Subgroups — #483939

Order 512.

432 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33734	C₂	1	1	yes	maximal, characteristic
2	#17958	C₂	1	2	yes	maximal
3	#12083	C₂	1	4	yes	maximal, complemented
4	C₈×Q₃₂	C₂	1	4	yes	maximal, complemented
5	#12022	C₂	1	1	yes	maximal, characteristic
6	#11672	C₂	1	2	yes	maximal
7	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
8	#3170	C₂²	1	2	yes	—
9	#2960	—	2	2	no	—
10	#2252	C₄	1	1	yes	characteristic
11	#2247	C₄	1	1	yes	characteristic
12	#2241	C₄	1	2	yes	—
13	#2216	C₄	1	4	yes	—
14	C₄×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1591	—	2	2	no	—
18	#1591	C₂²	1	4	yes	—
19	C₈×Q₁₆	—	2	2	no	—
20	C₈×Q₁₆	C₂²	1	4	yes	—
21	#1567	C₂²	1	2	yes	—
22	#1493	—	2	2	no	—
23	C₂×C₈²	C₂²	1	1	yes	characteristic
24	#1371	C₂²	1	4	yes	—
25	#1332	C₂²	1	8	yes	—
26	C₈×C₁₆	C₂²	1	2	yes	—
27	C₈×C₁₆	C₂²	1	2	yes	—
28	#533	—	2	2	no	—
29	C₂×Q₃₂	C₂×C₄	1	4	yes	—
30	C₂²×C₁₆	—	2	1	no	—
31	C₂²×C₁₆	C₂×C₄	1	2	yes	—
32	#500	C₂×C₄	1	2	yes	—
33	#494	C₂×C₄	1	2	yes	—
34	#487	—	2	2	no	—
35	#467	—	2	2	no	—
36	#467	C₂×C₄	1	4	yes	—
37	#446	—	2	4	no	—
38	#446	—	4	2	no	—
39	C₈×Q₈	—	2	4	no	—
40	C₈×Q₈	—	4	2	no	—
41	C₄×Q₁₆	—	2	2	no	—
42	C₄×Q₁₆	C₂³	1	4	yes	—
43	#424	—	4	2	no	—
44	#423	—	2	2	no	—
45	#405	—	4	2	no	—
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	#366	C₂×C₄	1	4	yes	—
50	#358	C₂×C₄	1	8	yes	—
51	C₄×C₁₆	—	2	1	no	—
52	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
53	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
54	C₄×C₁₆	C₂³	1	2	yes	—
55	C₄×C₁₆	C₂³	1	2	yes	—
56	#335	—	2	2	no	—
57	#335	C₂³	1	4	yes	—
58	Q₈⋊C₈	—	2	4	no	—
59	Q₈⋊C₈	—	2	4	no	—
60	C₈²	D₄	1	1	yes	characteristic
61	C₈²	D₄	1	1	yes	characteristic
62	C₈²	C₂³	1	1	yes	characteristic
63	C₈²	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆	—	2	2	no	—
65	C₂×Q₁₆	C₂²×C₄	1	4	yes	—
66	C₂×M₁₆	—	4	2	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	D₈	1	1	yes	characteristic
69	C₂²×C₈	D₈	1	1	yes	characteristic
70	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₈	#41	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	—	2	2	no	—
74	#125	—	4	2	no	—
75	C₂²⋊Q₈	—	2	4	no	—
76	C₂²⋊Q₈	—	4	2	no	—
77	C₄×Q₈	—	2	4	no	—
78	C₄×Q₈	—	4	2	no	—
79	#117	—	4	2	no	—
80	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
81	Q₃₂	—	2	8	no	—
82	C₂×C₁₆	—	2	4	no	—
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	—	2	4	no	—
85	C₂×C₁₆	C₂²×C₄	1	2	yes	—
86	C₂×C₁₆	C₂²×C₄	1	2	yes	—
87	#107	—	2	2	no	—
88	#107	C₂²×C₄	1	4	yes	—
89	C₄⋊C₈	—	2	4	no	—
90	C₄⋊C₈	—	4	4	no	—
91	C₄⋊C₈	—	4	4	no	—
92	C₄⋊C₈	—	4	2	no	—
93	#103	—	2	4	no	—
94	#103	—	2	4	no	—
95	#97	—	4	4	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	4	4	no	—
