Subgroups — #23150

Order 256.

390 conjugacy classes of subgroups.

The subgroup lattice has 219 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	C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	#1984	C₂	1	1	yes	maximal, characteristic
5	#1984	C₂	1	1	yes	maximal, characteristic
6	#1940	C₂	1	2	yes	maximal
7	#1917	C₂	1	2	yes	maximal
8	#1916	C₂	1	2	yes	maximal
9	#1771	C₂	1	2	yes	maximal
10	#1767	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
15	#446	—	2	4	no	—
16	C₈×Q₈	—	2	4	no	—
17	C₂×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×#97	C₂²	1	2	yes	—
24	C₂×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	2	yes	—
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	#391	C₄	1	1	yes	characteristic
29	#391	C₄	1	1	yes	characteristic
30	#389	C₄	1	2	yes	—
31	#389	C₄	1	2	yes	—
32	#384	C₄	1	1	yes	characteristic
33	C₄×C₄⋊C₄	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄	C₂²	1	2	yes	—
35	C₄³	C₂²	1	1	yes	characteristic
36	#336	C₂²	1	1	yes	characteristic
37	#336	C₂²	1	2	yes	—
38	#336	C₂²	1	2	yes	—
39	#336	C₂²	1	2	yes	—
40	#336	C₂²	1	1	yes	characteristic
41	#336	C₂²	1	2	yes	—
42	#336	C₂²	1	2	yes	—
43	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×Q₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	2	yes	—
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₈	C₂³	1	1	yes	characteristic
48	C₂²×C₈	C₂³	1	2	yes	—
49	C₂²×C₈	C₂³	1	1	yes	characteristic
50	C₂²×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊Q₈	—	2	4	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	—	2	2	no	—
55	C₄×Q₈	—	2	2	no	—
56	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
64	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²	C₂³	1	2	yes	—
69	C₂×C₄²	C₂³	1	2	yes	—
70	C₂×C₄²	C₂³	1	1	yes	characteristic
71	C₂×C₄²	C₂³	1	1	yes	characteristic
72	C₂×C₄²	C₂³	1	1	yes	characteristic
73	C₄⋊C₈	—	2	4	no	—
74	C₄⋊C₈	—	2	4	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	D₄	1	4	yes	—
78	C₄⋊C₈	Q₈	1	4	yes	complemented
79	#97	—	2	4	no	—
80	#97	—	2	4	no	—
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	2	2	no	—
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	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	M₁₆	1	4	yes	—
94	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄	C₂²×C₄	1	2	yes	—
96	C₂²×C₄	C₂²×C₄	1	2	yes	—
97	C₂²×C₄	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	2	yes	—
102	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄	C₂⁴	1	1	yes	characteristic
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	C₂×D₄	1	4	yes	—
121	C₂×C₈	C₂×Q₈	1	2	yes	—
122	C₂×C₈	C₂×Q₈	1	2	yes	—
123	C₄⋊C₄	—	2	4	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	—	2	2	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	2	4	no	—
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	—	2	2	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	2	2	no	—
138	C₄²	—	2	1	no	—
139	C₄²	—	2	1	no	—
140	C₄²	—	4	1	no	—
141	C₄²	C₂×D₄	1	2	yes	—
142	C₄²	C₂×Q₈	1	2	yes	—
143	C₄²	#41	1	2	yes	—
144	C₄²	#41	1	2	yes	—
145	C₂³	C₂³×C₄	1	1	yes	characteristic
146	Q₈	—	2	8	no	—
147	Q₈	—	4	4	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	1	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	1	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	—	2	1	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	C₄×D₄	1	2	yes	—
171	C₂×C₄	C₄×D₄	1	2	yes	—
172	C₂×C₄	C₄×Q₈	1	2	yes	—
173	C₂×C₄	C₄×Q₈	1	2	yes	—
174	C₂×C₄	C₂×M₁₆	1	4	yes	—
175	C₂×C₄	C₂×M₁₆	1	2	yes	—
176	C₂×C₄	#131	1	2	yes	—
177	C₂×C₄	#131	1	2	yes	—
178	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₄	C₂×#41	1	1	yes	characteristic
181	C₂×C₄	C₂×#41	1	1	yes	characteristic
182	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
183	C₂×C₄	2₋¹⁺⁴	1	1	yes	characteristic
184	C₈	—	2	4	no	—
185	C₈	—	2	2	no	—
186	C₈	—	2	2	no	—
187	C₈	—	4	1	no	—
188	C₈	—	4	1	no	—
189	C₈	—	4	2	no	—
190	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
191	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
192	C₂²	#519	1	1	yes	characteristic
193	C₂²	C₂²×M₁₆	1	1	yes	characteristic
194	C₂²	C₂×#131	1	1	yes	characteristic
195	C₂²	C₂×#131	1	1	yes	characteristic
196	C₂²	#568	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	1	no	—
206	C₄	#435	1	4	yes	—
207	C₄	#547	1	1	yes	characteristic
208	C₄	D₄×Q₈	1	1	yes	characteristic
209	C₄	#556	1	1	yes	characteristic
210	C₄	#557	1	1	yes	characteristic
211	C₂	#2365	1	1	yes	characteristic
212	C₂	C₂×#435	1	1	yes	characteristic
213	C₂	#2935	1	1	yes	characteristic
214	C₂	#2960	1	1	yes	characteristic
215	C₂	Q₈×M₁₆	1	1	yes	characteristic
216	C₂	#2984	1	1	yes	characteristic
217	C₂	#2985	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
78. 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
C₄×C₄⋊Q₈	2
#1984	2
#1940	2
#1917	2
#1916	2
#1771	2
#1767	2
C₄×C₄⋊C₈	2