Subgroups — #21717

Order 256.

383 conjugacy classes of subgroups.

The subgroup lattice has 189 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	#2399	C₂	1	1	yes	maximal, characteristic
3	#1984	C₂	1	2	yes	maximal
4	#1940	C₂	1	1	yes	maximal, characteristic
5	#1940	C₂	1	1	yes	maximal, characteristic
6	#1939	C₂	1	2	yes	maximal
7	#1843	C₂	1	1	yes	maximal, characteristic
8	#1771	C₂	1	2	yes	maximal
9	#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#1726	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈	C₂²	1	2	yes	—
13	#446	—	2	8	no	—
14	C₂×C₄⋊C₈	C₂²	1	2	yes	—
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	2	yes	—
19	C₂×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×#97	C₂²	1	2	yes	—
21	C₂×#97	C₂²	1	1	yes	characteristic
22	C₂×#97	C₂²	1	1	yes	characteristic
23	C₂×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
28	#398	C₄	1	2	yes	—
29	#398	C₄	1	2	yes	—
30	#395	C₄	1	2	yes	—
31	#395	C₄	1	2	yes	—
32	C₄×C₄⋊C₄	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	#376	C₂²	1	1	yes	characteristic
36	#336	C₂²	1	2	yes	—
37	#336	C₂²	1	2	yes	—
38	#336	C₂²	1	2	yes	—
39	#336	C₂²	1	2	yes	—
40	C₂²×Q₈	C₂×C₄	1	2	yes	—
41	C₂²×C₈	C₂³	1	2	yes	—
42	C₂²×C₈	C₂³	1	2	yes	—
43	C₂²×C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₈	C₂³	1	1	yes	characteristic
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	C₄×Q₈	—	2	4	no	—
48	C₄×Q₈	—	2	4	no	—
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×C₄⋊C₄	C₂×C₄	1	2	yes	—
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	C₂×C₄²	C₂³	1	1	yes	characteristic
61	C₄⋊C₈	—	2	4	no	—
62	C₄⋊C₈	—	2	4	no	—
63	C₄⋊C₈	—	2	4	no	—
64	C₄⋊C₈	—	2	4	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	#97	—	2	4	no	—
68	#97	—	2	2	no	—
69	#97	D₄	1	4	yes	—
70	#97	Q₈	1	4	yes	complemented
71	C₄×C₈	—	2	4	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	Q₈	1	4	yes	complemented
75	#95	C₂×C₄	1	2	yes	—
76	#95	C₂×C₄	1	2	yes	—
77	#95	C₂×C₄	1	1	yes	characteristic
78	#95	C₂×C₄	1	2	yes	—
79	#95	C₂×C₄	1	1	yes	characteristic
80	#95	C₂×C₄	1	2	yes	—
81	#95	C₂×C₄	1	1	yes	characteristic
82	#95	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₈	—	2	4	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂×Q₈	—	4	2	no	—
86	C₂²×C₄	C₂²×C₄	1	2	yes	—
87	C₂²×C₄	C₂²×C₄	1	2	yes	—
88	C₂²×C₄	C₂²×C₄	1	2	yes	—
89	C₂²×C₄	C₂²×C₄	1	2	yes	—
90	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄	C₂²×C₄	1	2	yes	—
92	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×C₄	1	2	yes	—
94	C₂²×C₄	C₂⁴	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	C₂×D₄	1	2	yes	—
108	C₂×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₈	C₂×Q₈	1	2	yes	—
110	C₂×C₈	C₂×Q₈	1	2	yes	—
111	C₂×C₈	C₂×Q₈	1	2	yes	—
112	C₂×C₈	C₂×Q₈	1	2	yes	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	C₂×D₄	1	2	yes	—
125	C₄²	C₂×Q₈	1	2	yes	—
126	C₄²	C₂×Q₈	1	2	yes	—
127	C₄²	#41	1	2	yes	—
128	C₄²	#41	1	2	yes	—
129	C₄²	#41	1	2	yes	—
130	C₂³	C₂³×C₄	1	1	yes	characteristic
131	Q₈	—	4	8	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	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₄×D₄	1	4	yes	—
150	C₂×C₄	C₄×Q₈	1	2	yes	—
151	C₂×C₄	C₄×Q₈	1	2	yes	—
152	C₂×C₄	C₄×Q₈	1	4	yes	—
153	C₂×C₄	#131	1	4	yes	—
154	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄	C₂×#41	1	1	yes	characteristic
158	C₂×C₄	C₂×#41	1	1	yes	characteristic
159	C₂×C₄	C₂×#41	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	4	2	no	—
164	C₈	—	4	2	no	—
165	C₈	C₄⋊Q₈	1	4	yes	—
166	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
168	C₂²	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₂²	C₂×#131	1	2	yes	—
170	C₂²	#568	1	2	yes	—
171	C₄	—	2	4	no	—
172	C₄	—	2	4	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	4	no	—
176	C₄	—	4	2	no	—
177	C₄	—	4	2	no	—
178	C₄	—	4	2	no	—
179	C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
180	C₄	#532	1	1	yes	characteristic
181	C₄	#533	1	1	yes	characteristic
182	C₄	#533	1	1	yes	characteristic
183	C₂	C₄×C₄⋊Q₈	1	1	yes	characteristic
184	C₂	#2933	1	2	yes	—
185	C₂	#2960	1	1	yes	characteristic
186	C₂	#2961	1	1	yes	characteristic
187	C₂	#2962	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
70. #97	Q₈	semidirect
74. 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
#2399	2
#1984	2
#1940	2
#1939	2
#1843	2
#1771	2
#1770	2
C₈×C₄⋊C₄	2
#1726	2