Subgroups — #493255

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2183	C₂	1	2	yes	maximal
2	C₄×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	#12663	C₂	1	4	yes	maximal
4	C₁₆×C₄⋊C₄	C₂	1	2	yes	maximal
5	#12653	C₂	1	2	yes	maximal
6	#12653	C₂	1	1	yes	maximal, characteristic
7	C₄×#346	C₂	1	2	yes	maximal
8	C₄×#346	C₂	1	1	yes	maximal, characteristic
9	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
10	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
11	C₄²×Q₈	C₄	1	1	yes	characteristic
12	#2183	C₄	1	16	yes	complemented
13	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
15	C₂×#346	C₂²	1	2	yes	—
16	C₂×#346	C₂²	1	1	yes	characteristic
17	C₂×#346	C₂²	1	2	yes	—
18	C₂×#346	C₂²	1	4	yes	—
19	C₂×#346	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	4	yes	—
21	C₂×C₄×C₁₆	C₂²	1	2	yes	—
22	C₈×C₄⋊C₄	C₄	1	2	yes	—
23	C₈×C₄⋊C₄	C₄	1	1	yes	characteristic
24	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₈×C₄⋊C₄	C₂²	1	2	yes	—
26	C₄×C₄⋊C₈	C₄	1	2	yes	—
27	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
28	C₄²×C₈	C₂²	1	2	yes	—
29	C₄²×C₈	C₂²	1	1	yes	characteristic
30	#1374	C₂²	1	4	yes	—
31	#1374	C₂²	1	2	yes	—
32	C₂×C₄×Q₈	C₈	1	2	yes	—
33	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₁₆	C₂³	1	4	yes	—
35	C₂²×C₁₆	C₂³	1	2	yes	—
36	C₂²×C₁₆	C₂³	1	2	yes	—
37	C₈×Q₈	C₈	1	8	yes	—
38	C₈×Q₈	C₂×C₄	1	8	yes	—
39	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	2	yes	—
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	2	yes	—
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	C₄×C₄⋊C₄	C₈	1	4	yes	—
55	C₄×C₄⋊C₄	C₈	1	2	yes	—
56	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
57	C₄×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
58	C₄³	C₂×C₄	1	1	yes	characteristic
59	C₄³	C₂×C₄	1	2	yes	—
60	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
61	C₄⋊C₁₆	C₂×C₄	1	16	yes	—
62	#346	—	2	4	no	—
63	#346	—	2	4	no	—
64	#346	—	2	2	no	—
65	#346	C₂×C₄	1	16	yes	—
66	#346	Q₈	1	4	yes	complemented
67	C₄×C₁₆	—	2	8	no	—
68	C₄×C₁₆	C₂×C₄	1	8	yes	—
69	#336	C₂×C₄	1	2	yes	—
70	#336	C₂×C₄	1	2	yes	—
71	#336	C₂×C₄	1	2	yes	—
72	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₄	1	2	yes	—
75	C₂²×C₈	C₂²×C₄	1	2	yes	—
76	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂⁴	1	1	yes	characteristic
79	C₄×Q₈	—	2	8	no	—
80	C₄×Q₈	C₄²	1	4	yes	—
81	C₄×Q₈	C₂×C₈	1	4	yes	—
82	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
83	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
84	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
86	C₂×C₄²	C₂×C₈	1	2	yes	—
87	C₂×C₄²	C₂×C₈	1	4	yes	—
88	C₂×C₄²	C₂×C₈	1	2	yes	—
89	C₂×C₄²	C₂×C₈	1	4	yes	—
90	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄²	C₂²×C₄	1	2	yes	—
92	C₂×C₄²	C₂²×C₄	1	2	yes	—
93	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₁₆	—	2	4	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	2	4	no	—
98	C₂×C₁₆	—	2	2	no	—
99	C₂×C₁₆	C₂²×C₄	1	4	yes	—
100	C₂×C₁₆	C₂²×C₄	1	8	yes	—
101	C₂×C₁₆	C₂²×C₄	1	4	yes	—
102	C₂×C₁₆	C₂×Q₈	1	4	yes	—
103	C₂×C₁₆	#41	1	4	yes	—
104	C₄⋊C₈	C₄²	1	8	yes	—
105	C₄⋊C₈	C₄²	1	4	yes	—
106	C₄⋊C₈	C₂×C₈	1	8	yes	—
107	C₄⋊C₈	C₂×C₈	1	4	yes	—
108	C₄×C₈	—	2	4	no	—
109	C₄×C₈	—	2	2	no	—
