Subgroups — #484071

Order 512.

440 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	M₁₆×Q₁₆	C₂	1	2	yes	maximal
3	#12085	C₂	1	2	yes	maximal, complemented
4	#12076	C₂	1	4	yes	maximal, complemented
5	#12044	C₂	1	2	yes	maximal, complemented
6	#12023	C₂	1	1	yes	maximal, characteristic
7	#11658	C₂	1	2	yes	maximal
8	#11408	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₃₂	C₄	1	1	yes	characteristic
10	Q₈×M₁₆	—	2	2	no	—
11	C₂×C₄×Q₁₆	C₂²	1	2	yes	—
12	C₄×Q₃₂	C₄	1	2	yes	—
13	C₄×Q₃₂	C₄	1	1	yes	characteristic
14	C₄×Q₃₂	C₄	1	1	yes	characteristic
15	C₄×Q₃₂	C₂²	1	1	yes	characteristic
16	C₄×Q₃₂	C₂²	1	2	yes	—
17	C₄×Q₃₂	C₂²	1	1	yes	characteristic
18	#2162	C₂²	1	1	yes	characteristic
19	C₂×#366	C₄	1	1	yes	characteristic
20	C₂×#358	C₄	1	2	yes	—
21	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	#1591	C₂²	1	4	yes	—
23	#1584	—	2	4	no	—
24	C₈×Q₁₆	C₂²	1	4	yes	—
25	#1567	C₂²	1	2	yes	—
26	#1487	—	2	2	no	—
27	C₈×M₁₆	C₂²	1	1	yes	characteristic
28	#1372	C₂²	1	2	yes	—
29	#1372	C₂²	1	2	yes	—
30	#1332	C₂²	1	4	yes	—
31	#1332	C₂²	1	4	yes	—
32	#1312	C₂²	1	2	yes	—
33	#1311	C₂²	1	2	yes	—
34	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₄×Q₈	—	2	2	no	—
36	C₂×Q₃₂	—	2	4	no	—
37	C₂×Q₃₂	C₂×C₄	1	2	yes	—
38	C₂×Q₃₂	C₂×C₄	1	1	yes	characteristic
39	C₂×Q₃₂	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	#446	—	4	4	no	—
44	C₈×Q₈	—	2	4	no	—
45	C₄×Q₁₆	—	2	2	no	—
46	C₄×Q₁₆	—	2	2	no	—
47	C₄×Q₁₆	C₂×C₄	1	2	yes	—
48	C₄×Q₁₆	C₂×C₄	1	2	yes	—
49	C₄×Q₁₆	C₂³	1	2	yes	—
50	C₄×Q₁₆	C₂³	1	2	yes	—
51	#431	D₄	1	1	yes	characteristic
52	C₂×#107	C₂×C₄	1	2	yes	—
53	#423	—	2	2	no	—
54	#423	—	4	2	no	—
55	C₂×#103	—	2	2	no	—
56	C₄×M₁₆	—	4	2	no	—
57	C₄×M₁₆	D₄	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	#366	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	#358	C₂×C₄	1	2	yes	—
66	#358	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	Q₈⋊C₈	—	2	4	no	—
77	#322	D₄	1	2	yes	—
78	C₈²	C₂³	1	2	yes	—
79	C₂²×Q₈	—	2	2	no	—
80	C₂×Q₁₆	—	2	4	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	4	2	no	—
83	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
84	C₂×Q₁₆	C₂²×C₄	1	2	yes	—
85	C₂×M₁₆	—	4	2	no	—
86	C₂×M₁₆	D₈	1	2	yes	—
87	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₈	#41	1	1	yes	characteristic
89	C₄×Q₈	—	2	2	no	—
90	C₄×Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	C₄×Q₈	—	2	2	no	—
93	C₄×Q₈	—	4	2	no	—
94	C₄×Q₈	—	4	2	no	—
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	4	2	no	—
97	C₂×C₄²	—	4	2	no	—
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	Q₃₂	—	2	4	no	—
100	Q₃₂	—	2	2	no	—
101	Q₃₂	M₁₆	1	4	yes	complemented
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	4	1	no	—
105	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
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	#107	—	2	2	no	—
110	#107	C₂²×C₄	1	2	yes	—
111	#107	C₂²×C₄	1	2	yes	—
