Subgroups — #484084

Order 512.

455 conjugacy classes of subgroups.

The subgroup lattice has 239 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#508	C₂	1	1	yes	maximal, characteristic
2	#18055	C₂	1	2	yes	maximal, complemented
3	#12083	C₂	1	2	yes	maximal, complemented
4	#12074	C₂	1	4	yes	maximal, complemented
5	C₈×D₁₆	C₂	1	2	yes	maximal, complemented
6	#12022	C₂	1	1	yes	maximal, characteristic, complemented
7	#11660	C₂	1	2	yes	maximal, complemented
8	C₁₆×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#508	C₄	1	1	yes	characteristic
10	#2990	—	2	2	no	—
11	C₄×#135	C₂²	1	2	yes	—
12	C₄×Q₃₂	C₄	1	1	yes	characteristic
13	C₄×Q₃₂	C₂²	1	1	yes	characteristic
14	C₄×SD₃₂	C₄	1	2	yes	—
15	C₄×SD₃₂	C₂²	1	2	yes	—
16	C₄×D₁₆	C₄	1	1	yes	characteristic
17	C₄×D₁₆	C₂²	1	1	yes	characteristic
18	#2162	C₂²	1	1	yes	characteristic
19	#2158	C₄	1	1	yes	characteristic
20	#2138	C₄	1	2	yes	—
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	#1591	C₂²	1	4	yes	—
23	#1585	—	2	4	no	—
24	C₈×D₈	C₂²	1	4	yes	—
25	#1567	C₂²	1	2	yes	—
26	#1489	—	2	2	no	—
27	C₈×M₁₆	C₂²	1	1	yes	characteristic
28	#1371	C₂²	1	2	yes	—
29	#1371	C₂²	1	2	yes	—
30	#1332	C₂²	1	4	yes	—
31	#1331	C₂²	1	4	yes	—
32	#1312	C₂²	1	2	yes	—
33	C₈×C₁₆	C₂²	1	2	yes	—
34	C₂×#135	C₂×C₄	1	2	yes	—
35	C₄×#41	—	2	2	no	—
36	#508	—	2	4	no	—
37	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
38	C₂×SD₃₂	C₂×C₄	1	2	yes	—
39	C₂×D₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
42	#446	—	2	4	no	—
43	C₄×Q₁₆	C₂×C₄	1	2	yes	—
44	C₄×Q₁₆	C₂³	1	2	yes	—
45	C₄×SD₁₆	—	2	2	no	—
46	C₄×SD₁₆	—	2	2	no	—
47	C₄×D₈	C₂×C₄	1	2	yes	—
48	C₄×D₈	C₂³	1	2	yes	—
49	#435	—	4	4	no	—
50	C₈×D₄	—	2	4	no	—
51	#432	—	4	2	no	—
52	#431	D₄	1	1	yes	characteristic
53	#427	C₂×C₄	1	2	yes	—
54	#423	—	2	2	no	—
55	C₂×C₄⋊C₈	—	4	2	no	—
56	#416	—	2	2	no	—
57	C₄×M₁₆	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	#367	C₂×C₄	1	2	yes	—
60	#366	C₂×C₄	1	1	yes	characteristic
61	#366	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₁₆	—	2	1	no	—
63	#358	C₂×C₄	1	2	yes	—
64	#358	C₂×C₄	1	2	yes	—
65	#357	C₂×C₄	1	2	yes	—
66	#357	C₂×C₄	1	2	yes	—
67	C₂²⋊C₁₆	—	2	1	no	—
68	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
69	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
70	C₄×C₁₆	C₂³	1	2	yes	—
71	C₄×C₁₆	C₂³	1	1	yes	characteristic
72	C₄×C₁₆	C₂³	1	1	yes	characteristic
73	#335	—	2	2	no	—
74	#335	C₂³	1	4	yes	—
75	Q₈⋊C₈	—	2	4	no	—
76	D₄⋊C₈	—	2	4	no	—
77	#322	D₄	1	2	yes	—
78	C₈²	C₂³	1	2	yes	—
79	C₂×#41	—	2	2	no	—
80	#135	—	2	4	no	—
81	#135	—	4	2	no	—
82	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
83	C₂×SD₁₆	—	2	2	no	—
84	C₂×D₈	C₂²×C₄	1	2	yes	—
85	C₂×M₁₆	—	2	1	no	—
86	C₂×M₁₆	—	4	2	no	—
87	C₂²×C₈	—	4	4	no	—
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	#41	1	1	yes	characteristic
90	C₄×Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	C₄×D₄	—	2	2	no	—
93	C₄×D₄	—	2	2	no	—
94	C₄×D₄	—	4	2	no	—
95	C₄×D₄	—	4	2	no	—
96	#117	—	2	2	no	—
97	#117	—	4	2	no	—
98	C₂×C₄²	—	4	2	no	—
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	Q₃₂	—	2	2	no	—
101	SD₃₂	—	2	4	no	—
102	D₁₆	M₁₆	1	4	yes	complemented
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	4	1	no	—