110	C₄×C₈	—	2	4	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	—	2	2	no	—
113	C₄×C₈	—	2	4	no	—
114	C₄×C₈	C₂×C₈	1	8	yes	—
115	C₄×C₈	C₂×C₈	1	4	yes	—
116	C₄×C₈	C₂²×C₄	1	4	yes	—
117	C₄×C₈	C₂²×C₄	1	8	yes	—
118	C₄×C₈	C₂×Q₈	1	2	yes	—
119	C₄×C₈	#41	1	2	yes	—
120	#95	C₂×C₈	1	4	yes	—
121	#95	C₂×C₈	1	2	yes	—
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	C₄×C₈	1	4	yes	—
124	C₂²×C₄	C₂²×C₈	1	2	yes	—
125	C₂²×C₄	C₂²×C₈	1	2	yes	—
126	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
127	C₂²×C₄	C₂²×C₈	1	2	yes	—
128	C₂²×C₄	C₂²×C₈	1	2	yes	—
129	C₂²×C₄	C₂²×C₈	1	4	yes	—
130	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
131	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₂×C₄²	1	2	yes	—
141	C₂×C₈	C₂×C₄²	1	4	yes	—
142	C₂×C₈	C₄×Q₈	1	2	yes	—
143	C₂×C₈	C₂²×C₈	1	4	yes	—
144	C₂×C₈	C₂²×C₈	1	2	yes	—
145	C₂×C₈	C₂²×C₈	1	2	yes	—
146	C₂×C₈	#131	1	2	yes	—
147	C₂×C₈	C₂³×C₄	1	2	yes	—
148	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
149	C₂×C₈	C₂×#41	1	2	yes	—
150	C₂×C₈	C₂×#41	1	1	yes	characteristic
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	C₄×C₈	1	8	yes	—
153	C₄⋊C₄	C₄×C₈	1	4	yes	—
154	C₄⋊C₄	M₃₂	1	8	yes	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	1	no	—
157	C₄²	—	2	2	no	—
158	C₄²	—	2	1	no	—
159	C₄²	—	2	8	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	4	4	no	—
162	C₄²	—	4	2	no	—
163	C₄²	C₂×C₄²	1	4	yes	—
164	C₄²	C₂×C₄²	1	2	yes	—
165	C₄²	C₄×Q₈	1	2	yes	—
166	C₄²	C₂²×C₈	1	4	yes	—
167	C₄²	C₂²×C₈	1	2	yes	—
168	C₄²	#131	1	2	yes	—
169	C₁₆	—	2	8	no	—
170	C₁₆	—	2	4	no	—
171	C₁₆	C₄×Q₈	1	8	yes	complemented
172	C₂³	C₂³×C₈	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	C₂×C₄×C₈	1	4	yes	—
185	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
186	C₂×C₄	C₂×C₄×C₈	1	4	yes	—
187	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
188	C₂×C₄	C₈×Q₈	1	4	yes	—
189	C₂×C₄	C₂×M₃₂	1	8	yes	—
190	C₂×C₄	C₂×M₃₂	1	4	yes	—
191	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
193	C₂×C₄	C₄×#41	1	1	yes	characteristic
194	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
195	C₂×C₄	C₂×#131	1	1	yes	characteristic
196	C₂×C₄	C₂×#131	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	2	4	no	—
199	C₈	—	2	4	no	—
200	C₈	—	2	2	no	—
201	C₈	C₈×Q₈	1	4	yes	—
202	C₈	C₂×C₄×Q₈	1	2	yes	—
203	C₈	C₄×#41	1	2	yes	—
204	C₂²	C₂²×C₄×C₈	1	1	yes	characteristic
205	C₂²	C₂×C₈×Q₈	1	1	yes	characteristic
206	C₂²	C₈×#41	1	1	yes	characteristic
207	C₂²	C₂²×M₃₂	1	2	yes	—
208	C₂²	#3407	1	2	yes	—
209	C₄	—	2	4	no	—
210	C₄	—	2	4	no	—
211	C₄	—	2	2	no	—
212	C₄	—	2	4	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	2	no	—
215	C₄	C₄×M₃₂	1	4	yes	—
216	C₄	C₄²×Q₈	1	1	yes	characteristic
217	C₄	C₄×#131	1	1	yes	characteristic
218	C₄	C₂×C₈×Q₈	1	1	yes	characteristic
219	C₄	C₈×#41	1	1	yes	characteristic
220	C₂	C₄×C₈×Q₈	1	1	yes	characteristic
221	C₂	C₂×C₄×M₃₂	1	1	yes	characteristic
222	C₂	#33326	1	1	yes	characteristic
223	C₂	Q₈×M₃₂	1	2	yes	—
224	C₂	#33484	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
12. #2183	C₄	semidirect
66. #346	Q₈	semidirect
171. C₁₆	C₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2183	2
C₄×C₈×Q₈	2
#12663	2
C₁₆×C₄⋊C₄	2
#12653	2
C₄×#346	2