Subgroups — #483712

Order 512.

442 conjugacy classes of subgroups.

The subgroup lattice has 211 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33495	C₂	1	2	yes	maximal, complemented
2	#17956	C₂	1	1	yes	maximal, characteristic
3	#12063	C₂	1	4	yes	maximal, complemented
4	#12038	C₂	1	4	yes	maximal, complemented
5	#12013	C₂	1	1	yes	maximal, characteristic, complemented
6	#11632	C₂	1	2	yes	maximal, complemented
7	#11411	C₂	1	1	yes	maximal, characteristic, complemented
8	#3169	C₄	1	1	yes	characteristic
9	#2931	C₂²	1	2	yes	—
10	#2169	C₂²	1	4	yes	—
11	#2168	—	2	4	no	—
12	D₄×C₁₆	C₂²	1	4	yes	—
13	#2163	C₂²	1	2	yes	—
14	#2162	C₂²	1	1	yes	characteristic
15	#2162	C₂²	1	1	yes	characteristic
16	#2113	—	2	4	no	—
17	#1589	C₄	1	2	yes	—
18	#1589	C₄	1	2	yes	—
19	C₈×D₈	C₂²	1	2	yes	—
20	C₈×D₈	C₂²	1	2	yes	—
21	#1567	C₄	1	1	yes	characteristic
22	#1492	C₄	1	2	yes	—
23	C₂×C₈²	C₂²	1	1	yes	characteristic
24	#1370	C₂²	1	2	yes	—
25	#1367	C₂²	1	2	yes	—
26	D₄⋊C₁₆	C₂²	1	8	yes	—
27	#1312	C₂²	1	2	yes	—
28	#1311	C₂²	1	2	yes	—
29	C₂×#131	—	2	2	no	—
30	#532	C₂×C₄	1	2	yes	—
31	C₂×M₃₂	—	2	4	no	—
32	C₂²×C₁₆	—	2	4	no	—
33	#500	C₈	1	1	yes	characteristic
34	#493	C₈	1	1	yes	characteristic
35	#486	C₈	1	2	yes	—
36	#466	C₈	1	2	yes	—
37	#466	C₈	1	2	yes	—
38	C₄×D₈	C₂×C₄	1	2	yes	—
39	C₄×D₈	C₂×C₄	1	2	yes	—
40	#436	—	2	2	no	—
41	#436	C₂×C₄	1	4	yes	—
42	C₈×D₄	—	2	2	no	—
43	C₈×D₄	C₂³	1	4	yes	—
44	#423	C₂×C₄	1	2	yes	—
45	#408	—	2	2	no	—
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₄⋊C₁₆	—	2	1	no	—
50	C₄⋊C₁₆	D₄	1	2	yes	—
51	C₄⋊C₁₆	C₂³	1	4	yes	—
52	C₂²⋊C₁₆	—	2	2	no	—
53	C₂²⋊C₁₆	—	2	1	no	—
54	C₂²⋊C₁₆	—	2	4	no	—
55	C₂²⋊C₁₆	—	2	4	no	—
56	C₂²⋊C₁₆	D₄	1	2	yes	—
57	#346	—	2	2	no	—
58	C₄×C₁₆	C₂³	1	2	yes	—
59	C₄×C₁₆	C₂³	1	2	yes	—
60	#335	C₂×C₄	1	2	yes	—
61	#335	C₂×C₄	1	2	yes	—
62	D₄⋊C₈	C₂×C₄	1	4	yes	—
63	D₄⋊C₈	C₂×C₄	1	4	yes	—
64	C₈²	C₂×C₄	1	1	yes	characteristic
65	C₈²	C₂×C₄	1	1	yes	characteristic
66	C₈²	C₂³	1	1	yes	characteristic
67	C₈²	C₂³	1	1	yes	characteristic
68	C₂×#41	—	2	2	no	—
69	C₂×D₈	C₂×C₈	1	2	yes	—
70	C₂×D₈	C₂×C₈	1	2	yes	—
71	#131	—	8	4	no	—
72	C₂×M₁₆	—	2	2	no	—
73	C₂×M₁₆	—	2	4	no	—
74	C₂²×C₈	—	2	1	no	—
75	C₂²×C₈	—	2	4	no	—
76	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₈	#41	1	1	yes	characteristic
80	C₄⋊Q₈	C₂×C₈	1	2	yes	—
81	#127	C₂×C₈	1	2	yes	—
82	#124	—	2	2	no	—
83	#121	—	2	2	no	—
84	#121	C₂×C₈	1	4	yes	—
85	C₄×D₄	—	2	2	no	—
86	C₄×D₄	C₂²×C₄	1	4	yes	—
87	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
88	M₃₂	—	8	4	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	8	4	no	—
92	C₂×C₁₆	C₂×D₄	1	4	yes	—
93	#107	C₂×C₈	1	2	yes	—
94	#107	C₂×C₈	1	2	yes	—
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	C₂²×C₄	1	4	yes	—
