Subgroups — #483743

Order 512.

393 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33500	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	#12053	C₂	1	4	yes	maximal, complemented
6	#12012	C₂	1	1	yes	maximal, characteristic
7	#11638	C₂	1	2	yes	maximal
8	#11411	C₂	1	1	yes	maximal, characteristic
9	#3170	C₄	1	1	yes	characteristic
10	#2960	C₂²	1	2	yes	—
11	#2183	—	2	2	no	—
12	#2183	C₂²	1	4	yes	—
13	Q₈×C₁₆	—	2	2	no	—
14	Q₈×C₁₆	C₂²	1	4	yes	—
15	#2164	—	2	4	no	—
16	#2162	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	1	yes	characteristic
28	#1367	C₂²	1	1	yes	characteristic
29	Q₈⋊C₁₆	C₂²	1	4	yes	—
30	Q₈⋊C₁₆	C₂²	1	4	yes	—
31	#1312	C₂²	1	2	yes	—
32	#1311	C₂²	1	2	yes	—
33	#533	C₂×C₄	1	2	yes	—
34	#500	C₈	1	1	yes	characteristic
35	#494	C₈	1	1	yes	characteristic
36	#487	C₈	1	2	yes	—
37	#467	C₈	1	2	yes	—
38	#467	C₈	1	2	yes	—
39	#446	—	2	2	no	—
40	#446	C₂×C₄	1	4	yes	—
41	C₈×Q₈	—	2	2	no	—
42	C₈×Q₈	C₂³	1	4	yes	—
43	C₄×Q₁₆	C₂×C₄	1	2	yes	—
44	C₄×Q₁₆	C₂×C₄	1	2	yes	—
45	#424	—	2	2	no	—
46	#423	C₂×C₄	1	2	yes	—
47	#405	—	2	2	no	—
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	4	no	—
52	C₄⋊C₁₆	—	2	4	no	—
53	C₄⋊C₁₆	—	2	1	no	—
54	C₄⋊C₁₆	D₄	1	2	yes	—
55	C₄⋊C₁₆	C₂³	1	2	yes	—
56	C₄⋊C₁₆	C₂³	1	2	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	C₄×C₁₆	—	2	4	no	—
62	C₄×C₁₆	—	2	2	no	—
63	C₄×C₁₆	C₂³	1	2	yes	—
64	C₄×C₁₆	C₂³	1	2	yes	—
65	#335	C₂×C₄	1	2	yes	—
66	#335	C₂×C₄	1	2	yes	—
67	Q₈⋊C₈	C₂×C₄	1	4	yes	—
68	Q₈⋊C₈	C₂×C₄	1	4	yes	—
69	C₈²	C₂×C₄	1	1	yes	characteristic
70	C₈²	C₂×C₄	1	1	yes	characteristic
71	C₈²	C₂³	1	1	yes	characteristic
72	C₈²	C₂³	1	1	yes	characteristic
73	C₂×Q₁₆	C₂×C₈	1	2	yes	—
74	C₂×Q₁₆	C₂×C₈	1	2	yes	—
75	C₂×M₁₆	—	2	2	no	—
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₈	#41	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×C₈	1	2	yes	—
82	C₄⋊Q₈	C₂×C₈	1	2	yes	—
83	#125	—	2	2	no	—
84	C₂²⋊Q₈	—	2	2	no	—
85	C₂²⋊Q₈	C₂×C₈	1	4	yes	—
86	C₄×Q₈	—	2	2	no	—
87	C₄×Q₈	C₂²×C₄	1	4	yes	—
88	#117	—	2	2	no	—
89	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	C₂×D₄	1	4	yes	—
94	#107	C₂×C₈	1	2	yes	—
95	#107	C₂×C₈	1	2	yes	—
96	C₄⋊C₈	—	2	4	no	—
97	C₄⋊C₈	—	2	4	no	—
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	C₂²×C₄	1	4	yes	—
100	#103	C₂×C₈	1	4	yes	—
101	#103	C₂×C₈	1	4	yes	—
102	#97	—	2	4	no	—
103	C₄×C₈	—	2	4	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	C₂×C₈	1	1	yes	characteristic
106	C₄×C₈	C₂×C₈	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	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
113	C₄×C₈	#41	1	1	yes	characteristic
114	C₄×C₈	C₂⁴	1	1	yes	characteristic
115	C₂×Q₈	—	2	2	no	—
116	C₂×Q₈	C₂²×C₈	1	4	yes	—
117	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
118	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
119	C₂²×C₄	#131	1	1	yes	characteristic
120	Q₁₆	—	2	4	no	—
121	Q₁₆	—	4	2	no	—
122	M₁₆	—	8	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	4	1	no	—
129	C₂×C₈	—	4	1	no	—
130	C₂×C₈	—	4	2	no	—
131	C₂×C₈	C₄×D₄	1	1	yes	characteristic
132	C₂×C₈	C₄×D₄	1	1	yes	characteristic
133	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₈	C₂×#41	1	1	yes	characteristic
139	C₄⋊C₄	—	2	4	no	—
140	C₄⋊C₄	—	2	2	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	C₂²×C₈	1	4	yes	—
145	C₂²⋊C₄	—	2	4	no	—
146	C₄²	—	2	4	no	—
147	C₄²	C₄×D₄	1	1	yes	characteristic
148	C₄²	C₂²×C₈	1	1	yes	characteristic
149	C₄²	#131	1	1	yes	characteristic
150	C₄²	C₂³×C₄	1	1	yes	characteristic
151	C₁₆	—	2	4	no	—
152	C₁₆	—	4	2	no	—
153	C₁₆	—	4	2	no	—
154	C₁₆	—	4	2	no	—
155	C₂³	C₈×D₄	1	1	yes	characteristic
156	Q₈	—	2	4	no	—
157	Q₈	—	2	4	no	—
158	Q₈	—	4	4	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	C₈×D₄	1	1	yes	characteristic
167	C₂×C₄	C₈×D₄	1	1	yes	characteristic
168	C₂×C₄	C₈×D₄	1	1	yes	characteristic
169	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#131	1	1	yes	characteristic
172	C₈	—	2	1	no	—
173	C₈	—	2	1	no	—
174	C₈	—	2	1	no	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	4	1	no	—
179	C₈	—	4	1	no	—
180	C₈	—	4	4	no	—
181	C₈	#504	1	2	yes	—
182	C₈	#578	1	2	yes	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	1	no	—
186	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	4	4	no	—
191	C₄	—	4	4	no	—
192	C₄	#2171	1	2	yes	—
193	C₄	#2950	1	1	yes	characteristic
194	C₄	#2957	1	1	yes	characteristic
195	C₄	C₂×#504	1	1	yes	characteristic
196	C₄	#3407	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	#17856	1	1	yes	characteristic
199	C₂	#33438	1	1	yes	characteristic
200	C₂	C₂×#2171	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. #12053	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33500	2
#17958	2
#12061	2
#12059	2
#12053	2
#12012	2
#11638	2
#11411	2