112	C₄⋊C₈	—	2	1	no	—
113	C₄⋊C₈	—	2	4	no	—
114	C₄⋊C₈	—	4	4	no	—
115	C₄⋊C₈	—	4	4	no	—
116	C₄⋊C₈	—	4	2	no	—
117	#103	—	2	2	no	—
118	#103	—	2	2	no	—
119	#103	—	2	2	no	—
120	#103	—	2	2	no	—
121	C₂²⋊C₈	—	2	1	no	—
122	#97	—	4	4	no	—
123	#97	D₈	1	2	yes	—
124	C₄×C₈	—	4	4	no	—
125	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₄×C₈	C₂×D₄	1	2	yes	—
127	C₄×C₈	C₂×D₄	1	1	yes	characteristic
128	C₄×C₈	C₂×D₄	1	2	yes	—
129	C₄×C₈	#41	1	1	yes	characteristic
130	C₄×C₈	C₂⁴	1	1	yes	characteristic
131	C₂×Q₈	—	2	2	no	—
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	4	4	no	—
134	C₂×Q₈	—	4	2	no	—
135	C₂×Q₈	—	8	2	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
140	C₂²×C₄	#135	1	1	yes	characteristic
141	Q₁₆	—	2	2	no	—
142	Q₁₆	—	2	2	no	—
143	Q₁₆	—	2	2	no	—
144	Q₁₆	—	4	4	no	—
145	Q₁₆	C₂×M₁₆	1	4	yes	—
146	M₁₆	—	2	4	no	—
147	M₁₆	—	8	4	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	—	4	4	no	—
153	C₂×C₈	C₄×D₄	1	1	yes	characteristic
154	C₂×C₈	C₄×D₄	1	1	yes	characteristic
155	C₂×C₈	C₂×D₈	1	4	yes	—
156	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₈	C₂×#41	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	—	4	2	no	—
169	C₄²	C₄×D₄	1	1	yes	characteristic
170	C₄²	C₂×D₈	1	1	yes	characteristic
171	C₄²	#135	1	1	yes	characteristic
172	C₄²	C₂²×D₄	1	1	yes	characteristic
173	C₁₆	—	2	1	no	—
174	C₁₆	—	2	1	no	—
175	C₁₆	—	2	1	no	—
176	C₁₆	—	4	2	no	—
177	C₁₆	C₂×M₁₆	1	2	yes	—
178	C₂³	C₄×D₈	1	1	yes	characteristic
179	Q₈	—	2	4	no	—
180	Q₈	—	4	2	no	—
181	Q₈	—	4	2	no	—
182	Q₈	—	4	2	no	—
183	Q₈	—	8	4	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	—	8	4	no	—
192	C₂×C₄	—	16	2	no	—
193	C₂×C₄	C₄×D₈	1	1	yes	characteristic
194	C₂×C₄	C₄×D₈	1	1	yes	characteristic
195	C₂×C₄	C₄×D₈	1	1	yes	characteristic
196	C₂×C₄	#510	1	4	yes	—
197	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
199	C₂×C₄	C₂×#135	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	2	1	no	—
204	C₈	—	4	2	no	—
205	C₈	—	8	4	no	—
206	C₈	C₂²×M₁₆	1	1	yes	characteristic
207	C₈	#568	1	1	yes	characteristic
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	#2176	1	2	yes	—
211	C₂²	C₂×C₄×D₈	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	4	no	—
216	C₄	—	8	2	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	#2176	1	2	yes	—
220	C₄	D₄×M₁₆	1	1	yes	characteristic
221	C₄	#2957	1	1	yes	characteristic
222	C₄	C₂×#510	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	M₁₆×D₈	1	1	yes	characteristic
225	C₂	C₂×#2176	1	1	yes	characteristic
226	C₂	#33465	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	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ 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
3. #12085	C₂	semidirect
4. #12076	C₂	semidirect
5. #12044	C₂	semidirect
101. Q₃₂	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₃₂	2
M₁₆×Q₁₆	2
#12085	2
#12076	2
#12044	2
#12023	2
#11658	2
#11408	2