Subgroups — #140388

Order 512.

268 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12653	C₂	1	1	yes	maximal, characteristic
2	C₂×#1368	C₂	1	2	yes	maximal
3	#8805	C₂	1	1	yes	maximal, characteristic
4	C₄×#334	C₂	1	1	yes	maximal, characteristic
5	#7156	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
8	C₂×#346	C₂²	1	2	yes	—
9	#1842	—	2	1	no	—
10	C₄×C₄⋊C₈	—	2	1	no	—
11	C₄²×C₈	C₂²	1	1	yes	characteristic
12	C₂×#334	—	2	1	no	—
13	C₂×#334	C₂²	1	2	yes	—
14	C₂×#334	C₂²	1	1	yes	characteristic
15	C₂×#334	C₂²	1	1	yes	characteristic
16	#1374	C₄	1	2	yes	—
17	#1368	—	2	4	no	—
18	#1277	—	2	1	no	—
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₈	—	2	1	no	—
22	C₂×C₄⋊C₈	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	—	4	1	no	—
25	C₂×C₄⋊C₈	—	4	1	no	—
26	C₂×C₄×C₈	—	4	1	no	—
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	C₄³	D₄	1	1	yes	characteristic
34	C₄⋊C₁₆	—	2	2	no	—
35	C₄⋊C₁₆	C₈	1	4	yes	complemented
36	#346	—	2	2	no	—
37	#346	—	2	2	no	—
38	#336	—	4	1	no	—
39	#334	—	2	2	no	—
40	#334	—	2	1	no	—
41	#334	—	2	1	no	—
42	#334	—	2	2	no	—
43	#334	—	4	2	no	—
44	#334	D₄	1	2	yes	—
45	#334	Q₈	1	2	yes	—
46	C₂²×C₈	—	4	1	no	—
47	C₂²×C₈	—	4	1	no	—
48	C₂²×C₈	—	4	2	no	—
49	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄²	SD₁₆	1	1	yes	characteristic
56	C₂×C₄²	SD₁₆	1	1	yes	characteristic
57	C₂×C₄²	SD₁₆	1	1	yes	characteristic
58	C₂×C₄²	SD₁₆	1	1	yes	characteristic
59	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
60	C₂×C₁₆	—	2	2	no	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	C₂×C₈	1	4	yes	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	4	2	no	—
69	C₄⋊C₈	—	4	1	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	4	1	no	—
72	C₄⋊C₈	—	8	2	no	—
73	C₄⋊C₈	—	8	2	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	4	1	no	—
80	C₄×C₈	—	8	2	no	—
81	C₄×C₈	C₂×C₈	1	2	yes	—
82	C₄×C₈	M₁₆	1	2	yes	—
83	C₄×C₈	C₂×D₄	1	1	yes	characteristic
84	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
85	C₄×C₈	#41	1	1	yes	characteristic
86	C₄×C₈	#41	1	1	yes	characteristic
87	C₂²×C₄	#106	1	1	yes	characteristic
88	C₂²×C₄	#106	1	1	yes	characteristic
89	C₂²×C₄	#106	1	1	yes	characteristic
90	C₂²×C₄	#106	1	1	yes	characteristic
91	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
93	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	8	1	no	—
104	C₂×C₈	—	8	1	no	—
105	C₂×C₈	—	8	1	no	—
106	C₂×C₈	—	8	1	no	—
107	C₂×C₈	—	8	2	no	—
108	C₂×C₈	—	8	2	no	—
109	C₂×C₈	—	8	2	no	—
110	C₂×C₈	C₄⋊C₈	1	2	yes	—
111	C₂×C₈	C₄×D₄	1	1	yes	characteristic
112	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₈	#125	1	1	yes	characteristic
114	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₈	#131	1	1	yes	characteristic
118	C₂×C₈	#131	1	1	yes	characteristic
119	C₄²	—	2	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	4	1	no	—
125	C₄²	C₄⋊C₈	1	2	yes	—
126	C₄²	#108	1	2	yes	—
127	C₄²	#121	1	1	yes	characteristic
128	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
129	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₄²	#137	1	1	yes	characteristic
131	C₁₆	—	2	4	no	—
132	C₁₆	—	4	2	no	—
133	C₂³	C₂×#106	1	1	yes	characteristic
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	#334	1	2	yes	—
146	C₂×C₄	#334	1	2	yes	—
147	C₂×C₄	#365	1	2	yes	—
148	C₂×C₄	#365	1	2	yes	—
149	C₂×C₄	#365	1	2	yes	—
150	C₂×C₄	#389	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₄	#424	1	1	yes	characteristic
153	C₂×C₄	#428	1	1	yes	characteristic
154	C₂×C₄	C₂×#108	1	1	yes	characteristic
155	C₂×C₄	#430	1	1	yes	characteristic
156	C₂×C₄	#460	1	1	yes	characteristic
157	C₂×C₄	#461	1	1	yes	characteristic
158	C₂×C₄	#475	1	1	yes	characteristic
159	C₂×C₄	#476	1	1	yes	characteristic
160	C₈	—	2	2	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	4	1	no	—
164	C₈	—	8	2	no	—
165	C₈	—	8	2	no	—
166	C₈	—	16	2	no	—
167	C₈	C₈×D₄	1	1	yes	characteristic
168	C₈	#436	1	1	yes	characteristic
169	C₈	C₈×Q₈	1	1	yes	characteristic
170	C₈	#446	1	1	yes	characteristic
171	C₂²	C₂×#334	1	1	yes	characteristic
172	C₂²	#1570	1	1	yes	characteristic
173	C₂²	#1948	1	1	yes	characteristic
174	C₂²	C₂×#365	1	1	yes	characteristic
175	C₂²	C₂×#365	1	1	yes	characteristic
176	C₂²	C₂×#365	1	1	yes	characteristic
177	C₂²	#2155	1	1	yes	characteristic
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	4	1	no	—
185	C₄	#1368	1	4	yes	complemented
186	C₄	#1565	1	1	yes	characteristic
187	C₄	#1574	1	1	yes	characteristic
188	C₄	#1939	1	1	yes	characteristic
189	C₄	#1950	1	1	yes	characteristic
190	C₂	#9602	1	1	yes	characteristic
191	C₂	C₂×#1368	1	1	yes	characteristic
192	C₂	#12000	1	1	yes	characteristic
193	C₂	#12029	1	1	yes	characteristic
194	C₂	#12030	1	1	yes	characteristic
195	C₂	#13183	1	1	yes	characteristic
196	C₂	#13185	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
35. C₄⋊C₁₆	C₈	semidirect
185. C₄	#1368	semidirect

Maximal subgroups

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

subgroup	index
#12653	2
C₂×#1368	2
#8805	2
C₄×#334	2
#7156	2