Subgroups — #146993

Order 512.

656 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#20797	C₂	1	2	yes	maximal
3	C₂×#1384	C₂	1	4	yes	maximal
4	C₂²×M₃₂	C₂²	1	2	yes	—
5	C₂×#427	—	2	2	no	—
6	C₂×#427	C₂²	1	4	yes	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	#2285	—	2	2	no	—
9	C₄×M₃₂	—	2	2	no	—
10	C₄×M₃₂	—	2	2	no	—
11	C₂×#346	C₄	1	2	yes	complemented
12	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
13	#1846	—	2	2	no	—
14	#1846	C₄	1	4	yes	complemented
15	#1845	—	2	2	no	—
16	#1844	C₄	1	4	yes	complemented
17	#1384	—	2	8	no	—
18	C₂³×C₈	D₄	1	1	yes	characteristic
19	C₂³×C₈	C₂³	1	1	yes	characteristic
20	C₂×#117	—	2	4	no	—
21	C₂²×C₄²	D₄	1	1	yes	characteristic
22	C₂×M₃₂	—	2	4	no	—
23	C₂×M₃₂	—	2	4	no	—
24	C₂×M₃₂	D₄	1	4	yes	—
25	C₂×M₃₂	Q₈	1	4	yes	—
26	C₂²×C₁₆	C₂×C₄	1	4	yes	—
27	#427	—	2	8	no	—
28	#427	—	2	8	no	—
29	#427	—	4	4	no	—
30	#427	—	4	4	no	—
31	C₂×#107	—	2	4	no	—
32	C₂×#107	—	2	4	no	—
33	C₂×#106	—	2	4	no	—
34	C₂×#106	C₂×C₄	1	8	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₄×C₈	D₄	1	2	yes	—
40	C₂×C₄×C₈	D₄	1	2	yes	—
41	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
42	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
43	#384	—	2	4	no	—
44	#382	—	2	4	no	—
45	#380	—	4	4	no	—
46	C₄×C₄⋊C₄	—	4	4	no	—
47	#346	—	2	4	no	—
48	C₄×C₁₆	—	2	2	no	—
49	C₄×C₁₆	—	2	2	no	—
50	C₂³×C₄	D₈	1	2	yes	—
51	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	D₈	1	2	yes	—
57	C₂²×C₈	SD₁₆	1	4	yes	—
58	C₂²×C₈	Q₁₆	1	2	yes	—
59	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₈	C₂²×C₄	1	2	yes	—
61	C₂²×C₈	C₂×D₄	1	2	yes	—
62	C₂²×C₈	C₂×D₄	1	2	yes	—
63	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₈	C₂×Q₈	1	2	yes	—
65	C₂²×C₈	#41	1	4	yes	—
66	#117	—	4	4	no	—
67	#117	—	4	4	no	—
68	#117	—	4	4	no	—
69	#117	—	4	4	no	—
70	C₂×C₄⋊C₄	—	2	8	no	—
71	C₂×C₄⋊C₄	—	4	8	no	—
72	C₂×C₂²⋊C₄	—	4	4	no	—
73	C₂×C₄²	—	4	4	no	—
74	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
76	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄²	SD₁₆	1	4	yes	—
79	C₂×C₄²	Q₁₆	1	2	yes	—
80	M₃₂	—	2	8	no	—
81	M₃₂	—	2	8	no	—
82	C₂×C₁₆	—	2	4	no	—
83	C₂×C₁₆	—	2	4	no	—
84	C₂×C₁₆	C₄⋊C₄	1	8	yes	complemented
85	#107	—	4	8	no	—
86	#107	—	4	8	no	—
87	#106	—	2	16	no	—
88	#106	—	4	8	no	—
89	C₄×C₈	C₄²	1	1	yes	characteristic
90	C₄×C₈	C₄²	1	1	yes	characteristic
91	C₄×C₈	C₄²	1	2	yes	—
92	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊C₄	1	2	yes	—
95	C₄×C₈	C₄⋊C₄	1	2	yes	—
96	C₄×C₈	C₄⋊C₄	1	2	yes	—
97	C₄×C₈	C₄⋊C₄	1	2	yes	—
98	C₄×C₈	C₄⋊C₄	1	2	yes	—
99	#95	—	4	8	no	—
100	C₂⁴	C₂×D₈	1	1	yes	characteristic
101	C₂²×C₄	—	4	4	no	—
102	C₂²×C₄	—	4	8	no	—
103	C₂²×C₄	#102	1	4	yes	—
104	C₂²×C₄	#102	1	4	yes	—
105	C₂²×C₄	#107	1	2	yes	—
106	C₂²×C₄	#107	1	2	yes	—
107	C₂²×C₄	C₂×C₂²⋊C₄	1	2	yes	—
108	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄	#124	1	2	yes	—
110	C₂²×C₄	#127	1	1	yes	characteristic
111	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
113	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
114	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
