Subgroups — #483413

Order 512.

403 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33428	C₂	1	2	yes	maximal
2	C₄×#504	C₂	1	1	yes	maximal, characteristic
3	#12027	C₂	1	4	yes	maximal, complemented
4	#12026	C₂	1	4	yes	maximal, complemented
5	C₂×#1373	C₂	1	2	yes	maximal, complemented
6	#12002	C₂	1	2	yes	maximal, complemented
7	C₂×#504	C₄	1	1	yes	characteristic
8	C₂×#504	C₄	1	1	yes	characteristic
9	C₄×#131	C₂²	1	1	yes	characteristic
10	#2166	C₂²	1	8	yes	—
11	#2165	—	2	4	no	—
12	C₂×#364	C₂²	1	4	yes	—
13	#2115	—	2	2	no	—
14	#2110	—	2	2	no	—
15	#2108	C₄	1	1	yes	characteristic
16	#2108	C₄	1	1	yes	characteristic
17	#2108	C₂²	1	2	yes	—
18	#2108	C₂²	1	2	yes	—
19	C₄×M₃₂	C₄	1	2	yes	—
20	C₄×M₃₂	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₁₆	C₄	1	2	yes	—
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1373	C₂²	1	4	yes	—
24	#1373	C₂²	1	4	yes	—
25	#1369	C₂²	1	8	yes	—
26	C₂×#131	C₂×C₄	1	1	yes	characteristic
27	C₂×#131	D₄	1	1	yes	characteristic
28	C₄×#41	Q₈	1	1	yes	characteristic
29	#504	—	2	4	no	—
30	#504	—	2	4	no	—
31	C₂×M₃₂	—	2	4	no	—
32	C₂×M₃₂	C₂×C₄	1	2	yes	—
33	C₂×M₃₂	C₂×C₄	1	2	yes	—
34	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
35	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₁₆	C₂×C₄	1	2	yes	—
37	C₂²×C₁₆	C₂×C₄	1	2	yes	—
38	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
40	#405	C₂×C₄	1	1	yes	characteristic
41	#405	D₄	1	1	yes	characteristic
42	#405	Q₈	1	2	yes	—
43	#405	C₂³	1	2	yes	—
44	C₄×M₁₆	D₄	1	2	yes	—
45	C₄×M₁₆	Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
48	#364	—	2	4	no	—
49	#364	C₂³	1	8	yes	—
50	#349	—	2	8	no	—
51	#347	—	2	8	no	—
52	#346	C₂×C₄	1	2	yes	—
53	#346	C₂×C₄	1	2	yes	—
54	#346	C₂³	1	2	yes	—
55	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
56	C₄×C₁₆	C₂×C₄	1	2	yes	—
57	C₄×C₁₆	C₂×C₄	1	2	yes	—
58	C₄×C₁₆	C₂×C₄	1	2	yes	—
59	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
60	C₄×C₁₆	C₂³	1	1	yes	characteristic
61	C₄×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
63	#131	—	2	2	no	—
64	#131	—	2	2	no	—
65	#131	—	2	2	no	—
66	#131	—	2	2	no	—
67	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
68	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
69	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
70	C₂×M₁₆	C₂×D₄	1	2	yes	—
71	C₂²×C₈	C₄⋊C₄	1	2	yes	—
72	C₂²×C₈	C₂²×C₄	1	2	yes	—
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×Q₈	—	2	1	no	—
76	C₄×D₄	—	2	2	no	—
77	C₄×D₄	—	2	1	no	—
78	#117	C₄⋊C₄	1	1	yes	characteristic
79	#117	C₂×Q₈	1	2	yes	—
80	C₂×C₄²	C₄⋊C₄	1	2	yes	—
81	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
82	M₃₂	—	2	2	no	—
83	M₃₂	—	2	4	no	—
84	M₃₂	—	2	4	no	—
85	M₃₂	—	2	8	no	—
86	M₃₂	—	4	4	no	—
87	M₃₂	M₁₆	1	4	yes	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	—	4	4	no	—
92	C₂×C₁₆	M₁₆	1	4	yes	—
