Subgroups — #146987

Order 512.

642 conjugacy classes of subgroups.

The subgroup lattice has 191 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	2	yes	maximal
3	C₂×#1381	C₂	1	4	yes	maximal
4	C₂³×C₁₆	C₂²	1	2	yes	—
5	C₂²×#108	—	2	2	no	—
6	C₂²×#108	C₂²	1	4	yes	—
7	C₂×#423	—	2	2	no	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	—	2	4	no	—
10	C₂×C₄×C₁₆	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₄	1	2	yes	—
12	#1848	—	2	4	no	—
13	#1848	C₄	1	8	yes	—
14	#1381	—	2	8	no	—
15	C₂²×M₁₆	—	2	4	no	—
16	C₂³×C₈	D₄	1	1	yes	characteristic
17	C₂³×C₈	C₂³	1	1	yes	characteristic
18	C₂²×C₄²	D₄	1	1	yes	characteristic
19	C₂²×C₁₆	—	2	4	no	—
20	C₂²×C₁₆	—	2	4	no	—
21	C₂²×C₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₁₆	D₄	1	4	yes	—
23	C₂²×C₁₆	Q₈	1	4	yes	—
24	C₂×#108	—	2	8	no	—
25	C₂×#108	—	2	8	no	—
26	C₂×#108	—	2	8	no	—
27	C₂×#108	—	2	4	no	—
28	C₂×#108	—	4	4	no	—
29	C₂×#108	—	4	4	no	—
30	C₂×#108	C₂×C₄	1	8	yes	—
31	#423	—	2	8	no	—
32	#423	—	4	4	no	—
33	C₂×C₄⋊C₈	—	4	4	no	—
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
37	C₂×C₄×C₈	D₄	1	4	yes	—
38	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
40	C₄×C₁₆	—	2	4	no	—
41	C₄×C₁₆	—	2	4	no	—
42	C₂³×C₄	Q₁₆	1	2	yes	—
43	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
44	C₂×M₁₆	—	2	8	no	—
45	C₂×M₁₆	—	4	4	no	—
46	C₂×M₁₆	—	4	4	no	—
47	C₂×M₁₆	—	4	4	no	—
48	C₂×M₁₆	—	4	4	no	—
49	C₂×M₁₆	—	4	4	no	—
50	C₂²×C₈	—	4	4	no	—
51	C₂²×C₈	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	D₈	1	2	yes	—
55	C₂²×C₈	SD₁₆	1	4	yes	—
56	C₂²×C₈	Q₁₆	1	2	yes	—
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂²×C₄	1	2	yes	—
59	C₂²×C₈	C₂×D₄	1	2	yes	—
60	C₂²×C₈	C₂×D₄	1	2	yes	—
61	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₈	C₂×Q₈	1	2	yes	—
63	C₂²×C₈	#41	1	4	yes	—
64	C₂×C₄²	C₂²⋊C₄	1	4	yes	—
65	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	D₈	1	2	yes	—
68	C₂×C₄²	SD₁₆	1	4	yes	—
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	—	2	8	no	—
72	C₂×C₁₆	—	2	8	no	—
73	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
74	#108	—	2	16	no	—
75	#108	—	4	16	no	—
76	#108	—	4	8	no	—
77	C₄⋊C₈	—	4	16	no	—
78	C₄×C₈	C₄²	1	2	yes	—
79	C₄×C₈	C₄²	1	2	yes	—
80	C₄×C₈	C₂²⋊C₄	1	2	yes	—
81	C₄×C₈	C₂²⋊C₄	1	2	yes	—
82	C₄×C₈	C₄⋊C₄	1	4	yes	—
83	C₄×C₈	C₄⋊C₄	1	4	yes	—
84	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
85	C₂²×C₄	—	2	8	no	—
86	C₂²×C₄	#103	1	8	yes	—
87	C₂²×C₄	#107	1	2	yes	—
88	C₂²×C₄	#107	1	2	yes	—
89	C₂²×C₄	C₂×C₂²⋊C₄	1	2	yes	—
90	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	#124	1	2	yes	—
92	C₂²×C₄	#127	1	1	yes	characteristic
93	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
96	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
