Subgroups — #146926

Order 512.

484 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2108	C₂	1	1	yes	maximal, characteristic
2	#12563	C₂	1	2	yes	maximal
3	C₂×#1382	C₂	1	2	yes	maximal
4	#10867	C₂	1	1	yes	maximal, characteristic
5	#10866	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₃₂	C₂²	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂×#427	C₂²	1	2	yes	—
10	C₂×#405	C₂²	1	1	yes	characteristic
11	#2108	—	2	4	no	—
12	C₂×#346	C₄	1	2	yes	complemented
13	C₂×C₄×C₁₆	C₄	1	2	yes	complemented
14	#1742	—	2	1	no	—
15	#1740	—	2	1	no	—
16	C₂×#338	—	2	1	no	—
17	C₂×H(3,ℤ/4)	—	2	1	no	—
18	#1383	—	2	4	no	—
19	#1382	C₄	1	8	yes	complemented
20	C₂²×M₁₆	—	2	2	no	—
21	C₂²×M₁₆	D₄	1	1	yes	characteristic
22	C₂³×C₈	C₂³	1	1	yes	characteristic
23	C₂×#117	—	2	2	no	—
24	C₂×#117	D₄	1	1	yes	characteristic
25	C₂×M₃₂	—	2	1	no	—
26	C₂×M₃₂	—	2	1	no	—
27	C₂×M₃₂	—	2	2	no	—
28	C₂×M₃₂	D₄	1	4	yes	—
29	C₂²×C₁₆	—	2	2	no	—
30	C₂²×C₁₆	—	2	1	no	—
31	C₂²×C₁₆	—	2	1	no	—
32	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₁₆	C₂×C₄	1	2	yes	—
36	C₂×#108	—	2	2	no	—
37	C₂×#108	—	2	2	no	—
38	C₂×#108	—	2	2	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	C₂×C₄	1	4	yes	—
42	#427	—	2	2	no	—
43	#427	—	2	2	no	—
44	#427	—	2	2	no	—
45	#427	C₂×C₄	1	4	yes	—
46	C₂×#107	—	2	2	no	—
47	C₂×#106	—	2	2	no	—
48	#405	—	2	2	no	—
49	#405	—	2	2	no	—
50	C₂×#97	Q₈	1	2	yes	—
51	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
52	#346	—	2	4	no	—
53	C₄×C₁₆	—	2	4	no	—
54	#343	—	4	2	no	—
55	#342	—	4	2	no	—
56	#338	—	4	2	no	—
57	H(3,ℤ/4)	—	4	2	no	—
58	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×M₁₆	—	2	2	no	—
60	C₂×M₁₆	—	2	2	no	—
61	C₂×M₁₆	—	2	4	no	—
62	C₂×M₁₆	—	4	2	no	—
63	C₂×M₁₆	—	4	2	no	—
64	C₂×M₁₆	—	4	2	no	—
65	C₂×M₁₆	—	4	2	no	—
66	C₂×M₁₆	—	4	2	no	—
67	C₂×M₁₆	D₈	1	2	yes	—
68	C₂×M₁₆	Q₁₆	1	2	yes	—
69	C₂²×C₈	—	4	2	no	—
70	C₂²×C₈	C₄⋊C₄	1	2	yes	—
71	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂²×C₄	1	2	yes	—
73	C₂²×C₈	C₂×D₄	1	2	yes	—
74	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₈	#41	1	2	yes	—
76	C₂²×C₈	#41	1	1	yes	characteristic
77	C₂²×C₈	#41	1	1	yes	characteristic
78	C₂²×C₈	#41	1	1	yes	characteristic
79	C₂²×C₈	#41	1	1	yes	characteristic
80	C₂²×C₈	#41	1	2	yes	—
81	#117	—	2	2	no	—
82	#117	—	2	2	no	—
83	#117	—	2	4	no	—
84	#117	—	4	2	no	—
85	#117	—	4	2	no	—
86	#117	—	4	2	no	—
87	C₂×C₄⋊C₄	—	4	2	no	—
88	C₂×C₄⋊C₄	SD₁₆	1	2	yes	—
89	C₂×C₂²⋊C₄	—	4	2	no	—
90	C₂×C₂²⋊C₄	SD₁₆	1	2	yes	—
91	C₂×C₄²	—	4	2	no	—
92	C₂×C₄²	C₄⋊C₄	1	2	yes	—
93	M₃₂	—	2	4	no	—
94	M₃₂	—	2	4	no	—
95	C₂×C₁₆	—	2	1	no	—
96	C₂×C₁₆	—	2	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	2	1	no	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	C₄²	1	2	yes	—
104	C₂×C₁₆	C₄²	1	2	yes	—
105	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
106	#108	—	2	4	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	4	2	no	—
110	#107	—	2	4	no	—
111	#107	—	4	2	no	—
112	#106	—	2	4	no	—
113	#106	—	4	2	no	—
114	#97	—	2	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₂⁴	C₂×SD₁₆	1	1	yes	characteristic
118	C₂²×C₄	—	8	2	no	—
119	C₂²×C₄	#106	1	4	yes	—
120	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄	C₄×D₄	1	2	yes	—
122	C₂²×C₄	#121	1	2	yes	—
123	C₂²×C₄	#124	1	1	yes	characteristic
