Subgroups — #129193

Order 512.

244 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12012	C₂	1	1	yes	maximal, characteristic
2	#11408	C₂	1	1	yes	maximal, characteristic
3	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×#1311	C₂	1	1	yes	maximal, characteristic
5	C₈×#108	C₂	1	1	yes	maximal, characteristic
6	#7163	C₂	1	1	yes	maximal, characteristic
7	#7163	C₂	1	1	yes	maximal, characteristic
8	#2162	C₂²	1	1	yes	characteristic
9	#2162	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×#108	C₄	1	1	yes	characteristic
13	#1564	C₄	1	1	yes	characteristic
14	C₈×M₁₆	C₂²	1	1	yes	characteristic
15	C₈×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	#1370	C₄	1	2	yes	—
18	#1367	C₄	1	2	yes	—
19	#1312	—	2	1	no	—
20	#1312	—	2	1	no	—
21	#1311	—	2	1	no	—
22	#1311	C₄	1	2	yes	—
23	#1311	C₄	1	2	yes	—
24	C₈×C₁₆	—	2	1	no	—
25	#1278	C₄	1	1	yes	characteristic
26	#1278	C₄	1	1	yes	characteristic
27	C₂²×C₁₆	—	2	1	no	—
28	C₂²×C₁₆	Q₈	1	2	yes	—
29	#431	C₂×C₄	1	1	yes	characteristic
30	#431	C₂×C₄	1	1	yes	characteristic
31	C₂×#108	C₈	1	2	yes	—
32	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₄×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
38	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
39	C₂²⋊C₁₆	—	2	1	no	—
40	C₂²⋊C₁₆	—	2	1	no	—
41	C₄×C₁₆	—	2	1	no	—
42	C₄×C₁₆	—	2	1	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	C₂×C₄	1	2	yes	—
45	C₄×C₁₆	C₂×C₄	1	2	yes	—
46	C₄×C₁₆	D₄	1	1	yes	characteristic
47	C₄×C₁₆	D₄	1	1	yes	characteristic
48	#341	C₈	1	2	yes	—
49	#335	C₈	1	2	yes	—
50	#334	C₈	1	2	yes	—
51	#322	—	2	1	no	—
52	#322	—	2	1	no	—
53	C₈²	—	2	1	no	—
54	C₈²	—	2	1	no	—
55	C₈²	C₂×C₄	1	1	yes	characteristic
56	C₈²	C₂×C₄	1	1	yes	characteristic
57	C₈²	C₂×C₄	1	1	yes	characteristic
58	C₈²	C₂×C₄	1	1	yes	characteristic
59	C₂×M₁₆	C₂×C₈	1	2	yes	—
60	C₂×M₁₆	C₂×C₈	1	2	yes	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₈	#41	1	1	yes	characteristic
66	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₁₆	—	2	4	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
73	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
74	#108	—	2	4	no	—
75	C₄⋊C₈	C₂×C₈	1	2	yes	—
76	C₄⋊C₈	C₂×C₈	1	2	yes	—
77	C₂²⋊C₈	—	2	1	no	—
78	C₂²⋊C₈	—	2	1	no	—
79	#97	—	2	1	no	—
80	#97	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	—	2	1	no	—
84	C₄×C₈	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	C₄²	1	1	yes	characteristic
87	C₄×C₈	C₄²	1	1	yes	characteristic
88	C₄×C₈	C₄²	1	1	yes	characteristic
89	C₄×C₈	C₄²	1	1	yes	characteristic
90	C₄×C₈	C₂×C₈	1	1	yes	characteristic
91	C₄×C₈	C₂×C₈	1	1	yes	characteristic
92	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₄×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄×C₈	#41	1	1	yes	characteristic
96	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
97	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
98	C₂²×C₄	#131	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	2	2	no	—
101	M₁₆	—	2	2	no	—
102	M₁₆	M₃₂	1	4	yes	—
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	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	C₄×C₈	1	1	yes	characteristic
115	C₂×C₈	C₄×C₈	1	1	yes	characteristic
116	C₂×C₈	C₄×C₈	1	1	yes	characteristic
117	C₂×C₈	C₄×C₈	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₈	#117	1	1	yes	characteristic
120	C₂×C₈	C₄×D₄	1	1	yes	characteristic
121	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
122	C₄²	C₂×C₄²	1	1	yes	characteristic
123	C₄²	C₄×D₄	1	1	yes	characteristic
124	C₄²	C₂²×C₈	1	1	yes	characteristic
125	C₄²	#131	1	1	yes	characteristic
126	C₁₆	—	2	4	no	—
127	C₁₆	—	2	2	no	—
128	C₁₆	—	4	2	no	—
129	C₁₆	—	4	2	no	—
130	C₁₆	#108	1	4	yes	complemented
131	C₂³	C₈×Q₈	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
137	C₂×C₄	#405	1	1	yes	characteristic
138	C₂×C₄	C₈×D₄	1	1	yes	characteristic
139	C₂×C₄	C₈×D₄	1	1	yes	characteristic
140	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
141	C₂×C₄	C₂×M₃₂	1	2	yes	—
142	C₂×C₄	#504	1	2	yes	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	4	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	4	2	no	—
153	C₈	—	4	2	no	—
154	C₈	—	4	2	no	—
155	C₈	C₂×#108	1	2	yes	—
156	C₈	#443	1	2	yes	—
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
160	C₂²	#2183	1	2	yes	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	#1772	1	1	yes	characteristic
164	C₄	C₄×#108	1	1	yes	characteristic
165	C₄	C₄×M₃₂	1	1	yes	characteristic
166	C₄	#2108	1	1	yes	characteristic
167	C₄	#2168	1	2	yes	—
168	C₄	#2171	1	2	yes	—
169	C₂	—	2	2	no	—
170	C₂	C₈×#108	1	1	yes	characteristic
171	C₂	#12663	1	1	yes	characteristic
172	C₂	#12665	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₄⋊C₄
prime radical	C₂³	C₈×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
130. C₁₆	#108	semidirect

Maximal subgroups

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

subgroup	index
#12012	2
#11408	2
C₁₆×M₁₆	2
C₂×#1311	2
C₈×#108	2
#7163	2