98	C₄×C₈	D₈	1	1	yes	characteristic
99	C₄×C₈	D₈	1	1	yes	characteristic
100	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄×C₈	C₂×D₄	1	1	yes	characteristic
103	C₄×C₈	C₂×D₄	1	1	yes	characteristic
104	C₄×C₈	C₂×D₄	1	1	yes	characteristic
105	C₄×C₈	C₂×D₄	1	1	yes	characteristic
106	C₄×C₈	#41	1	1	yes	characteristic
107	C₄×C₈	C₂⁴	1	1	yes	characteristic
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	4	2	no	—
110	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
112	C₂²×C₄	#135	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	4	4	no	—
115	M₁₆	—	8	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	—	4	4	no	—
123	C₂×C₈	Q₃₂	1	4	yes	complemented
124	C₂×C₈	C₄×D₄	1	1	yes	characteristic
125	C₂×C₈	C₄×D₄	1	1	yes	characteristic
126	C₂×C₈	C₂×D₈	1	1	yes	characteristic
127	C₂×C₈	C₂×D₈	1	1	yes	characteristic
128	C₂×C₈	C₂×D₈	1	1	yes	characteristic
129	C₂×C₈	C₂×D₈	1	1	yes	characteristic
130	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₈	C₂×#41	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	2	no	—
138	C₂²⋊C₄	—	4	4	no	—
139	C₄²	—	4	4	no	—
140	C₄²	C₄×D₄	1	1	yes	characteristic
141	C₄²	C₂×D₈	1	1	yes	characteristic
142	C₄²	#135	1	1	yes	characteristic
143	C₄²	C₂²×D₄	1	1	yes	characteristic
144	C₁₆	—	2	4	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	#131	1	4	yes	—
147	C₂³	C₄×D₈	1	1	yes	characteristic
148	Q₈	—	4	4	no	—
149	Q₈	—	4	4	no	—
150	Q₈	—	8	4	no	—
151	C₂×C₄	—	2	1	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	C₄×D₈	1	1	yes	characteristic
157	C₂×C₄	C₄×D₈	1	1	yes	characteristic
158	C₂×C₄	C₄×D₈	1	1	yes	characteristic
159	C₂×C₄	C₂×Q₃₂	1	2	yes	—
160	C₂×C₄	#508	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
162	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
163	C₂×C₄	C₂×#135	1	1	yes	characteristic
164	C₈	—	2	2	no	—
165	C₈	—	2	2	no	—
166	C₈	—	2	1	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	8	4	no	—
171	C₈	#443	1	2	yes	—
172	C₈	C₂×Q₃₂	1	2	yes	—
173	C₈	C₂×Q₃₂	1	2	yes	—
174	C₈	C₂×#131	1	2	yes	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	C₄×Q₃₂	1	2	yes	—
178	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	8	4	no	—
182	C₄	—	8	4	no	—
183	C₄	C₄×Q₃₂	1	2	yes	—
184	C₄	#2178	1	2	yes	—
185	C₄	#2931	1	1	yes	characteristic
186	C₄	C₂×#443	1	1	yes	characteristic
187	C₄	C₂²×Q₃₂	1	1	yes	characteristic
188	C₄	C₂×#508	1	1	yes	characteristic
189	C₂	—	2	2	no	—
190	C₂	#17956	1	1	yes	characteristic
191	C₂	C₂×C₄×Q₃₂	1	1	yes	characteristic
192	C₂	C₂×#2178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12083	C₂	semidirect
4. C₈×Q₃₂	C₂	semidirect
123. C₂×C₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#33734	2
#17958	2
#12083	2
C₈×Q₃₂	2
#12022	2
#11672	2
C₂×C₈×C₁₆	2