Subgroups — #497950

Order 512.

488 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×M₃₂	C₂	1	1	yes	maximal, characteristic
2	Q₈×#108	C₂	1	1	yes	maximal, characteristic
3	Q₈×#106	C₂	1	1	yes	maximal, characteristic
4	#13185	C₂	1	3	yes	maximal
5	#13183	C₂	1	3	yes	maximal
6	#12886	C₂	1	3	yes	maximal
7	C₄×#365	C₂	1	3	yes	maximal
8	Q₈×M₁₆	—	2	1	no	—
9	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
10	Q₈×C₄⋊C₄	—	2	1	no	—
11	#2183	C₄	1	6	yes	complemented
12	Q₈×C₁₆	C₄	1	2	yes	complemented
13	C₂×#365	C₂²	1	3	yes	—
14	C₂×#365	C₂²	1	3	yes	—
15	#2150	C₂²	1	3	yes	—
16	C₄×M₃₂	C₂²	1	3	yes	—
17	#1950	—	2	3	no	—
18	#1948	—	2	3	no	—
19	#1848	C₂²	1	3	yes	—
20	#1844	C₂²	1	3	yes	—
21	C₄×#108	C₂²	1	3	yes	—
22	C₄×#106	C₂²	1	3	yes	—
23	#1383	C₂²	1	3	yes	—
24	#1383	C₂²	1	6	yes	—
25	#1383	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈	—	4	1	no	—
27	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
28	C₂×M₃₂	C₂³	1	1	yes	characteristic
29	C₂×M₃₂	C₂³	1	3	yes	—
30	#446	—	4	3	no	—
31	C₈×Q₈	—	4	1	no	—
32	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₈×Q₈	C₂×C₄	1	3	yes	—
34	C₈×Q₈	Q₈	1	3	yes	—
35	C₈×Q₈	Q₈	1	1	yes	characteristic
36	C₂×#108	—	2	3	no	—
37	C₂×#108	C₂³	1	3	yes	—
38	C₂×#108	C₂³	1	1	yes	characteristic
39	C₂×#106	—	2	3	no	—
40	C₂×#106	C₂³	1	3	yes	—
41	C₂×#106	C₂³	1	1	yes	characteristic
42	#423	—	2	3	no	—
43	C₂×C₄⋊C₈	D₄	1	3	yes	—
44	C₄×M₁₆	—	2	3	no	—
45	C₂×C₄×C₈	C₂³	1	3	yes	—
46	#389	—	4	3	no	—
47	#384	—	2	3	no	—
48	C₄×C₄⋊C₄	—	2	3	no	—
49	#365	—	2	6	no	—
50	#365	—	2	6	no	—
51	C₄⋊C₁₆	C₂×C₄	1	6	yes	—
52	C₄⋊C₁₆	C₂×C₄	1	6	yes	—
53	#346	C₂×C₄	1	6	yes	—
54	C₄×C₁₆	C₂×C₄	1	6	yes	—
55	#341	—	2	3	no	—
56	#340	—	2	3	no	—
57	C₂²×Q₈	SD₁₆	1	1	yes	characteristic
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	2	3	no	—
60	C₂²×C₈	C₂×D₄	1	3	yes	—
61	C₂²×C₈	C₂⁴	1	1	yes	characteristic
62	C₄×Q₈	—	8	4	no	—
63	C₄×Q₈	C₄⋊C₄	1	3	yes	—
64	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
65	C₄×Q₈	SD₁₆	1	1	yes	characteristic
66	C₄×Q₈	SD₁₆	1	3	yes	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	2	3	no	—
69	C₂×C₄⋊C₄	—	4	3	no	—
70	C₂×C₄⋊C₄	—	4	3	no	—
71	C₂×C₄⋊C₄	SD₁₆	1	3	yes	—
72	C₂×C₄²	—	4	3	no	—
73	C₂×C₄²	C₂×D₄	1	3	yes	—
74	M₃₂	—	2	3	no	—
75	M₃₂	—	2	3	no	—
76	M₃₂	—	2	1	no	—
77	M₃₂	—	2	1	no	—
78	C₂×C₁₆	C₂²×C₄	1	2	yes	—
79	C₂×C₁₆	C₂²×C₄	1	6	yes	—
80	#108	—	2	3	no	—
81	#108	—	2	1	no	—
82	#108	—	2	3	no	—
83	#108	—	8	3	no	—
84	#108	#41	1	2	yes	—
85	#106	—	2	1	no	—
86	#106	—	2	3	no	—
87	#106	—	2	3	no	—
88	#106	—	8	3	no	—
89	#106	#41	1	2	yes	—
90	C₄⋊C₈	—	4	3	no	—
91	C₄⋊C₈	—	4	3	no	—
92	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
93	C₄⋊C₈	C₄⋊C₄	1	6	yes	—
94	C₄⋊C₈	C₄⋊C₄	1	3	yes	—
95	#97	—	4	3	no	—
96	C₄×C₈	—	4	3	no	—
97	C₄×C₈	C₂²×C₄	1	3	yes	—
98	C₄×C₈	C₂²×C₄	1	3	yes	—
99	C₄×C₈	C₂×Q₈	1	3	yes	—
