Subgroups — #497961

Order 512.

446 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	C₂×#2183	C₂	1	1	yes	maximal, characteristic
2	Q₈×#108	C₂	1	2	yes	maximal
3	#13179	C₂	1	2	yes	maximal
4	#13179	C₂	1	4	yes	maximal
5	#12892	C₂	1	2	yes	maximal
6	#12891	C₂	1	1	yes	maximal, characteristic
7	#12676	C₂	1	2	yes	maximal
8	C₄×#368	C₂	1	1	yes	maximal, characteristic
9	Q₈×M₁₆	—	2	2	no	—
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	#2183	C₄	1	4	yes	—
12	#2183	C₄	1	4	yes	—
13	C₂×#368	C₂²	1	2	yes	—
14	C₂×#368	C₂²	1	4	yes	—
15	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×#346	C₂²	1	2	yes	—
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	#1950	—	2	4	no	—
20	#1950	—	2	2	no	—
21	#1848	C₂²	1	2	yes	—
22	#1848	C₂²	1	4	yes	—
23	C₄×#108	C₂²	1	2	yes	—
24	C₄×#108	C₂²	1	4	yes	—
25	#1381	C₂²	1	2	yes	—
26	#1381	C₂²	1	2	yes	—
27	#1381	C₂²	1	1	yes	characteristic
28	#1381	C₂²	1	4	yes	—
29	#1381	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	C₂²×C₁₆	C₂³	1	1	yes	characteristic
33	C₂²×C₁₆	C₂³	1	2	yes	—
34	#446	—	4	2	no	—
35	#446	—	4	4	no	—
36	C₈×Q₈	—	4	2	no	—
37	C₈×Q₈	C₂×C₄	1	2	yes	—
38	C₈×Q₈	C₂×C₄	1	2	yes	—
39	C₈×Q₈	Q₈	1	2	yes	—
40	C₈×Q₈	Q₈	1	2	yes	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	—	2	4	no	—
43	C₂×#108	C₂³	1	4	yes	—
44	C₂×#108	C₂³	1	2	yes	—
45	C₂×#108	C₂³	1	2	yes	—
46	#423	—	2	2	no	—
47	#423	—	2	4	no	—
48	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₈	D₄	1	2	yes	—
50	C₄×M₁₆	—	2	2	no	—
51	C₄×M₁₆	—	2	4	no	—
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	2	yes	—
54	#368	—	2	4	no	—
55	#368	—	2	8	no	—
56	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
57	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
58	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
59	#346	C₂×C₄	1	8	yes	—
60	C₄×C₁₆	C₂×C₄	1	2	yes	—
61	C₄×C₁₆	C₂×C₄	1	2	yes	—
62	#341	—	2	2	no	—
63	#341	—	2	4	no	—
64	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
65	C₂×M₁₆	—	2	2	no	—
66	C₂×M₁₆	—	2	2	no	—
67	C₂×M₁₆	—	2	4	no	—
68	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₄	1	2	yes	—
70	C₂²×C₈	C₂⁴	1	1	yes	characteristic
71	C₄×Q₈	C₄⋊C₄	1	2	yes	—
72	C₄×Q₈	C₄⋊C₄	1	2	yes	—
73	C₄×Q₈	D₈	1	2	yes	—
74	C₄×Q₈	D₈	1	2	yes	—
75	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
76	C₂×C₄⋊C₄	Q₁₆	1	2	yes	—
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄²	C₂×D₄	1	2	yes	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	4	no	—
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	C₂²×C₄	1	2	yes	—
83	C₂×C₁₆	C₂²×C₄	1	2	yes	—
84	C₂×C₁₆	C₂²×C₄	1	4	yes	—
85	#108	—	2	2	no	—
86	#108	—	2	4	no	—
87	#108	—	2	2	no	—
88	#108	—	2	4	no	—
89	#108	—	2	2	no	—
90	#108	—	8	2	no	—
91	#108	—	8	4	no	—
92	#108	#41	1	4	yes	—
93	C₄⋊C₈	—	4	2	no	—
94	C₄⋊C₈	—	4	4	no	—
95	C₄⋊C₈	—	4	4	no	—
96	C₄⋊C₈	—	4	2	no	—
97	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
98	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
99	C₄⋊C₈	C₄⋊C₄	1	4	yes	—
100	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
101	#97	—	4	4	no	—
102	#97	—	4	2	no	—
103	C₄×C₈	—	4	2	no	—
104	C₄×C₈	—	4	4	no	—
105	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₄	1	4	yes	—
108	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
109	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₄×C₈	C₂×Q₈	1	4	yes	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	#107	1	2	yes	—
113	C₂×Q₈	#107	1	2	yes	—
114	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
115	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
116	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
117	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
118	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
119	M₁₆	—	2	4	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	4	no	—
123	M₁₆	—	4	4	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
133	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
135	C₂×C₈	C₂×#41	1	2	yes	—
136	C₂×C₈	2₋¹⁺⁴	1	2	yes	—
137	C₄⋊C₄	#107	1	4	yes	—
138	C₄⋊C₄	#107	1	2	yes	—
139	C₄⋊C₄	#107	1	4	yes	—
140	C₄⋊C₄	#107	1	2	yes	—
141	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₄²	C₂×C₄⋊C₄	1	4	yes	—
143	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₄²	C₂×D₈	1	1	yes	characteristic
145	C₄²	C₂×D₈	1	4	yes	—
146	C₄²	C₂×D₈	1	1	yes	characteristic
147	C₁₆	—	2	2	no	—
148	C₁₆	—	2	4	no	—
149	C₁₆	—	2	2	no	—
150	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
151	Q₈	—	2	4	no	—
152	Q₈	—	4	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	C₂×#107	1	1	yes	characteristic
158	C₂×C₄	C₂×#107	1	4	yes	—
159	C₂×C₄	C₂×#107	1	1	yes	characteristic
160	C₂×C₄	C₂×#107	1	1	yes	characteristic
161	C₂×C₄	C₂×#107	1	4	yes	—
162	C₂×C₄	C₂×#107	1	1	yes	characteristic
163	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄	#548	1	2	yes	—
165	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
166	C₂×C₄	#577	1	2	yes	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	2	4	no	—
170	C₈	—	4	4	no	—
171	C₈	—	8	2	no	—
172	C₈	—	8	4	no	—
173	C₈	C₄×#41	1	1	yes	characteristic
174	C₈	#519	1	1	yes	characteristic
175	C₈	#557	1	2	yes	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	C₂²×#107	1	1	yes	characteristic
179	C₂²	#3363	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	4	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	1	no	—
185	C₄	#2161	1	2	yes	—
186	C₄	#2351	1	1	yes	characteristic
187	C₄	#2912	1	1	yes	characteristic
188	C₄	#3358	1	2	yes	—
189	C₂	—	2	2	no	—
190	C₂	#22740	1	1	yes	characteristic
191	C₂	C₂×#2161	1	1	yes	characteristic
192	C₂	#33414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂³	C₂²×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#2183	2
Q₈×#108	2
#13179	2
#12892	2
#12891	2
#12676	2
C₄×#368	2