Subgroups — #483339

Order 512.

533 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33398	C₂	1	2	yes	maximal
2	C₈×#131	C₂	1	1	yes	maximal, characteristic
3	#12005	C₂	1	6	yes	maximal, complemented
4	#12004	C₂	1	3	yes	maximal, complemented
5	C₂×#1370	C₂	1	3	yes	maximal, complemented
6	C₂×#504	—	2	2	no	—
7	C₈×#41	C₄	1	1	yes	characteristic
8	C₈×#41	C₄	1	1	yes	characteristic
9	C₄×#131	C₂²	1	1	yes	characteristic
10	#2151	—	2	6	no	—
11	#2150	C₂²	1	6	yes	—
12	C₂×C₄⋊C₁₆	C₂²	1	6	yes	—
13	#1450	C₂²	1	3	yes	—
14	C₈×M₁₆	C₄	1	3	yes	—
15	C₈×M₁₆	C₄	1	3	yes	—
16	C₂×C₈²	C₂²	1	3	yes	—
17	#1370	C₂²	1	1	yes	characteristic
18	#1370	C₂²	1	3	yes	—
19	#1370	C₂²	1	6	yes	—
20	#1370	C₂²	1	6	yes	—
21	C₂×#131	D₄	1	1	yes	characteristic
22	C₂×#131	Q₈	1	1	yes	characteristic
23	C₄×#41	C₂×C₄	1	1	yes	characteristic
24	#504	—	4	8	no	—
25	C₂×M₃₂	—	2	6	no	—
26	C₂²×C₁₆	—	2	6	no	—
27	C₈×Q₈	—	2	1	no	—
28	C₈×Q₈	C₈	1	1	yes	characteristic
29	C₈×Q₈	C₈	1	1	yes	characteristic
30	C₈×D₄	—	2	3	no	—
31	C₈×D₄	C₈	1	3	yes	—
32	C₈×D₄	C₈	1	3	yes	—
33	#431	C₂×C₄	1	3	yes	—
34	#431	C₂×C₄	1	3	yes	—
35	#405	D₄	1	3	yes	—
36	#405	Q₈	1	3	yes	—
37	C₄×M₁₆	C₂×C₄	1	3	yes	—
38	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
39	C₂×C₄×C₈	C₂×C₄	1	3	yes	—
40	C₂×C₄×C₈	C₂³	1	3	yes	—
41	C₄⋊C₁₆	—	2	6	no	—
42	C₄⋊C₁₆	—	2	6	no	—
43	C₄⋊C₁₆	C₂³	1	2	yes	—
44	C₄⋊C₁₆	C₂³	1	6	yes	—
45	#322	C₂×C₄	1	6	yes	—
46	C₈²	C₂×C₄	1	3	yes	—
47	C₈²	C₂×C₄	1	3	yes	—
48	C₈²	C₂³	1	1	yes	characteristic
49	C₈²	C₂³	1	3	yes	—
50	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
51	#131	—	2	4	no	—
52	#131	D₈	1	4	yes	—
53	#131	Q₁₆	1	4	yes	—
54	C₂×M₁₆	C₄⋊C₄	1	3	yes	—
55	C₂×M₁₆	C₄⋊C₄	1	3	yes	—
56	C₂²×C₈	—	2	3	no	—
57	C₂²×C₈	C₂×C₈	1	3	yes	—
58	C₂²×C₈	C₂×C₈	1	3	yes	—
59	C₂²×C₈	C₂×D₄	1	3	yes	—
60	C₂²×C₈	C₂×Q₈	1	3	yes	—
61	C₄×Q₈	C₂×C₈	1	1	yes	characteristic
62	C₄×Q₈	M₁₆	1	1	yes	characteristic
63	C₄×D₄	C₂×C₈	1	3	yes	—
64	C₄×D₄	M₁₆	1	3	yes	—
65	#117	C₄⋊C₄	1	3	yes	—
66	C₂×C₄²	C₂²×C₄	1	3	yes	—
67	M₃₂	—	4	12	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	6	no	—
70	C₂×C₁₆	—	4	12	no	—
71	C₄⋊C₈	—	2	3	no	—
72	C₄⋊C₈	C₂×C₈	1	3	yes	—
73	C₄⋊C₈	C₂×C₈	1	3	yes	—
74	C₂²⋊C₈	—	2	3	no	—
75	C₂²⋊C₈	C₂×C₈	1	6	yes	—
76	#97	C₄⋊C₄	1	6	yes	—
77	C₄×C₈	—	2	3	no	—
78	C₄×C₈	C₂×C₈	1	3	yes	—
79	C₄×C₈	C₂×C₈	1	3	yes	—
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂²×C₄	1	3	yes	—
82	C₄×C₈	C₂²×C₄	1	3	yes	—
83	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₄×C₈	C₂²×C₄	1	3	yes	—
85	C₄×C₈	C₂×D₄	1	3	yes	—
86	C₄×C₈	C₂×Q₈	1	3	yes	—
87	C₄×C₈	C₂⁴	1	1	yes	characteristic
88	#41	#107	1	4	yes	—
