Subgroups — #140338

Order 512.

274 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13159	C₂	1	2	yes	maximal
2	C₂×#1370	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	—	2	2	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×#107	C₄	1	1	yes	characteristic
11	C₈×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#335	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1370	—	2	2	no	—
16	C₈×C₁₆	—	2	2	no	—
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂×#107	C₈	1	1	yes	characteristic
23	C₂×#107	C₈	1	1	yes	characteristic
24	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	—	2	2	no	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₄⋊C₁₆	—	2	2	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	—	4	4	no	—
33	C₄×C₁₆	—	2	1	no	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	—	2	1	no	—
36	C₄×C₁₆	Q₈	1	2	yes	—
37	#340	C₈	1	2	yes	—
38	#336	—	2	2	no	—
39	#335	—	2	2	no	—
40	C₈²	D₄	1	2	yes	—
41	C₈²	Q₈	1	2	yes	—
42	C₂²×C₈	—	2	1	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₈	#41	1	1	yes	characteristic
48	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
49	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
50	C₂×C₄²	—	2	2	no	—
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	2	1	no	—
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	4	2	no	—
60	C₂×C₁₆	—	4	2	no	—
61	C₂×C₁₆	—	4	2	no	—
62	C₂×C₁₆	D₈	1	2	yes	—
63	C₂×C₁₆	Q₁₆	1	2	yes	—
64	#107	—	2	2	no	—
65	#107	C₁₆	1	4	yes	complemented
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	4	2	no	—
70	C₄×C₈	—	4	2	no	—
71	C₄×C₈	C₄⋊C₄	1	2	yes	—
72	C₄×C₈	C₄⋊C₄	1	2	yes	—
73	C₄×C₈	M₁₆	1	2	yes	—
74	C₄×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₄×C₈	#41	1	1	yes	characteristic
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
83	C₂²×C₄	#131	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	C₄⋊C₈	1	2	yes	—
95	C₂×C₈	C₄⋊C₈	1	2	yes	—
96	C₂×C₈	C₂×C₁₆	1	2	yes	—
97	C₂×C₈	M₃₂	1	2	yes	—
98	C₂×C₈	#121	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₈	#125	1	1	yes	characteristic
101	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₈	C₂×D₈	1	1	yes	characteristic
103	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₈	#135	1	1	yes	characteristic
105	C₂×C₈	#135	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	C₂×C₁₆	1	4	yes	—
110	C₄²	—	4	2	no	—
111	C₄²	—	4	2	no	—
112	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₄²	C₄×Q₈	1	1	yes	characteristic
114	C₄²	C₂×M₁₆	1	1	yes	characteristic
115	C₄²	#131	1	1	yes	characteristic
116	C₁₆	—	2	4	no	—
117	C₁₆	—	2	2	no	—
118	C₁₆	—	2	2	no	—
119	C₁₆	—	8	2	no	—
120	C₂³	C₈×D₄	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	#389	1	1	yes	characteristic
130	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
131	C₂×C₄	#424	1	1	yes	characteristic
132	C₂×C₄	#436	1	1	yes	characteristic
133	C₂×C₄	C₄×D₈	1	1	yes	characteristic
134	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
135	C₂×C₄	#443	1	1	yes	characteristic
136	C₂×C₄	#443	1	1	yes	characteristic
137	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
138	C₂×C₄	#446	1	1	yes	characteristic
139	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
140	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
141	C₂×C₄	#504	1	1	yes	characteristic
142	C₂×C₄	#504	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₈	—	2	2	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	8	2	no	—
148	C₈	C₄⋊C₁₆	1	4	yes	complemented
149	C₈	#466	1	1	yes	characteristic
150	C₈	#467	1	1	yes	characteristic
151	C₈	#499	1	1	yes	characteristic
152	C₈	#500	1	1	yes	characteristic
153	C₂²	C₈×D₈	1	1	yes	characteristic
154	C₂²	C₈×Q₁₆	1	1	yes	characteristic
155	C₂²	#1939	1	1	yes	characteristic
156	C₂²	D₄×C₁₆	1	1	yes	characteristic
157	C₂²	#2169	1	1	yes	characteristic
158	C₂²	#2170	1	1	yes	characteristic
159	C₂²	#2170	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	4	4	no	—
163	C₄	—	8	2	no	—
164	C₄	#1589	1	1	yes	characteristic
165	C₄	#1591	1	1	yes	characteristic
166	C₄	#1942	1	1	yes	characteristic
167	C₄	#1945	1	1	yes	characteristic
168	C₄	C₂×C₄⋊C₁₆	1	1	yes	characteristic
169	C₄	#2151	1	1	yes	characteristic
170	C₄	Q₈×C₁₆	1	1	yes	characteristic
171	C₄	#2183	1	1	yes	characteristic
172	C₂	#9594	1	1	yes	characteristic
173	C₂	C₁₆×D₈	1	1	yes	characteristic
174	C₂	#12039	1	1	yes	characteristic
175	C₂	C₁₆×Q₁₆	1	1	yes	characteristic
176	C₂	#12054	1	1	yes	characteristic
177	C₂	#13159	1	1	yes	characteristic
178	C₂	#13161	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
65. #107	C₁₆	semidirect
148. C₈	C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#13159	2
C₂×#1370	2
C₂×C₈×C₁₆	2
C₈×#107	2
#7167	2