100	C₄×C₈	C₂×Q₈	1	3	yes	—
101	#95	—	4	3	no	—
102	C₂×Q₈	—	2	4	no	—
103	C₂×Q₈	#106	1	3	yes	—
104	C₂×Q₈	#106	1	1	yes	characteristic
105	C₂²×C₄	—	4	1	no	—
106	C₂²×C₄	—	4	3	no	—
107	C₂²×C₄	C₂×SD₁₆	1	3	yes	—
108	C₂²×C₄	C₂×SD₁₆	1	3	yes	—
109	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
110	M₁₆	—	2	2	no	—
111	M₁₆	—	4	1	no	—
112	M₁₆	—	4	3	no	—
113	M₁₆	—	4	3	no	—
114	C₂×C₈	—	2	3	no	—
115	C₂×C₈	—	2	3	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	3	no	—
118	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
119	C₂×C₈	C₂×C₄⋊C₄	1	3	yes	—
120	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₈	C₂×#41	1	1	yes	characteristic
123	C₂×C₈	C₂×#41	1	1	yes	characteristic
124	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	4	3	no	—
129	C₄⋊C₄	—	4	3	no	—
130	C₄⋊C₄	—	8	6	no	—
131	C₄⋊C₄	—	16	3	no	—
132	C₄⋊C₄	#106	1	6	yes	—
133	C₄⋊C₄	#106	1	3	yes	—
134	C₄⋊C₄	#106	1	3	yes	—
135	C₄²	—	8	6	no	—
136	C₄²	C₂×C₄⋊C₄	1	3	yes	—
137	C₄²	C₂×C₄⋊C₄	1	3	yes	—
138	C₄²	C₂×SD₁₆	1	3	yes	—
139	C₄²	C₂×SD₁₆	1	3	yes	—
140	C₁₆	—	2	6	no	—
141	C₁₆	—	2	2	no	—
142	C₂³	C₂²×SD₁₆	1	1	yes	characteristic
143	Q₈	—	2	6	no	—
144	Q₈	#365	1	4	yes	complemented
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	4	3	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	8	1	no	—
150	C₂×C₄	—	8	3	no	—
151	C₂×C₄	—	8	3	no	—
152	C₂×C₄	—	8	3	no	—
153	C₂×C₄	C₂×#106	1	3	yes	—
154	C₂×C₄	C₂×#106	1	3	yes	—
155	C₂×C₄	C₂×#106	1	3	yes	—
156	C₂×C₄	C₂×#106	1	3	yes	—
157	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂×C₄	#548	1	1	yes	characteristic
159	C₂×C₄	#548	1	1	yes	characteristic
160	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
161	C₂×C₄	#576	1	1	yes	characteristic
162	C₂×C₄	#578	1	1	yes	characteristic
163	C₈	—	2	3	no	—
164	C₈	—	2	3	no	—
165	C₈	—	4	2	no	—
166	C₈	—	8	3	no	—
167	C₈	C₄×#41	1	1	yes	characteristic
168	C₈	#519	1	1	yes	characteristic
169	C₈	#557	1	1	yes	characteristic
170	C₈	#557	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	C₂²×#106	1	1	yes	characteristic
174	C₂²	#3359	1	1	yes	characteristic
175	C₂²	#3362	1	1	yes	characteristic
176	C₄	—	2	3	no	—
177	C₄	—	2	3	no	—
178	C₄	—	2	3	no	—
179	C₄	—	8	2	no	—
180	C₄	—	16	3	no	—
181	C₄	C₂×#365	1	6	yes	—
182	C₄	#2351	1	1	yes	characteristic
183	C₄	#2913	1	1	yes	characteristic
184	C₄	#3359	1	1	yes	characteristic
185	C₄	#3362	1	1	yes	characteristic
186	C₂	—	2	1	no	—
187	C₂	—	2	1	no	—
188	C₂	#22738	1	1	yes	characteristic
189	C₂	C₂²×#365	1	1	yes	characteristic
190	C₂	#33404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂²×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
11. #2183	C₄	semidirect
12. Q₈×C₁₆	C₄	semidirect
144. Q₈	#365	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₃₂	2
Q₈×#108	2
Q₈×#106	2
#13185	2
#13183	2
#12886	2
C₄×#365	2