Subgroups — #101939

Order 512.

304 conjugacy classes of subgroups.

The subgroup lattice has 203 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#9601	C₂	1	2	yes	maximal
2	#8805	C₂	1	1	yes	maximal, characteristic
3	C₄×#334	C₂	1	1	yes	maximal, characteristic
4	#8261	C₂	1	1	yes	maximal, characteristic
5	#7031	C₂	1	2	yes	maximal
6	#1842	C₂²	1	2	yes	—
7	C₄×C₄⋊C₈	C₂²	1	2	yes	—
8	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₄×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	—	2	2	no	—
12	C₂×#334	—	2	1	no	—
13	C₂×#334	—	2	1	no	—
14	C₂×#334	C₄	1	2	yes	—
15	C₂×#322	—	2	1	no	—
16	#1277	—	2	2	no	—
17	#1277	C₄	1	1	yes	characteristic
18	#1277	C₄	1	1	yes	characteristic
19	#1273	—	2	1	no	—
20	C₂×C₄⋊C₈	—	2	2	no	—
21	C₂×C₄⋊C₈	—	2	2	no	—
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄⋊C₈	D₄	1	2	yes	—
27	C₂×C₄×C₈	—	2	2	no	—
28	C₂×C₄×C₈	—	2	1	no	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
35	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
36	C₄³	C₂³	1	1	yes	characteristic
37	#336	—	2	2	no	—
38	#336	—	2	1	no	—
39	#336	Q₈	1	2	yes	—
40	#335	—	4	4	no	—
41	#334	—	2	4	no	—
42	#334	—	4	2	no	—
43	#334	—	4	2	no	—
44	#322	—	4	2	no	—
45	C₂²×C₈	—	2	2	no	—
46	C₂²×C₈	—	2	2	no	—
47	C₂²×C₈	—	2	2	no	—
48	C₂²×C₈	—	2	2	no	—
49	C₂²×C₈	—	4	2	no	—
50	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄²	#41	1	1	yes	characteristic
60	C₂×C₄²	#41	1	1	yes	characteristic
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	4	1	no	—
66	C₄⋊C₈	—	4	1	no	—
67	C₄⋊C₈	—	4	2	no	—
68	C₄⋊C₈	—	4	2	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	8	2	no	—
72	C₄⋊C₈	M₁₆	1	4	yes	—
73	C₄⋊C₈	SD₁₆	1	4	yes	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	4	1	no	—
80	C₄×C₈	—	4	1	no	—
81	C₄×C₈	—	4	1	no	—
82	C₄×C₈	—	4	2	no	—
83	C₄×C₈	—	4	2	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	D₈	1	2	yes	—
87	C₄×C₈	Q₁₆	1	2	yes	—
88	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
90	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
91	C₂²×C₄	#121	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
93	C₂²×C₄	#125	1	1	yes	characteristic
94	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	8	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	—	8	2	no	—
119	C₂×C₈	#107	1	2	yes	—
120	C₂×C₈	#107	1	2	yes	—
121	C₂×C₈	#108	1	2	yes	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	2	no	—
127	C₄²	—	4	1	no	—
128	C₄²	C₂×M₁₆	1	2	yes	—
129	C₄²	#131	1	2	yes	—
130	C₄²	C₂×D₈	1	1	yes	characteristic
131	C₄²	C₂×SD₁₆	1	2	yes	—
132	C₄²	C₂×Q₁₆	1	1	yes	characteristic
133	C₄²	#135	1	2	yes	—
134	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₄²	#137	1	1	yes	characteristic
136	C₂³	#389	1	1	yes	characteristic
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	#423	1	1	yes	characteristic
147	C₂×C₄	#424	1	1	yes	characteristic
148	C₂×C₄	C₂×#107	1	1	yes	characteristic
149	C₂×C₄	#428	1	1	yes	characteristic
150	C₂×C₄	C₂×#108	1	1	yes	characteristic
151	C₂×C₄	#430	1	1	yes	characteristic
152	C₂×C₄	#435	1	2	yes	—
153	C₂×C₄	C₄×SD₁₆	1	2	yes	—
154	C₂×C₄	#444	1	1	yes	characteristic
155	C₂×C₄	#444	1	1	yes	characteristic
156	C₂×C₄	#446	1	2	yes	—
157	C₂×C₄	#459	1	1	yes	characteristic
158	C₂×C₄	#462	1	1	yes	characteristic
159	C₂×C₄	#465	1	2	yes	—
160	C₂×C₄	#471	1	2	yes	—
161	C₂×C₄	#474	1	1	yes	characteristic
162	C₂×C₄	#477	1	1	yes	characteristic
163	C₂×C₄	#498	1	1	yes	characteristic
164	C₂×C₄	#499	1	1	yes	characteristic
165	C₈	—	2	2	no	—
166	C₈	—	2	2	no	—
167	C₈	—	2	2	no	—
168	C₈	—	4	1	no	—
169	C₈	—	4	4	no	—
170	C₈	—	4	4	no	—
171	C₈	—	8	2	no	—
172	C₈	—	8	2	no	—
173	C₈	—	16	2	no	—
174	C₂²	#1940	1	1	yes	characteristic
175	C₂²	#1941	1	1	yes	characteristic
176	C₂²	#1946	1	1	yes	characteristic
177	C₂²	#1947	1	1	yes	characteristic
178	C₂²	#1950	1	1	yes	characteristic
179	C₂²	#1951	1	1	yes	characteristic
180	C₂²	#1953	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	2	1	no	—
185	C₄	—	2	1	no	—
186	C₄	—	4	1	no	—
187	C₄	#1567	1	1	yes	characteristic
188	C₄	#1569	1	1	yes	characteristic
189	C₄	#1571	1	1	yes	characteristic
190	C₄	#1573	1	1	yes	characteristic
191	C₄	#1582	1	2	yes	—
192	C₄	#1583	1	2	yes	—
193	C₄	#1665	1	2	yes	—
194	C₄	#1672	1	2	yes	—
195	C₂	#9597	1	1	yes	characteristic
196	C₂	#9604	1	1	yes	characteristic
197	C₂	#9609	1	1	yes	characteristic
198	C₂	#9611	1	1	yes	characteristic
199	C₂	#9623	1	1	yes	characteristic
200	C₂	#9630	1	1	yes	characteristic
201	C₂	#9635	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#389
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Maximal subgroups

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

subgroup	index
#9601	2
#8805	2
C₄×#334	2
#8261	2
#7031	2