89	#41	#108	1	4	yes	—
90	C₂×Q₈	C₄⋊C₈	1	1	yes	characteristic
91	C₂×D₄	C₄⋊C₈	1	3	yes	—
92	C₂²×C₄	C₂×C₄⋊C₄	1	3	yes	—
93	C₂²×C₄	C₂²×C₈	1	3	yes	—
94	C₂²×C₄	C₂×M₁₆	1	3	yes	—
95	M₁₆	—	2	6	no	—
96	M₁₆	#107	1	12	yes	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	6	no	—
99	C₂×C₈	—	2	6	no	—
100	C₂×C₈	—	2	3	no	—
101	C₂×C₈	—	4	3	no	—
102	C₂×C₈	—	4	6	no	—
103	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
104	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
105	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₈	C₂²×C₈	1	3	yes	—
107	C₂×C₈	C₂²×C₈	1	3	yes	—
108	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₈	C₂×D₈	1	6	yes	—
110	C₂×C₈	C₂×Q₁₆	1	6	yes	—
111	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
113	C₄⋊C₄	C₄⋊C₈	1	3	yes	—
114	C₄⋊C₄	C₄⋊C₈	1	3	yes	—
115	C₂²⋊C₄	C₄⋊C₈	1	6	yes	—
116	C₄²	C₂×C₄⋊C₄	1	3	yes	—
117	C₄²	C₂²×C₈	1	3	yes	—
118	C₄²	C₂×M₁₆	1	3	yes	—
119	C₄²	C₂³×C₄	1	1	yes	characteristic
120	C₁₆	—	4	2	no	—
121	C₁₆	—	4	6	no	—
122	C₂³	C₂×C₄⋊C₈	1	3	yes	—
123	Q₈	#335	1	4	yes	—
124	D₄	#335	1	12	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	4	3	no	—
128	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
129	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
130	C₂×C₄	C₂×C₄⋊C₈	1	3	yes	—
131	C₂×C₄	C₂×#107	1	6	yes	—
132	C₂×C₄	C₂×#108	1	6	yes	—
133	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
135	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	—	2	3	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	3	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	3	no	—
142	C₈	—	4	3	no	—
143	C₈	C₂×#107	1	6	yes	—
144	C₈	#504	1	2	yes	complemented
145	C₈	C₂²×D₈	1	1	yes	characteristic
146	C₈	C₂²×Q₁₆	1	1	yes	characteristic
147	C₂²	—	2	3	no	—
148	C₂²	—	2	3	no	—
149	C₂²	C₂×#335	1	6	yes	—
150	C₂²	C₂²×C₄⋊C₈	1	1	yes	characteristic
151	C₄	—	2	3	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	3	no	—
154	C₄	C₂×#335	1	6	yes	—
155	C₄	C₂²×#107	1	1	yes	characteristic
156	C₄	C₂²×#108	1	1	yes	characteristic
157	C₄	C₂×#504	1	1	yes	characteristic
158	C₄	#3407	1	1	yes	characteristic
159	C₂	—	2	6	no	—
160	C₂	C₂²×#335	1	1	yes	characteristic
161	C₂	#33398	1	1	yes	characteristic
162	C₂	#33401	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×#131 ◁ 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. #12005	C₂	semidirect
4. #12004	C₂	semidirect
5. C₂×#1370	C₂	semidirect
144. C₈	#504	semidirect

Maximal subgroups

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

subgroup	index
#33398	2
C₈×#131	2
#12005	2
#12004	2
C₂×#1370	2