106	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
110	#107	C₂²×C₄	1	2	yes	—
111	#107	C₂²×C₄	1	2	yes	—
112	#106	—	2	2	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	4	4	no	—
115	C₄⋊C₈	—	4	2	no	—
116	C₄⋊C₈	D₈	1	1	yes	characteristic
117	C₄⋊C₈	D₈	1	1	yes	characteristic
118	#103	—	2	2	no	—
119	#103	—	2	2	no	—
120	#102	—	2	2	no	—
121	#102	—	2	2	no	—
122	C₂²⋊C₈	—	4	4	no	—
123	C₂²⋊C₈	D₈	1	2	yes	—
124	#97	—	2	1	no	—
125	#97	—	4	4	no	—
126	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
127	C₄×C₈	C₂×D₄	1	2	yes	—
128	C₄×C₈	C₂×D₄	1	1	yes	characteristic
129	C₄×C₈	C₂×D₄	1	2	yes	—
130	C₄×C₈	#41	1	1	yes	characteristic
131	C₄×C₈	C₂⁴	1	1	yes	characteristic
132	#41	—	4	4	no	—
133	#41	—	8	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×D₄	—	2	2	no	—
136	C₂×D₄	—	4	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
142	C₂²×C₄	#135	1	1	yes	characteristic
143	Q₁₆	—	2	2	no	—
144	Q₁₆	—	2	2	no	—
145	SD₁₆	—	4	4	no	—
146	D₈	—	2	2	no	—
147	D₈	C₂×M₁₆	1	4	yes	—
148	M₁₆	—	2	4	no	—
149	M₁₆	—	16	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	—	16	4	no	—
156	C₂×C₈	C₄×D₄	1	1	yes	characteristic
157	C₂×C₈	C₄×D₄	1	1	yes	characteristic
158	C₂×C₈	C₂×D₈	1	2	yes	—
159	C₂×C₈	C₂×D₈	1	2	yes	—
160	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₈	C₂×#41	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₂²⋊C₄	—	4	2	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	—	4	2	no	—
171	C₄²	—	4	2	no	—
172	C₄²	C₄×D₄	1	1	yes	characteristic
173	C₄²	C₂×D₈	1	1	yes	characteristic
174	C₄²	#135	1	1	yes	characteristic
175	C₄²	C₂²×D₄	1	1	yes	characteristic
176	C₁₆	—	2	1	no	—
177	C₁₆	—	2	1	no	—
178	C₁₆	—	2	1	no	—
179	C₁₆	—	4	2	no	—
180	C₁₆	C₂×M₁₆	1	2	yes	—
181	C₂³	—	4	2	no	—
182	C₂³	C₄×D₈	1	1	yes	characteristic
183	Q₈	—	4	2	no	—
184	Q₈	—	4	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	2	no	—
187	D₄	—	8	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	C₄×D₈	1	1	yes	characteristic
201	C₂×C₄	C₄×D₈	1	1	yes	characteristic
202	C₂×C₄	C₄×D₈	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
205	C₂×C₄	C₂×#135	1	1	yes	characteristic
206	C₈	—	2	4	no	—
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	8	4	no	—
213	C₈	C₂²×M₁₆	1	1	yes	characteristic
214	C₈	#568	1	1	yes	characteristic
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	4	4	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	2	no	—
221	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
222	C₄	—	2	1	no	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	#2178	1	2	yes	—
229	C₄	D₄×M₁₆	1	1	yes	characteristic
230	C₄	#2957	1	1	yes	characteristic
231	C₄	#3415	1	1	yes	characteristic
232	C₄	#3417	1	1	yes	characteristic
233	C₂	—	4	1	no	—
234	C₂	—	8	4	no	—
235	C₂	M₁₆×D₈	1	1	yes	characteristic
236	C₂	#33459	1	1	yes	characteristic
237	C₂	C₂×#2178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #18055	C₂	semidirect
3. #12083	C₂	semidirect
4. #12074	C₂	semidirect
5. C₈×D₁₆	C₂	semidirect
6. #12022	C₂	semidirect
7. #11660	C₂	semidirect
8. C₁₆×M₁₆	C₂	semidirect
102. D₁₆	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×#508	2
#18055	2
#12083	2
#12074	2
C₈×D₁₆	2
#12022	2
#11660	2
C₁₆×M₁₆	2