Subgroups — #140383

Order 512.

226 conjugacy classes of subgroups.

The subgroup lattice has 177 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12900	C₂	1	1	yes	maximal, characteristic
2	C₄×#364	C₂	1	1	yes	maximal, characteristic
3	C₄×#364	C₂	1	1	yes	maximal, characteristic
4	#12011	C₂	1	2	yes	maximal
5	#7162	C₂	1	2	yes	maximal
6	C₂×#364	—	2	1	no	—
7	C₂×#364	C₄	1	1	yes	characteristic
8	C₂×#364	C₄	1	1	yes	characteristic
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	1	yes	characteristic
12	#2150	—	2	1	no	—
13	C₄×M₃₂	—	2	1	no	—
14	C₄×M₃₂	C₂²	1	2	yes	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	#1376	—	2	1	no	—
17	#1376	C₄	1	2	yes	—
18	#1373	—	2	4	no	—
19	C₂×M₃₂	—	2	1	no	—
20	C₂×M₃₂	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	C₂×C₄	1	2	yes	—
23	C₂×M₃₂	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₄³	Q₈	1	1	yes	characteristic
31	#364	—	2	1	no	—
32	#364	—	2	1	no	—
33	#364	—	2	2	no	—
34	#364	—	2	1	no	—
35	#364	—	2	1	no	—
36	#364	—	2	2	no	—
37	#364	—	4	1	no	—
38	#364	—	4	1	no	—
39	#364	C₈	1	2	yes	—
40	#364	C₈	1	2	yes	—
41	#364	D₄	1	2	yes	—
42	#364	Q₈	1	2	yes	—
43	C₄⋊C₁₆	—	4	2	no	—
44	#346	—	2	2	no	—
45	#346	—	4	1	no	—
46	C₄×C₁₆	—	2	2	no	—
47	C₄×C₁₆	—	4	1	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×D₄	1	1	yes	characteristic
52	C₂×C₄²	—	4	1	no	—
53	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
56	M₃₂	—	2	2	no	—
57	M₃₂	—	2	2	no	—
58	M₃₂	—	2	2	no	—
59	M₃₂	—	2	2	no	—
60	M₃₂	—	2	2	no	—
61	M₃₂	—	4	1	no	—
62	M₃₂	—	4	1	no	—
63	M₃₂	—	4	2	no	—
64	M₃₂	—	4	2	no	—
65	M₃₂	C₂×C₈	1	4	yes	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	4	1	no	—
69	C₂×C₁₆	—	4	1	no	—
70	C₂×C₁₆	—	4	2	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	1	no	—
77	C₄×C₈	C₂×C₈	1	1	yes	characteristic
78	C₄×C₈	C₂×C₈	1	1	yes	characteristic
79	C₄×C₈	M₁₆	1	1	yes	characteristic
80	C₄×C₈	M₁₆	1	1	yes	characteristic
81	C₄×C₈	D₈	1	1	yes	characteristic
82	C₄×C₈	D₈	1	1	yes	characteristic
83	C₄×C₈	Q₁₆	1	1	yes	characteristic
84	C₄×C₈	Q₁₆	1	1	yes	characteristic
85	C₄×C₈	C₂×D₄	1	1	yes	characteristic
86	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₈	#107	1	1	yes	characteristic
99	C₂×C₈	#107	1	1	yes	characteristic
100	C₂×C₈	#108	1	1	yes	characteristic
101	C₂×C₈	#108	1	1	yes	characteristic
102	C₂×C₈	C₄×D₄	1	1	yes	characteristic
103	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₈	#121	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₈	#131	1	1	yes	characteristic
109	C₂×C₈	#131	1	1	yes	characteristic
110	C₂×C₈	C₂×D₈	1	1	yes	characteristic
111	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
112	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₈	#137	1	1	yes	characteristic
114	C₄²	—	2	1	no	—
115	C₄²	—	2	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	4	4	no	—
118	C₄²	C₄⋊C₈	1	1	yes	characteristic
119	C₄²	C₄⋊C₈	1	1	yes	characteristic
120	C₄²	#107	1	1	yes	characteristic
121	C₄²	#107	1	1	yes	characteristic
122	C₄²	#125	1	1	yes	characteristic
123	C₄²	C₄⋊Q₈	1	1	yes	characteristic
124	C₁₆	—	2	4	no	—
125	C₁₆	—	4	2	no	—
126	C₁₆	—	4	2	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	—	8	2	no	—
129	C₂³	#430	1	1	yes	characteristic
130	C₂×C₄	—	2	1	no	—
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	—	4	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	#389	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
141	C₂×C₄	#424	1	1	yes	characteristic
142	C₂×C₄	C₂×#107	1	1	yes	characteristic
143	C₂×C₄	#428	1	1	yes	characteristic
144	C₂×C₄	C₂×#108	1	1	yes	characteristic
145	C₈	—	4	1	no	—
146	C₈	#335	1	1	yes	characteristic
147	C₈	#335	1	1	yes	characteristic
148	C₈	#335	1	1	yes	characteristic
149	C₈	#335	1	1	yes	characteristic
150	C₈	C₈×D₄	1	1	yes	characteristic
151	C₈	#436	1	1	yes	characteristic
152	C₈	C₈×Q₈	1	1	yes	characteristic
153	C₈	#446	1	1	yes	characteristic
154	C₈	#459	1	1	yes	characteristic
155	C₈	#462	1	1	yes	characteristic
156	C₈	#474	1	1	yes	characteristic
157	C₈	#477	1	1	yes	characteristic
158	C₂²	—	4	1	no	—
159	C₂²	#1565	1	1	yes	characteristic
160	C₂²	#1571	1	1	yes	characteristic
161	C₂²	#1950	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	4	1	no	—
166	C₄	—	4	1	no	—
167	C₄	—	4	1	no	—
168	C₄	C₂×#335	1	1	yes	characteristic
169	C₄	#1574	1	1	yes	characteristic
170	C₄	#1939	1	1	yes	characteristic
171	C₄	#1947	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#9601	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9601
prime radical	C₂³	#430
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ 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
#12900	2
C₄×#364	2
#12011	2
#7162	2