	no	—
156	C₈	—	4	1	no	—
157	C₈	—	4	2	no	—
158	C₈	—	4	2	no	—
159	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
160	C₂²	#518	1	1	yes	characteristic
161	C₂²	#519	1	1	yes	characteristic
162	C₂²	C₂×#131	1	2	yes	—
163	C₂²	#568	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₄	—	4	2	no	—
168	C₄	—	4	2	no	—
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	4	2	no	—
172	C₄	#542	1	1	yes	characteristic
173	C₄	#542	1	1	yes	characteristic
174	C₄	#544	1	1	yes	characteristic
175	C₄	#544	1	1	yes	characteristic
176	C₂	#2331	1	1	yes	characteristic
177	C₂	#2960	1	1	yes	characteristic
178	C₂	#2961	1	1	yes	characteristic
179	C₂	#2976	1	2	yes	—
180	C₂	#2985	1	2	yes	—

	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
67. 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
#2399	2
#1985	2
#1984	2
#1940	2
#1917	2
#1916	2
#1832	2
#1770	2
C₈×C₄⋊C₄	2
#1746	2