97	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
98	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
99	M₁₆	—	4	8	no	—
100	M₁₆	—	4	8	no	—
101	M₁₆	—	8	4	no	—
102	M₁₆	—	8	4	no	—
103	C₂×C₈	—	4	8	no	—
104	C₂×C₈	—	8	4	no	—
105	C₂×C₈	—	8	4	no	—
106	C₂×C₈	#95	1	2	yes	—
107	C₂×C₈	#95	1	2	yes	—
108	C₂×C₈	#102	1	8	yes	—
109	C₂×C₈	#103	1	8	yes	—
110	C₂×C₈	#106	1	2	yes	—
111	C₂×C₈	#106	1	2	yes	—
112	C₂×C₈	#107	1	2	yes	—
113	C₂×C₈	#107	1	2	yes	—
114	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₈	#117	1	2	yes	—
119	C₂×C₈	C₄×D₄	1	4	yes	—
120	C₂×C₈	C₄×Q₈	1	4	yes	—
121	C₂×C₈	#121	1	4	yes	—
122	C₂×C₈	C₂²⋊Q₈	1	4	yes	—
123	C₂×C₈	#125	1	2	yes	—
124	C₂×C₈	C₄⋊Q₈	1	2	yes	—
125	C₄²	#95	1	2	yes	—
126	C₄²	#95	1	2	yes	—
127	C₄²	#102	1	8	yes	—
128	C₄²	#106	1	2	yes	—
129	C₄²	#106	1	2	yes	—
130	C₁₆	—	2	8	no	—
131	C₂³	—	2	4	no	—
132	C₂³	C₂×#103	1	2	yes	—
133	C₂³	C₂×#107	1	1	yes	characteristic
134	C₂³	#487	1	2	yes	—
135	C₂³	#494	1	1	yes	characteristic
136	C₂³	#500	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	16	no	—
139	C₂×C₄	#340	1	4	yes	—
140	C₂×C₄	#340	1	4	yes	—
141	C₂×C₄	#368	1	8	yes	—
142	C₂×C₄	C₂×#95	1	1	yes	characteristic
143	C₂×C₄	#382	1	1	yes	characteristic
144	C₂×C₄	#384	1	1	yes	characteristic
145	C₂×C₄	#390	1	2	yes	—
146	C₂×C₄	#391	1	2	yes	—
147	C₂×C₄	C₂×#102	1	2	yes	—
148	C₂×C₄	C₂×#102	1	2	yes	—
149	C₂×C₄	C₂×#103	1	2	yes	—
150	C₂×C₄	C₂×#106	1	1	yes	characteristic
151	C₂×C₄	C₂×#106	1	1	yes	characteristic
152	C₂×C₄	C₂×#107	1	1	yes	characteristic
153	C₂×C₄	#486	1	2	yes	—
154	C₂×C₄	#486	1	2	yes	—
155	C₂×C₄	#487	1	2	yes	—
156	C₂×C₄	#492	1	2	yes	—
157	C₂×C₄	#493	1	1	yes	characteristic
158	C₂×C₄	#498	1	2	yes	—
159	C₂×C₄	#500	1	1	yes	characteristic
160	C₈	—	8	8	no	—
161	C₈	#340	1	4	yes	—
162	C₈	#340	1	4	yes	—
163	C₈	C₄×C₄⋊C₄	1	2	yes	—
164	C₈	#380	1	2	yes	—
165	C₈	#389	1	4	yes	—
166	C₂²	—	2	4	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	C₂×#340	1	1	yes	characteristic
171	C₂²	#1834	1	1	yes	characteristic
172	C₂²	#1845	1	1	yes	characteristic
173	C₂²	#1956	1	2	yes	—
174	C₂²	#1990	1	2	yes	—
175	C₂²	C₂×#368	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	#1381	1	8	yes	complemented
178	C₄	C₄⋊#95	1	1	yes	characteristic
179	C₄	C₂×#340	1	1	yes	characteristic
180	C₄	#1834	1	1	yes	characteristic
181	C₄	#1844	1	1	yes	characteristic
182	C₄	#1957	1	2	yes	—
183	C₄	#1990	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	—	2	2	no	—
186	C₂	#10841	1	1	yes	characteristic
187	C₂	C₂×#1381	1	2	yes	—
188	C₂	#12891	1	2	yes	—
189	C₂	#12909	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
177. C₄	#1381	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₆	2
C₂×#1848	2
C₂×#1381	2