124	C₂²×C₄	#124	1	1	yes	characteristic
125	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
127	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
128	C₂²×C₄	#135	1	2	yes	—
129	C₂²×C₄	#135	1	2	yes	—
130	M₁₆	—	2	4	no	—
131	M₁₆	—	2	4	no	—
132	M₁₆	—	4	4	no	—
133	M₁₆	—	4	4	no	—
134	M₁₆	—	8	2	no	—
135	M₁₆	—	8	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	—	8	2	no	—
141	C₂×C₈	#107	1	4	yes	—
142	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₈	#117	1	1	yes	characteristic
145	C₂×C₈	#117	1	1	yes	characteristic
146	C₂×C₈	#117	1	2	yes	—
147	C₂×C₈	#117	1	2	yes	—
148	C₂×C₈	#117	1	1	yes	characteristic
149	C₂×C₈	#117	1	1	yes	characteristic
150	C₂×C₈	C₄×D₄	1	2	yes	—
151	C₂×C₈	C₄×D₄	1	2	yes	—
152	C₂×C₈	C₄×Q₈	1	2	yes	—
153	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
154	C₂×C₈	#123	1	2	yes	—
155	C₂×C₈	#123	1	2	yes	—
156	C₂×C₈	#125	1	1	yes	characteristic
157	C₂×C₈	#125	1	1	yes	characteristic
158	C₂×C₈	#126	1	2	yes	—
159	C₂×C₈	#126	1	2	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	—	8	2	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	4	4	no	—
165	C₂²⋊C₄	—	8	2	no	—
166	C₄²	—	2	2	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	4	4	no	—
169	C₄²	—	8	2	no	—
170	C₁₆	—	2	2	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	4	no	—
173	C₂³	—	4	2	no	—
174	C₂³	C₂×#106	1	1	yes	characteristic
175	C₂³	C₄×SD₁₆	1	2	yes	—
176	C₂³	#465	1	2	yes	—
177	C₂³	#486	1	1	yes	characteristic
178	C₂³	#487	1	1	yes	characteristic
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	8	4	no	—
182	C₂×C₄	—	16	4	no	—
183	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄	#380	1	1	yes	characteristic
185	C₂×C₄	#382	1	1	yes	characteristic
186	C₂×C₄	#388	1	2	yes	—
187	C₂×C₄	#388	1	2	yes	—
188	C₂×C₄	C₂×#107	1	1	yes	characteristic
189	C₂×C₄	#427	1	1	yes	characteristic
190	C₂×C₄	#427	1	1	yes	characteristic
191	C₂×C₄	C₄×D₈	1	2	yes	—
192	C₂×C₄	C₄×SD₁₆	1	2	yes	—
193	C₂×C₄	C₄×Q₁₆	1	2	yes	—
194	C₂×C₄	#465	1	2	yes	—
195	C₂×C₄	#466	1	2	yes	—
196	C₂×C₄	#467	1	2	yes	—
197	C₂×C₄	#486	1	1	yes	characteristic
198	C₂×C₄	#487	1	1	yes	characteristic
199	C₂×C₄	#488	1	2	yes	—
200	C₂×C₄	#488	1	2	yes	—
201	C₈	—	2	4	no	—
202	C₈	—	8	4	no	—
203	C₈	#376	1	1	yes	characteristic
204	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
205	C₈	#381	1	1	yes	characteristic
206	C₈	#383	1	1	yes	characteristic
207	C₈	#387	1	2	yes	—
208	C₈	#387	1	2	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	C₄×#106	1	1	yes	characteristic
215	C₂²	#1807	1	1	yes	characteristic
216	C₂²	#1834	1	1	yes	characteristic
217	C₂²	#1933	1	2	yes	—
218	C₂²	#1934	1	2	yes	—
219	C₄	—	4	4	no	—
220	C₄	—	16	4	no	—
221	C₄	#1434	1	1	yes	characteristic
222	C₄	C₄×#107	1	1	yes	characteristic
223	C₄	#1808	1	1	yes	characteristic
224	C₄	#1835	1	1	yes	characteristic
225	C₄	#1933	1	2	yes	—
226	C₄	#1934	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	—	2	1	no	—
229	C₂	—	2	2	no	—
230	C₂	—	2	1	no	—
231	C₂	#10820	1	1	yes	characteristic
232	C₂	#12832	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1807
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
12. C₂×#346	C₄	semidirect
13. C₂×C₄×C₁₆	C₄	semidirect
19. #1382	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2108	2
#12563	2
C₂×#1382	2
#10867	2
#10866	2