Subgroups — #140398

Order 512.

264 conjugacy classes of subgroups.

The subgroup lattice has 197 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1371	C₂	1	2	yes	maximal
3	#8806	C₂	1	1	yes	maximal, characteristic
4	#8274	C₂	1	1	yes	maximal, characteristic
5	#7154	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	#1842	—	2	1	no	—
9	#1767	—	2	1	no	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	—	2	1	no	—
12	C₂×#335	C₂²	1	2	yes	—
13	C₂×#335	C₂²	1	1	yes	characteristic
14	C₂×#335	C₂²	1	1	yes	characteristic
15	#1374	C₄	1	1	yes	characteristic
16	#1374	C₄	1	1	yes	characteristic
17	#1371	—	2	4	no	—
18	#1277	—	2	1	no	—
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₈	—	2	1	no	—
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	—	4	1	no	—
25	C₂×C₄⋊C₈	—	4	1	no	—
26	C₂×#97	—	4	1	no	—
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	C₄³	D₄	1	1	yes	characteristic
34	C₄⋊C₁₆	—	2	4	no	—
35	C₄×C₁₆	—	2	2	no	—
36	C₄×C₁₆	—	2	2	no	—
37	#336	—	4	1	no	—
38	#335	—	2	2	no	—
39	#335	—	2	1	no	—
40	#335	—	2	1	no	—
41	#335	—	2	2	no	—
42	#335	—	4	2	no	—
43	#335	D₄	1	2	yes	—
44	#335	Q₈	1	2	yes	—
45	C₂²×C₈	—	4	1	no	—
46	C₂²×C₈	—	4	1	no	—
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	D₈	1	1	yes	characteristic
55	C₂×C₄²	D₈	1	1	yes	characteristic
56	C₂×C₄²	Q₁₆	1	1	yes	characteristic
57	C₂×C₄²	Q₁₆	1	1	yes	characteristic
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	M₁₆	1	4	yes	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	C₄⋊C₈	—	4	1	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄⋊C₈	—	4	1	no	—
71	C₄⋊C₈	—	8	2	no	—
72	C₄⋊C₈	—	8	2	no	—
73	#97	—	8	2	no	—
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	C₂×D₄	1	1	yes	characteristic
81	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₂²×C₄	#107	1	1	yes	characteristic
85	C₂²×C₄	#107	1	1	yes	characteristic
86	C₂²×C₄	#107	1	1	yes	characteristic
87	C₂²×C₄	#107	1	1	yes	characteristic
88	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	8	1	no	—
101	C₂×C₈	—	8	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	—	8	1	no	—
104	C₂×C₈	—	8	2	no	—
105	C₂×C₈	—	8	2	no	—
106	C₂×C₈	—	8	2	no	—
107	C₂×C₈	#108	1	2	yes	—
108	C₂×C₈	C₄×D₄	1	1	yes	characteristic
109	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₈	#125	1	1	yes	characteristic
111	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
112	C₂×C₈	C₂×M₁₆	1	2	yes	—
113	C₂×C₈	#131	1	2	yes	—
114	C₄²	—	2	2	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	4	1	no	—
122	C₄²	D₁₆	1	2	yes	—
123	C₄²	Q₃₂	1	2	yes	—
124	C₄²	#121	1	1	yes	characteristic
125	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
126	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₄²	#137	1	1	yes	characteristic
128	C₁₆	—	2	4	no	—
129	C₁₆	—	4	2	no	—
130	C₂³	C₂×#107	1	1	yes	characteristic
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
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	#366	1	2	yes	—
143	C₂×C₄	#366	1	2	yes	—
144	C₂×C₄	#368	1	2	yes	—
145	C₂×C₄	#389	1	1	yes	characteristic
146	C₂×C₄	#423	1	1	yes	characteristic
147	C₂×C₄	#424	1	1	yes	characteristic
148	C₂×C₄	#428	1	1	yes	characteristic
149	C₂×C₄	C₂×#108	1	1	yes	characteristic
150	C₂×C₄	#430	1	1	yes	characteristic
151	C₂×C₄	#459	1	1	yes	characteristic
152	C₂×C₄	#462	1	1	yes	characteristic
153	C₂×C₄	#474	1	1	yes	characteristic
154	C₂×C₄	#477	1	1	yes	characteristic
155	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
156	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
157	C₂×C₄	#509	1	1	yes	characteristic
158	C₂×C₄	#510	1	1	yes	characteristic
159	C₈	—	2	2	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	4	1	no	—
163	C₈	—	8	2	no	—
164	C₈	—	8	2	no	—
165	C₈	—	16	2	no	—
166	C₈	#435	1	2	yes	—
167	C₈	#446	1	2	yes	—
168	C₂²	#1567	1	1	yes	characteristic
169	C₂²	#1571	1	1	yes	characteristic
170	C₂²	#1947	1	1	yes	characteristic
171	C₂²	C₂×#366	1	1	yes	characteristic
172	C₂²	#2159	1	1	yes	characteristic
173	C₂²	C₂×#368	1	1	yes	characteristic
174	C₂²	#2161	1	1	yes	characteristic
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	1	no	—
181	C₄	#1569	1	1	yes	characteristic
182	C₄	#1573	1	1	yes	characteristic
183	C₄	#1940	1	1	yes	characteristic
184	C₄	#1950	1	1	yes	characteristic
185	C₄	#2206	1	1	yes	characteristic
186	C₄	#2209	1	1	yes	characteristic
187	C₄	#2224	1	1	yes	characteristic
188	C₄	#2227	1	1	yes	characteristic
189	C₂	#9604	1	1	yes	characteristic
190	C₂	#12022	1	1	yes	characteristic
191	C₂	#12025	1	1	yes	characteristic
192	C₂	#12032	1	1	yes	characteristic
193	C₂	#12035	1	1	yes	characteristic
194	C₂	#13176	1	1	yes	characteristic
195	C₂	#13179	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₄×C₄⋊C₁₆	2
C₂×#1371	2
#8806	2
#8274	2
#7154	2