115	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
116	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₈	#95	1	1	yes	characteristic
118	C₂×C₈	#95	1	1	yes	characteristic
119	C₂×C₈	#95	1	2	yes	—
120	C₂×C₈	#102	1	4	yes	—
121	C₂×C₈	#102	1	4	yes	—
122	C₂×C₈	#103	1	4	yes	—
123	C₂×C₈	#103	1	4	yes	—
124	C₂×C₈	#106	1	2	yes	—
125	C₂×C₈	#106	1	2	yes	—
126	C₂×C₈	#107	1	2	yes	—
127	C₂×C₈	#107	1	2	yes	—
128	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₈	#117	1	2	yes	—
133	C₂×C₈	C₄×D₄	1	4	yes	—
134	C₂×C₈	C₄×Q₈	1	4	yes	—
135	C₂×C₈	#121	1	4	yes	—
136	C₂×C₈	C₂²⋊Q₈	1	4	yes	—
137	C₂×C₈	#125	1	2	yes	—
138	C₂×C₈	C₄⋊Q₈	1	2	yes	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	8	16	no	—
142	C₂²⋊C₄	—	8	4	no	—
143	C₂²⋊C₄	—	8	4	no	—
144	C₄²	—	8	4	no	—
145	C₄²	—	8	4	no	—
146	C₄²	#95	1	1	yes	characteristic
147	C₄²	#95	1	2	yes	—
148	C₄²	#95	1	1	yes	characteristic
149	C₄²	#103	1	4	yes	—
150	C₄²	#103	1	4	yes	—
151	C₄²	#106	1	2	yes	—
152	C₄²	#106	1	2	yes	—
153	C₁₆	—	2	8	no	—
154	C₂³	—	4	2	no	—
155	C₂³	C₂×#102	1	2	yes	—
156	C₂³	C₂×#107	1	1	yes	characteristic
157	C₂³	#486	1	2	yes	—
158	C₂³	#493	1	1	yes	characteristic
159	C₂³	#500	1	1	yes	characteristic
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	8	16	no	—
165	C₂×C₄	—	8	8	no	—
166	C₂×C₄	#340	1	4	yes	—
167	C₂×C₄	#340	1	2	yes	—
168	C₂×C₄	#340	1	2	yes	—
169	C₂×C₄	C₂×#95	1	1	yes	characteristic
170	C₂×C₄	#382	1	1	yes	characteristic
171	C₂×C₄	#384	1	1	yes	characteristic
172	C₂×C₄	#390	1	2	yes	—
173	C₂×C₄	#391	1	2	yes	—
174	C₂×C₄	C₂×#102	1	2	yes	—
175	C₂×C₄	C₂×#103	1	2	yes	—
176	C₂×C₄	C₂×#103	1	2	yes	—
177	C₂×C₄	C₂×#106	1	1	yes	characteristic
178	C₂×C₄	C₂×#106	1	1	yes	characteristic
179	C₂×C₄	C₂×#107	1	1	yes	characteristic
180	C₂×C₄	#486	1	2	yes	—
181	C₂×C₄	#487	1	2	yes	—
182	C₂×C₄	#487	1	2	yes	—
183	C₂×C₄	#492	1	2	yes	—
184	C₂×C₄	#494	1	1	yes	characteristic
185	C₂×C₄	#498	1	2	yes	—
186	C₂×C₄	#500	1	1	yes	characteristic
187	C₈	#340	1	2	yes	—
188	C₈	#340	1	4	yes	—
189	C₈	#340	1	2	yes	—
190	C₈	C₄×C₄⋊C₄	1	2	yes	—
191	C₈	#380	1	2	yes	—
192	C₈	#389	1	4	yes	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	C₂×#340	1	1	yes	characteristic
199	C₂²	#1834	1	1	yes	characteristic
200	C₂²	#1845	1	1	yes	characteristic
201	C₂²	#1957	1	2	yes	—
202	C₂²	#1990	1	2	yes	—
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	16	8	no	—
206	C₄	#1384	1	4	yes	complemented
207	C₄	C₄⋊#95	1	1	yes	characteristic
208	C₄	C₂×#340	1	1	yes	characteristic
209	C₄	#1834	1	1	yes	characteristic
210	C₄	#1844	1	1	yes	characteristic
211	C₄	#1956	1	2	yes	—
212	C₄	#1990	1	2	yes	—
213	C₂	—	2	2	no	—
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	2	no	—
217	C₂	#10841	1	1	yes	characteristic
218	C₂	C₂×#1384	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
11. C₂×#346	C₄	semidirect
12. C₂×C₄×C₁₆	C₄	semidirect
14. #1846	C₄	semidirect
16. #1844	C₄	semidirect
84. C₂×C₁₆	C₄⋊C₄	semidirect
206. C₄	#1384	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
#20797	2
C₂×#1384	2