93	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₁₆	C₂²×C₄	1	2	yes	—
95	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₁₆	C₂²×C₄	1	2	yes	—
98	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	#97	C₄⋊C₄	1	2	yes	—
100	#97	C₂×D₄	1	2	yes	—
101	#97	C₂×Q₈	1	2	yes	—
102	C₄×C₈	C₄⋊C₄	1	2	yes	—
103	C₄×C₈	C₄⋊C₄	1	2	yes	—
104	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₄×C₈	C₂²×C₄	1	2	yes	—
106	C₄×C₈	C₂×D₄	1	1	yes	characteristic
107	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
108	C₄×C₈	C₂⁴	1	1	yes	characteristic
109	#41	—	2	2	no	—
110	#41	—	2	2	no	—
111	C₂×Q₈	#108	1	1	yes	characteristic
112	C₂×D₄	#108	1	2	yes	—
113	C₂×D₄	#108	1	1	yes	characteristic
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
117	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
118	M₁₆	—	2	2	no	—
119	M₁₆	—	2	2	no	—
120	M₁₆	—	2	4	no	—
121	M₁₆	—	2	4	no	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₈	C₂×M₁₆	1	2	yes	—
131	C₂×C₈	C₂×M₁₆	1	2	yes	—
132	C₂×C₈	#135	1	2	yes	—
133	C₂×C₈	#135	1	2	yes	—
134	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
136	C₄⋊C₄	—	2	1	no	—
137	C₄⋊C₄	—	2	2	no	—
138	C₂²⋊C₄	—	2	1	no	—
139	C₂²⋊C₄	—	2	2	no	—
140	C₄²	—	2	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₄²	C₂×C₄⋊C₄	1	2	yes	—
144	C₄²	C₂²×Q₈	1	1	yes	characteristic
145	C₁₆	—	2	1	no	—
146	C₁₆	—	2	1	no	—
147	C₁₆	—	2	2	no	—
148	C₁₆	—	4	8	no	—
149	C₁₆	C₂×M₁₆	1	2	yes	—
150	C₁₆	C₂×M₁₆	1	4	yes	—
151	C₁₆	C₂×M₁₆	1	2	yes	—
152	C₂³	C₂×#108	1	2	yes	—
153	C₂³	C₂×#108	1	1	yes	characteristic
154	Q₈	—	2	2	no	—
155	D₄	—	2	4	no	—
156	D₄	—	2	2	no	—
157	C₂×C₄	—	2	1	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	4	4	no	—
164	C₂×C₄	#423	1	2	yes	—
165	C₂×C₄	#427	1	2	yes	—
166	C₂×C₄	C₂×#108	1	2	yes	—
167	C₂×C₄	C₂×#108	1	1	yes	characteristic
168	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	—	2	2	no	—
171	C₈	#423	1	2	yes	—
172	C₈	#427	1	1	yes	characteristic
173	C₈	#427	1	1	yes	characteristic
174	C₈	C₂²×M₁₆	1	1	yes	characteristic
175	C₈	C₂²×M₁₆	1	1	yes	characteristic
176	C₈	C₂×#135	1	2	yes	—
177	C₂²	—	2	2	no	—
178	C₂²	—	4	1	no	—
179	C₂²	—	4	2	no	—
180	C₂²	#1568	1	2	yes	—
181	C₂²	C₂²×#108	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	2	no	—
185	C₄	—	4	1	no	—
186	C₄	#1568	1	2	yes	—
187	C₄	C₂×#423	1	1	yes	characteristic
188	C₄	C₂×#427	1	1	yes	characteristic
189	C₂	—	2	1	no	—
190	C₂	—	2	2	no	—
191	C₂	—	4	2	no	—
192	C₂	C₂×#1568	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1568
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ #405 ◁ C₄×#131 ◁ 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. #12027	C₂	semidirect
4. #12026	C₂	semidirect
5. C₂×#1373	C₂	semidirect
6. #12002	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33428	2
C₄×#504	2
#12027	2
#12026	2
C₂×#1373	2
#12002	2