Subgroups — #483739

Order 512.

394 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33499	C₂	1	2	yes	maximal
2	#17958	C₂	1	1	yes	maximal, characteristic
3	#12061	C₂	1	2	yes	maximal, complemented
4	#12059	C₂	1	2	yes	maximal, complemented
5	#12054	C₂	1	2	yes	maximal, complemented
6	C₁₆×Q₁₆	C₂	1	2	yes	maximal, complemented
7	#12013	C₂	1	1	yes	maximal, characteristic
8	#11640	C₂	1	2	yes	maximal
9	#11410	C₂	1	1	yes	maximal, characteristic
10	#3170	C₄	1	1	yes	characteristic
11	#2960	C₂²	1	2	yes	—
12	#2183	—	2	4	no	—
13	#2183	C₂²	1	4	yes	—
14	Q₈×C₁₆	C₂²	1	4	yes	—
15	#2164	—	2	4	no	—
16	#2163	C₂²	1	2	yes	—
17	#2162	C₂²	1	1	yes	characteristic
18	#2162	C₂²	1	1	yes	characteristic
19	#1591	C₄	1	2	yes	—
20	#1591	C₄	1	2	yes	—
21	C₈×Q₁₆	C₂²	1	2	yes	—
22	C₈×Q₁₆	C₂²	1	2	yes	—
23	#1567	C₄	1	1	yes	characteristic
24	#1493	C₄	1	2	yes	—
25	C₂×C₈²	C₂²	1	1	yes	characteristic
26	#1370	C₂²	1	2	yes	—
27	#1367	C₂²	1	2	yes	—
28	Q₈⋊C₁₆	C₂²	1	4	yes	—
29	Q₈⋊C₁₆	C₂²	1	4	yes	—
30	#1312	C₂²	1	1	yes	characteristic
31	#1312	C₂²	1	1	yes	characteristic
32	C₈×C₁₆	C₂²	1	1	yes	characteristic
33	C₈×C₁₆	C₂²	1	1	yes	characteristic
34	#533	C₂×C₄	1	2	yes	—
35	#500	C₈	1	1	yes	characteristic
36	#494	C₈	1	1	yes	characteristic
37	#487	C₈	1	2	yes	—
38	#467	C₈	1	2	yes	—
39	#467	C₈	1	2	yes	—
40	#446	—	2	2	no	—
41	#446	C₂×C₄	1	4	yes	—
42	C₈×Q₈	—	2	2	no	—
43	C₈×Q₈	C₂³	1	4	yes	—
44	C₄×Q₁₆	C₂×C₄	1	2	yes	—
45	C₄×Q₁₆	C₂×C₄	1	2	yes	—
46	#424	—	2	2	no	—
47	#423	C₂×C₄	1	2	yes	—
48	#405	—	2	2	no	—
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₄⋊C₁₆	—	2	4	no	—
53	C₄⋊C₁₆	—	2	4	no	—
54	C₄⋊C₁₆	—	2	1	no	—
55	C₄⋊C₁₆	D₄	1	2	yes	—
56	C₄⋊C₁₆	C₂³	1	4	yes	—
57	C₂²⋊C₁₆	—	2	2	no	—
58	C₂²⋊C₁₆	—	2	1	no	—
59	C₂²⋊C₁₆	D₄	1	2	yes	—
60	#346	—	2	4	no	—
61	#346	—	2	2	no	—
62	C₄×C₁₆	—	2	4	no	—
63	C₄×C₁₆	C₂³	1	1	yes	characteristic
64	C₄×C₁₆	C₂³	1	1	yes	characteristic
65	C₄×C₁₆	C₂³	1	1	yes	characteristic
66	C₄×C₁₆	C₂³	1	1	yes	characteristic
67	#335	C₂×C₄	1	2	yes	—
68	#335	C₂×C₄	1	2	yes	—
69	Q₈⋊C₈	C₂×C₄	1	4	yes	—
70	Q₈⋊C₈	C₂×C₄	1	4	yes	—
71	C₈²	C₂×C₄	1	1	yes	characteristic
72	C₈²	C₂×C₄	1	1	yes	characteristic
73	C₈²	C₂³	1	1	yes	characteristic
74	C₈²	C₂³	1	1	yes	characteristic
75	C₂×Q₁₆	C₂×C₈	1	2	yes	—
76	C₂×Q₁₆	C₂×C₈	1	2	yes	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂²×C₈	—	2	1	no	—
79	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
80	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₈	#41	1	1	yes	characteristic
83	C₄⋊Q₈	C₂×C₈	1	2	yes	—
84	C₄⋊Q₈	C₂×C₈	1	2	yes	—
85	#125	—	2	2	no	—
86	C₂²⋊Q₈	—	2	2	no	—
87	C₂²⋊Q₈	C₂×C₈	1	4	yes	—
88	C₄×Q₈	—	2	2	no	—
89	C₄×Q₈	C₂²×C₄	1	4	yes	—
90	#117	—	2	2	no	—
91	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₁₆	—	2	4	no	—
