Subgroups — #22824

Order 256.

318 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2295	C₂	1	1	yes	maximal, characteristic
2	#1985	C₂	1	2	yes	maximal
3	#1917	C₂	1	2	yes	maximal
4	#1916	C₂	1	2	yes	maximal
5	#1842	C₂	1	1	yes	maximal, characteristic
6	#1832	C₂	1	2	yes	maximal
7	#1771	C₂	1	2	yes	maximal
8	#1768	C₂	1	2	yes	maximal
9	#1767	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈	C₂²	1	2	yes	—
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×#97	C₂²	1	2	yes	—
18	C₂×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	#391	C₄	1	1	yes	characteristic
21	#391	C₄	1	1	yes	characteristic
22	#389	C₄	1	1	yes	characteristic
23	#389	C₄	1	1	yes	characteristic
24	#387	C₄	1	2	yes	—
25	#387	C₄	1	2	yes	—
26	C₄×C₄⋊C₄	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	#376	C₂²	1	2	yes	—
29	C₄³	C₂²	1	1	yes	characteristic
30	#336	C₂²	1	2	yes	—
31	#336	C₂²	1	2	yes	—
32	#336	C₂²	1	2	yes	—
33	#336	C₂²	1	2	yes	—
34	#336	C₂²	1	2	yes	—
35	#336	C₂²	1	2	yes	—
36	#336	C₂²	1	2	yes	—
37	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂³	1	2	yes	—
39	C₂²×C₈	C₂³	1	2	yes	—
40	C₂²×C₈	C₂³	1	2	yes	—
41	C₂²×C₈	C₂³	1	2	yes	—
42	C₄×Q₈	—	2	2	no	—
43	C₄×Q₈	—	2	2	no	—
44	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
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	2	yes	—
57	C₂×C₄²	C₂³	1	1	yes	characteristic
58	C₂×C₄²	C₂³	1	1	yes	characteristic
59	C₄⋊C₈	—	2	4	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	2	4	no	—
62	C₄⋊C₈	—	4	1	no	—
63	C₄⋊C₈	—	4	1	no	—
64	C₄⋊C₈	Q₈	1	4	yes	complemented
65	#97	—	2	4	no	—
66	#97	—	4	2	no	—
67	C₄×C₈	—	4	2	no	—
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	2	yes	—
72	#95	C₂×C₄	1	2	yes	—
73	#95	C₂×C₄	1	2	yes	—
74	#95	C₂×C₄	1	2	yes	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	4	1	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂²×C₄	C₂²×C₄	1	2	yes	—
79	C₂²×C₄	C₂²×C₄	1	2	yes	—
80	C₂²×C₄	C₂²×C₄	1	2	yes	—
81	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄	C₂²×C₄	1	2	yes	—
83	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄	C₂⁴	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	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	2	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×Q₈	1	4	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	—	4	1	no	—
117	C₄²	C₂×Q₈	1	2	yes	—
118	C₄²	#41	1	2	yes	—
119	C₄²	#41	1	2	yes	—
120	C₄²	#41	1	2	yes	—
121	C₂³	C₂³×C₄	1	1	yes	characteristic
122	Q₈	—	4	4	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	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	1	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	#117	1	2	yes	—
149	C₂×C₄	#117	1	2	yes	—
150	C₂×C₄	C₄×Q₈	1	2	yes	—
151	C₂×C₄	C₄×Q₈	1	2	yes	—
152	C₂×C₄	#131	1	2	yes	—
153	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄	C₂×#41	1	1	yes	characteristic
155	C₂×C₄	C₂×#41	1	1	yes	characteristic
156	C₂×C₄	C₂×#41	1	1	yes	characteristic
157	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
159	C₈	—	2	4	no	—
160	C₈	—	4	2	no	—
161	C₈	—	4	2	no	—
162	C₈	—	4	2	no	—
163	C₂²	C₂×#117	1	1	yes	characteristic
164	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₂²	#520	1	1	yes	characteristic
166	C₂²	C₂×#131	1	1	yes	characteristic
167	C₂²	#568	1	1	yes	characteristic
168	C₂²	#568	1	1	yes	characteristic
169	C₂²	#568	1	1	yes	characteristic
170	C₄	—	2	2	no	—
171	C₄	—	2	2	no	—
172	C₄	—	2	2	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	1	no	—
179	C₄	—	4	2	no	—
180	C₄	#554	1	1	yes	characteristic
181	C₄	#556	1	1	yes	characteristic
182	C₄	#557	1	1	yes	characteristic
183	C₄	#559	1	1	yes	characteristic
184	C₂	#2355	1	1	yes	characteristic
185	C₂	#2921	1	1	yes	characteristic
186	C₂	#2925	1	1	yes	characteristic
187	C₂	#2961	1	1	yes	characteristic
188	C₂	#2962	1	1	yes	characteristic
189	C₂	#2992	1	1	yes	characteristic
190	C₂	#2995	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
64. 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
#2295	2
#1985	2
#1917	2
#1916	2
#1842	2
#1832	2
#1771	2
#1768	2
#1767	2