Subgroups — #22721

Order 256.

380 conjugacy classes of subgroups.

The subgroup lattice has 213 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	1	yes	maximal, characteristic
2	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#2295	C₂	1	1	yes	maximal, characteristic
4	#1940	C₂	1	2	yes	maximal
5	#1939	C₂	1	2	yes	maximal
6	#1939	C₂	1	1	yes	maximal, characteristic
7	#1939	C₂	1	1	yes	maximal, characteristic
8	#1843	C₂	1	2	yes	maximal
9	#1842	C₂	1	1	yes	maximal, characteristic
10	#1768	C₂	1	2	yes	maximal
11	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
13	#446	—	2	4	no	—
14	C₈×Q₈	—	2	4	no	—
15	C₂×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂²	1	2	yes	—
28	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
29	#391	C₄	1	1	yes	characteristic
30	#391	C₄	1	1	yes	characteristic
31	#389	C₄	1	1	yes	characteristic
32	#389	C₄	1	1	yes	characteristic
33	#387	C₄	1	2	yes	—
34	#387	C₄	1	2	yes	—
35	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄	C₂²	1	2	yes	—
37	#376	C₂²	1	2	yes	—
38	C₄³	C₂²	1	1	yes	characteristic
39	#336	C₂²	1	1	yes	characteristic
40	#336	C₂²	1	2	yes	—
41	#336	C₂²	1	2	yes	—
42	#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	2	yes	—
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	2	yes	—
48	C₂²×C₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₄×Q₈	D₄	1	4	yes	—
51	C₄×Q₈	Q₈	1	4	yes	—
52	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂³	1	1	yes	characteristic
63	C₂×C₄²	C₂³	1	2	yes	—
64	C₂×C₄²	C₂³	1	1	yes	characteristic
65	C₂×C₄²	C₂³	1	2	yes	—
66	C₂×C₄²	C₂³	1	1	yes	characteristic
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	4	no	—
69	C₄⋊C₈	—	2	2	no	—
70	C₄⋊C₈	—	2	4	no	—
71	C₄⋊C₈	—	2	4	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	4	2	no	—
74	C₄⋊C₈	—	4	2	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄⋊C₈	Q₈	1	4	yes	complemented
77	#97	—	2	4	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	4	no	—
80	C₄×C₈	—	2	2	no	—
81	#95	C₂×C₄	1	2	yes	—
82	#95	C₂×C₄	1	2	yes	—
83	#95	C₂×C₄	1	2	yes	—
84	#95	C₂×C₄	1	2	yes	—
85	#95	C₂×C₄	1	2	yes	—
86	#95	C₂×C₄	1	2	yes	—
87	#95	C₂×C₄	1	2	yes	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	C₄⋊C₄	1	4	yes	—
91	C₂²×C₄	C₂²×C₄	1	2	yes	—
92	C₂²×C₄	C₂²×C₄	1	2	yes	—
93	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂²×C₄	1	2	yes	—
95	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄	C₂²×C₄	1	2	yes	—
97	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄	C₂⁴	1	1	yes	characteristic
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	C₂×Q₈	1	2	yes	—
119	C₂×C₈	C₂×Q₈	1	2	yes	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	C₄⋊C₄	1	8	yes	—
125	C₄⋊C₄	C₄⋊C₄	1	4	yes	—
126	C₄²	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	2	1	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	1	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	4	1	no	—
134	C₄²	C₂×D₄	1	2	yes	—
135	C₄²	C₂×D₄	1	4	yes	—
136	C₄²	C₂×Q₈	1	4	yes	—
137	C₄²	C₂×Q₈	1	2	yes	—
138	C₄²	C₂×Q₈	1	2	yes	—
139	C₄²	#41	1	2	yes	—
140	C₂³	C₂³×C₄	1	1	yes	characteristic
141	Q₈	—	2	8	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
165	C₂×C₄	C₂×C₄⋊C₄	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
167	C₂×C₄	C₄×Q₈	1	2	yes	—
168	C₂×C₄	C₄×Q₈	1	2	yes	—
169	C₂×C₄	#131	1	2	yes	—
170	C₂×C₄	#131	1	2	yes	—
171	C₂×C₄	#131	1	2	yes	—
172	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
175	C₂×C₄	C₂×#41	1	1	yes	characteristic
176	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	2	2	no	—
180	C₈	—	4	1	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	2	no	—
184	C₂²	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂²	#520	1	1	yes	characteristic
187	C₂²	C₂×#131	1	1	yes	characteristic
188	C₂²	C₂×#131	1	1	yes	characteristic
189	C₂²	C₂×#131	1	1	yes	characteristic
190	C₂²	#568	1	1	yes	characteristic
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	—	2	4	no	—
194	C₄	—	2	2	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	1	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	#424	1	4	yes	—
201	C₄	D₄×Q₈	1	1	yes	characteristic
202	C₄	#550	1	1	yes	characteristic
203	C₄	#557	1	1	yes	characteristic
204	C₄	Q₈²	1	1	yes	characteristic
205	C₂	Q₈×C₄⋊C₄	1	1	yes	characteristic
206	C₂	C₂×#424	1	1	yes	characteristic
207	C₂	#2906	1	1	yes	characteristic
208	C₂	#2960	1	1	yes	characteristic
209	C₂	#2962	1	1	yes	characteristic
210	C₂	#2993	1	1	yes	characteristic
211	C₂	#2994	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
76. 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
C₂×#446	2
C₂×C₈×Q₈	2
#2295	2
#1940	2
#1939	2
#1843	2
#1842	2
#1768	2
C₄×C₄⋊C₈	2