97	#102	C₂×C₈	1	4	yes	—
98	#102	C₂×C₈	1	4	yes	—
99	C₂²⋊C₈	—	2	4	no	—
100	C₂²⋊C₈	—	2	4	no	—
101	C₄×C₈	—	2	1	no	—
102	C₄×C₈	C₂×C₈	1	1	yes	characteristic
103	C₄×C₈	C₂×C₈	1	1	yes	characteristic
104	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂×D₄	1	1	yes	characteristic
110	C₄×C₈	#41	1	1	yes	characteristic
111	C₄×C₈	C₂⁴	1	1	yes	characteristic
112	#41	—	8	2	no	—
113	#41	—	8	2	no	—
114	C₂×Q₈	—	2	2	no	—
115	C₂×D₄	—	2	4	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	2	2	no	—
118	C₂×D₄	C₂²×C₈	1	4	yes	—
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
121	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
122	C₂²×C₄	#131	1	1	yes	characteristic
123	D₈	—	4	2	no	—
124	D₈	—	4	2	no	—
125	M₁₆	—	8	4	no	—
126	M₁₆	—	8	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	4	4	no	—
137	C₂×C₈	C₄×D₄	1	1	yes	characteristic
138	C₂×C₈	C₄×D₄	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₈	C₂×#41	1	1	yes	characteristic
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	C₂²×C₈	1	4	yes	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	C₄×D₄	1	1	yes	characteristic
150	C₄²	C₂²×C₈	1	1	yes	characteristic
151	C₄²	#131	1	1	yes	characteristic
152	C₄²	C₂³×C₄	1	1	yes	characteristic
153	C₁₆	—	2	4	no	—
154	C₁₆	—	4	2	no	—
155	C₁₆	—	4	4	no	—
156	C₂³	—	2	4	no	—
157	C₂³	C₈×D₄	1	1	yes	characteristic
158	Q₈	—	8	2	no	—
159	D₄	—	2	4	no	—
160	D₄	—	2	4	no	—
161	D₄	—	8	2	no	—
162	D₄	—	8	2	no	—
163	D₄	—	8	4	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	C₈×D₄	1	1	yes	characteristic
173	C₂×C₄	C₈×D₄	1	1	yes	characteristic
174	C₂×C₄	C₈×D₄	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
177	C₂×C₄	C₂×#131	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	2	1	no	—
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	4	1	no	—
187	C₈	—	4	4	no	—
188	C₈	#576	1	2	yes	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	4	4	no	—
194	C₂²	—	4	4	no	—
195	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	1	no	—
200	C₄	—	4	4	no	—
201	C₄	#2948	1	1	yes	characteristic
202	C₄	#2957	1	1	yes	characteristic
203	C₄	#3407	1	1	yes	characteristic
204	C₄	#3407	1	1	yes	characteristic
205	C₂	—	4	1	no	—
206	C₂	—	8	4	no	—
207	C₂	#17854	1	1	yes	characteristic
208	C₂	#33435	1	1	yes	characteristic
209	C₂	#33443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3169	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ 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
1. #33495	C₂	semidirect
3. #12063	C₂	semidirect
4. #12038	C₂	semidirect
5. #12013	C₂	semidirect
6. #11632	C₂	semidirect
7. #11411	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33495	2
#17956	2
#12063	2
#12038	2
#12013	2
#11632	2
#11411	2