93	C₂×C₁₆	—	2	1	no	—
94	C₂×C₁₆	—	2	1	no	—
95	C₂×C₁₆	C₂×D₄	1	2	yes	—
96	C₂×C₁₆	C₂×D₄	1	2	yes	—
97	#107	C₂×C₈	1	2	yes	—
98	#107	C₂×C₈	1	2	yes	—
99	C₄⋊C₈	—	2	4	no	—
100	C₄⋊C₈	—	2	4	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	C₂²×C₄	1	4	yes	—
103	#103	C₂×C₈	1	4	yes	—
104	#103	C₂×C₈	1	4	yes	—
105	#97	—	2	4	no	—
106	C₄×C₈	—	2	4	no	—
107	C₄×C₈	—	2	1	no	—
108	C₄×C₈	C₂×C₈	1	1	yes	characteristic
109	C₄×C₈	C₂×C₈	1	1	yes	characteristic
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₄×C₈	C₂×D₄	1	1	yes	characteristic
116	C₄×C₈	#41	1	1	yes	characteristic
117	C₄×C₈	C₂⁴	1	1	yes	characteristic
118	C₂×Q₈	—	2	2	no	—
119	C₂×Q₈	C₂²×C₈	1	4	yes	—
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	Q₁₆	—	2	4	no	—
124	Q₁₆	—	2	4	no	—
125	M₁₆	—	8	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	C₄×D₄	1	1	yes	characteristic
135	C₂×C₈	C₄×D₄	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
138	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₈	C₂×#41	1	1	yes	characteristic
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	2	no	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	C₂²×C₈	1	4	yes	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₄²	—	2	4	no	—
150	C₄²	C₄×D₄	1	1	yes	characteristic
151	C₄²	C₂²×C₈	1	1	yes	characteristic
152	C₄²	#131	1	1	yes	characteristic
153	C₄²	C₂³×C₄	1	1	yes	characteristic
154	C₁₆	—	2	2	no	—
155	C₁₆	—	2	2	no	—
156	C₁₆	—	4	1	no	—
157	C₁₆	—	4	1	no	—
158	C₁₆	—	4	4	no	—
159	C₂³	C₈×D₄	1	1	yes	characteristic
160	Q₈	—	2	4	no	—
161	Q₈	—	2	4	no	—
162	Q₈	—	8	2	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₈×D₄	1	1	yes	characteristic
171	C₂×C₄	C₈×D₄	1	1	yes	characteristic
172	C₂×C₄	C₈×D₄	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
175	C₂×C₄	C₂×#131	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	2	1	no	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	2	1	no	—
182	C₈	—	4	1	no	—
183	C₈	—	4	4	no	—
184	C₈	#504	1	2	yes	—
185	C₈	#504	1	2	yes	—
186	C₈	#578	1	2	yes	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	—	2	1	no	—
195	C₄	—	4	4	no	—
196	C₄	—	4	4	no	—
197	C₄	#2950	1	1	yes	characteristic
198	C₄	#2957	1	1	yes	characteristic
199	C₄	C₂×#504	1	1	yes	characteristic
200	C₄	C₂×#504	1	1	yes	characteristic
201	C₂	—	4	1	no	—
202	C₂	#17856	1	1	yes	characteristic
203	C₂	#33434	1	1	yes	characteristic
204	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	C₂³	C₈×D₄
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
3. #12061	C₂	semidirect
4. #12059	C₂	semidirect
5. #12054	C₂	semidirect
6. C₁₆×Q₁₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33499	2
#17958	2
#12061	2
#12059	2
#12054	2
C₁₆×Q₁₆	2
#12013	2
#11640	2
#11410	2