Subgroups — #135644

Order 512.

322 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12898	C₂	1	2	yes	maximal
2	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
3	#7151	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₁₆	—	2	2	no	—
5	C₂×C₄⋊C₁₆	—	2	2	no	—
6	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
7	C₄²×C₈	C₄	1	1	yes	characteristic
8	C₄²×C₈	C₄	1	1	yes	characteristic
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₈²	C₄	1	2	yes	—
11	C₂×C₈²	C₂²	1	2	yes	—
12	C₂²×C₁₆	—	2	4	no	—
13	C₂×C₄×C₈	—	2	1	no	—
14	C₂×C₄×C₈	—	2	1	no	—
15	C₂×C₄×C₈	C₈	1	2	yes	—
16	C₂×C₄×C₈	C₈	1	1	yes	characteristic
17	C₂×C₄×C₈	C₈	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
24	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	C₄³	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₁₆	—	2	8	no	—
28	C₄⋊C₁₆	—	4	4	no	—
29	C₄⋊C₁₆	—	4	4	no	—
30	C₈²	—	2	4	no	—
31	C₈²	—	2	4	no	—
32	C₂²×C₈	—	2	2	no	—
33	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₈	C₂×C₈	1	2	yes	—
36	C₂²×C₈	C₂×C₈	1	2	yes	—
37	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄²	M₁₆	1	1	yes	characteristic
44	C₂×C₄²	M₁₆	1	1	yes	characteristic
45	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₁₆	—	4	4	no	—
47	C₂×C₁₆	—	4	4	no	—
48	C₂×C₁₆	—	4	4	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	—	2	4	no	—
51	C₄×C₈	—	2	4	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	C₁₆	1	4	yes	complemented
60	C₄×C₈	M₁₆	1	4	yes	—
61	C₄×C₈	D₈	1	2	yes	—
62	C₄×C₈	SD₁₆	1	4	yes	—
63	C₄×C₈	Q₁₆	1	2	yes	—
64	C₄×C₈	#41	1	4	yes	—
65	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
66	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
67	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
68	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
71	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	#106	1	2	yes	—
86	C₂×C₈	#107	1	2	yes	—
87	C₂×C₈	C₂×C₁₆	1	4	yes	—
88	C₂×C₈	M₃₂	1	4	yes	—
89	C₂×C₈	#124	1	2	yes	—
90	C₂×C₈	#125	1	2	yes	—
91	C₂×C₈	C₂×D₈	1	1	yes	characteristic
92	C₂×C₈	C₂×SD₁₆	1	2	yes	—
93	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
94	C₄²	—	2	2	no	—
95	C₄²	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	#106	1	2	yes	—
98	C₄²	#107	1	2	yes	—
99	C₄²	#108	1	2	yes	—
100	C₄²	#108	1	2	yes	—
101	C₄²	C₂×C₁₆	1	2	yes	—
102	C₄²	M₃₂	1	2	yes	—
103	C₄²	#117	1	2	yes	—
104	C₄²	C₂×M₁₆	1	2	yes	—
105	C₁₆	—	8	4	no	—
106	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	#334	1	2	yes	—
114	C₂×C₄	#334	1	2	yes	—
115	C₂×C₄	#335	1	2	yes	—
116	C₂×C₄	#335	1	2	yes	—
117	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
118	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
119	C₂×C₄	#364	1	2	yes	—
120	C₂×C₄	#364	1	2	yes	—
121	C₂×C₄	#382	1	1	yes	characteristic
122	C₂×C₄	#423	1	1	yes	characteristic
123	C₂×C₄	C₂×#106	1	1	yes	characteristic
124	C₂×C₄	C₂×#107	1	1	yes	characteristic
125	C₂×C₄	C₂×#108	1	1	yes	characteristic
126	C₂×C₄	C₂×#108	1	1	yes	characteristic
127	C₂×C₄	#431	1	2	yes	—
128	C₂×C₄	#432	1	2	yes	—
129	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
130	C₂×C₄	C₂×M₃₂	1	2	yes	—
131	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
132	C₈	—	2	8	no	—
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	—	2	4	no	—
137	C₈	—	2	4	no	—
138	C₈	#486	1	2	yes	—
139	C₈	#487	1	2	yes	—
140	C₂²	C₂×#334	1	1	yes	characteristic
141	C₂²	C₂×#335	1	1	yes	characteristic
142	C₂²	#1831	1	1	yes	characteristic
143	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
144	C₂²	#2150	1	1	yes	characteristic
145	C₂²	C₂×#364	1	1	yes	characteristic
146	C₂²	C₂×#364	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	#1367	1	4	yes	—
151	C₄	#1370	1	4	yes	—
152	C₄	#1566	1	1	yes	characteristic
153	C₄	#1567	1	1	yes	characteristic
154	C₄	#1834	1	1	yes	characteristic
155	C₄	#1837	1	1	yes	characteristic
156	C₄	#2162	1	2	yes	—
157	C₄	#2163	1	2	yes	—
158	C₂	#8784	1	1	yes	characteristic
159	C₂	C₂×#1367	1	1	yes	characteristic
160	C₂	#11996	1	1	yes	characteristic
161	C₂	C₂×#1370	1	1	yes	characteristic
162	C₂	#12004	1	1	yes	characteristic
163	C₂	#12879	1	1	yes	characteristic
164	C₂	#12883	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	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×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
59. C₄×C₈	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#12898	2
C₄×C₈²	2
#7151	2