Subgroups — #127435

Order 512.

396 conjugacy classes of subgroups.

The subgroup lattice has 187 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2162	C₂	1	1	yes	maximal, characteristic
2	C₂×#1568	C₂	1	2	yes	maximal
3	#7242	C₂	1	4	yes	maximal, complemented
4	C₂×#432	—	2	2	no	—
5	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
6	#2162	—	2	2	no	—
7	#2162	—	2	2	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	C₄	1	2	yes	—
11	#1568	—	2	2	no	—
12	#1568	—	2	2	no	—
13	#1568	C₄	1	4	yes	—
14	#1568	C₂²	1	4	yes	—
15	C₂×#335	—	2	2	no	—
16	C₂×#334	C₄	1	4	yes	—
17	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₈	Q₈	1	1	yes	characteristic
19	C₂²×C₄²	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	4	yes	—
21	#432	—	2	4	no	—
22	#432	—	2	4	no	—
23	#432	—	4	2	no	—
24	#432	—	4	2	no	—
25	C₂×C₄⋊C₈	—	2	4	no	—
26	C₂×C₂²⋊C₈	—	4	2	no	—
27	C₂×#97	—	4	2	no	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	2	yes	—
33	C₂×C₄×C₈	D₄	1	2	yes	—
34	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₄⋊C₁₆	—	2	4	no	—
37	C₂²⋊C₁₆	—	2	4	no	—
38	C₄×C₁₆	—	2	2	no	—
39	C₄×C₁₆	—	2	2	no	—
40	#335	—	2	4	no	—
41	#335	—	2	4	no	—
42	#334	—	2	4	no	—
43	#334	C₂×C₄	1	8	yes	—
44	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₄	D₈	1	1	yes	characteristic
46	C₂³×C₄	Q₁₆	1	1	yes	characteristic
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	—	4	2	no	—
49	C₂²×C₈	—	4	4	no	—
50	C₂²×C₈	C₄²	1	2	yes	—
51	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₄⋊C₄	1	2	yes	—
54	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₄⋊C₄	1	2	yes	—
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄²	D₈	1	1	yes	characteristic
62	C₂×C₄²	SD₁₆	1	2	yes	—
63	C₂×C₄²	SD₁₆	1	2	yes	—
64	C₂×C₄²	Q₁₆	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₁₆	—	2	4	no	—
67	C₂×C₁₆	—	4	4	no	—
68	C₄⋊C₈	—	2	8	no	—
69	C₄⋊C₈	—	4	4	no	—
70	C₂²⋊C₈	—	4	4	no	—
71	C₂²⋊C₈	—	4	4	no	—
72	#97	—	4	4	no	—
73	#97	—	4	4	no	—
74	C₄×C₈	C₄²	1	1	yes	characteristic
75	C₄×C₈	C₄²	1	1	yes	characteristic
76	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₈	C₂²⋊C₄	1	2	yes	—
79	C₄×C₈	C₂²⋊C₄	1	2	yes	—
80	C₄×C₈	C₄⋊C₄	1	2	yes	—
81	C₄×C₈	C₄⋊C₄	1	2	yes	—
82	C₄×C₈	C₂²×C₄	1	2	yes	—
83	C₄×C₈	C₂×D₄	1	2	yes	—
84	C₂⁴	#107	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	#95	1	2	yes	—
90	C₂²×C₄	#102	1	2	yes	—
91	C₂²×C₄	#102	1	2	yes	—
92	C₂²×C₄	#103	1	2	yes	—
93	C₂²×C₄	#103	1	2	yes	—
94	C₂²×C₄	#106	1	2	yes	—
95	C₂²×C₄	#107	1	1	yes	characteristic
96	C₂²×C₄	#107	1	1	yes	characteristic
97	C₂²×C₄	#107	1	1	yes	characteristic
98	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
100	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	—	4	1	no	—
109	C₂×C₈	—	4	8	no	—
110	C₂×C₈	—	8	8	no	—
111	C₂×C₈	#95	1	1	yes	characteristic
112	C₂×C₈	#95	1	2	yes	—
113	C₂×C₈	#95	1	1	yes	characteristic
114	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
117	C₄²	#95	1	1	yes	characteristic
118	C₄²	#95	1	1	yes	characteristic
119	C₄²	#102	1	2	yes	—
120	C₄²	#102	1	2	yes	—
121	C₄²	#103	1	2	yes	—
122	C₄²	#103	1	2	yes	—
123	C₄²	#106	1	1	yes	characteristic
124	C₄²	#106	1	1	yes	characteristic
125	C₄²	C₂×C₂²⋊C₄	1	2	yes	—
126	C₄²	C₂×SD₁₆	1	2	yes	—
127	C₁₆	—	4	4	no	—
128	C₂³	—	4	1	no	—
129	C₂³	—	4	1	no	—
130	C₂³	#340	1	2	yes	—
131	C₂³	#368	1	4	yes	—
132	C₂³	C₂×#107	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	—	4	1	no	—
147	C₂×C₄	#340	1	1	yes	characteristic
148	C₂×C₄	#340	1	1	yes	characteristic
149	C₂×C₄	#340	1	1	yes	characteristic
150	C₂×C₄	#340	1	2	yes	—
151	C₂×C₄	#340	1	1	yes	characteristic
152	C₂×C₄	C₂×#95	1	1	yes	characteristic
153	C₂×C₄	C₂×#102	1	2	yes	—
154	C₂×C₄	C₂×#103	1	2	yes	—
155	C₂×C₄	C₂×#106	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	2	no	—
158	C₈	—	2	2	no	—
159	C₈	—	2	1	no	—
160	C₈	—	8	8	no	—
161	C₈	#338	1	2	yes	—
162	C₈	#413	1	2	yes	—
163	C₂²	—	2	2	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	4	4	no	—
168	C₂²	#1381	1	4	yes	—
169	C₂²	C₂×#340	1	1	yes	characteristic
170	C₂²	C₂×#368	1	2	yes	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	2	2	no	—
174	C₄	—	2	2	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	2	no	—
177	C₄	—	2	1	no	—
178	C₄	#1384	1	2	yes	—
179	C₄	C₂×#338	1	1	yes	characteristic
180	C₄	#1742	1	1	yes	characteristic
181	C₂	—	2	4	no	—
182	C₂	—	2	2	no	—
183	C₂	#8001	1	1	yes	characteristic
184	C₂	C₂×#1381	1	1	yes	characteristic
185	C₂	C₂×#1384	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #7242	C₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#2162	2
C₂×#1568	2
#7242	2