Subgroups — #148941

Order 512.

434 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#1388	C₂	1	2	yes	maximal
3	C₂×#1381	C₂	1	1	yes	maximal, characteristic
4	#12545	C₂	1	1	yes	maximal, characteristic
5	#10873	C₂	1	2	yes	maximal
6	C₂²×M₃₂	C₂²	1	1	yes	characteristic
7	C₂³×C₁₆	C₂²	1	1	yes	characteristic
8	C₂²×#108	C₂²	1	2	yes	—
9	C₂²×#108	C₂²	1	2	yes	—
10	C₂×#405	C₂²	1	1	yes	characteristic
11	C₂×#368	—	2	2	no	—
12	C₂×#368	—	2	4	no	—
13	C₂×#344	—	2	2	no	—
14	#1738	—	2	2	no	—
15	#1388	—	2	4	no	—
16	#1381	C₄	1	4	yes	complemented
17	#1374	—	2	2	no	—
18	C₂²×M₁₆	—	2	2	no	—
19	C₂²×M₁₆	—	2	2	no	—
20	C₂²×M₁₆	D₄	1	1	yes	characteristic
21	C₂³×C₈	C₂³	1	1	yes	characteristic
22	C₂×#117	D₄	1	1	yes	characteristic
23	C₂×M₃₂	—	2	1	no	—
24	C₂×M₃₂	—	2	2	no	—
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	Q₈	1	2	yes	—
27	C₂²×C₁₆	—	2	2	no	—
28	C₂²×C₁₆	—	2	2	no	—
29	C₂²×C₁₆	—	2	1	no	—
30	C₂²×C₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₁₆	D₄	1	2	yes	—
32	C₂²×C₁₆	Q₈	1	2	yes	—
33	C₂×#108	—	2	2	no	—
34	C₂×#108	—	2	2	no	—
35	C₂×#108	—	2	2	no	—
36	C₂×#108	—	2	2	no	—
37	C₂×#108	—	2	2	no	—
38	C₂×#108	—	2	2	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	2	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	C₂×C₄	1	4	yes	—
44	C₂×#108	D₄	1	4	yes	—
45	#405	—	2	2	no	—
46	#405	—	2	2	no	—
47	C₂×#97	—	2	1	no	—
48	C₂×C₄×C₈	—	2	1	no	—
49	#368	—	2	8	no	—
50	#344	—	4	4	no	—
51	#341	—	4	4	no	—
52	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
53	C₂×M₁₆	—	2	1	no	—
54	C₂×M₁₆	—	2	2	no	—
55	C₂×M₁₆	—	2	2	no	—
56	C₂×M₁₆	—	2	4	no	—
57	C₂×M₁₆	—	2	4	no	—
58	C₂×M₁₆	—	4	2	no	—
59	C₂×M₁₆	—	4	2	no	—
60	C₂×M₁₆	—	4	2	no	—
61	C₂×M₁₆	—	4	2	no	—
62	C₂×M₁₆	—	4	2	no	—
63	C₂×M₁₆	—	4	2	no	—
64	C₂×M₁₆	—	4	2	no	—
65	C₂×M₁₆	—	4	2	no	—
66	C₂×M₁₆	—	4	2	no	—
67	C₂×M₁₆	—	4	2	no	—
68	C₂×M₁₆	D₈	1	2	yes	—
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	—	4	2	no	—
71	C₂²×C₈	—	4	2	no	—
72	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	2	yes	—
75	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₈	#41	1	2	yes	—
78	C₂²×C₈	#41	1	2	yes	—
79	C₂²×C₈	#41	1	2	yes	—
80	C₂²×C₈	#41	1	1	yes	characteristic
81	C₂²×C₈	#41	1	1	yes	characteristic
82	#117	—	2	2	no	—
83	#117	—	2	2	no	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₂²⋊C₄	Q₁₆	1	2	yes	—
86	C₂×C₄²	—	2	1	no	—
87	M₃₂	—	2	4	no	—
88	M₃₂	—	4	2	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	4	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	2	no	—
94	C₂×C₁₆	—	4	1	no	—
95	C₂×C₁₆	—	4	1	no	—
96	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
97	#108	—	2	4	no	—
98	#108	—	2	4	no	—
99	#108	—	2	4	no	—
100	#108	—	2	4	no	—
101	#108	—	4	2	no	—
102	#108	—	4	2	no	—
103	#108	—	4	2	no	—
104	#108	—	4	2	no	—
105	#97	—	4	2	no	—
106	C₄×C₈	—	4	1	no	—
107	C₄×C₈	—	4	1	no	—
108	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂²×C₄	#121	1	2	yes	—
112	C₂²×C₄	#121	1	1	yes	characteristic
113	C₂²×C₄	#123	1	2	yes	—
114	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
116	C₂²×C₄	#135	1	1	yes	characteristic
117	C₂²×C₄	#135	1	1	yes	characteristic
118	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
119	C₂²×C₄	#137	1	2	yes	—
120	M₁₆	—	2	4	no	—
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	4	no	—
123	M₁₆	—	4	4	no	—
124	M₁₆	—	4	4	no	—
125	M₁₆	—	4	4	no	—
126	M₁₆	—	8	2	no	—
127	M₁₆	—	8	2	no	—
128	M₁₆	—	8	2	no	—
129	M₁₆	—	8	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	8	2	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	—	8	2	no	—
138	C₂×C₈	—	8	2	no	—
139	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₈	#117	1	2	yes	—
141	C₂×C₈	C₄×D₄	1	2	yes	—
142	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
143	C₂×C₈	#120	1	1	yes	characteristic
144	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₈	#123	1	1	yes	characteristic
148	C₂×C₈	#123	1	2	yes	—
149	C₂×C₈	#124	1	1	yes	characteristic
150	C₂×C₈	#124	1	1	yes	characteristic
151	C₂×C₈	#124	1	2	yes	—
152	C₂×C₈	#125	1	2	yes	—
153	C₂×C₈	#125	1	2	yes	—
154	C₂×C₈	#125	1	1	yes	characteristic
155	C₂×C₈	#126	1	2	yes	—
156	C₂×C₈	#126	1	2	yes	—
157	C₄⋊C₄	—	4	2	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₁₆	—	2	4	no	—
162	C₁₆	—	4	2	no	—
163	C₂³	—	4	2	no	—
164	C₂³	C₄×Q₁₆	1	1	yes	characteristic
165	C₂³	#462	1	2	yes	—
166	C₂³	#467	1	1	yes	characteristic
167	C₂³	#484	1	2	yes	—
168	C₂³	#500	1	1	yes	characteristic
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	#388	1	2	yes	—
172	C₂×C₄	#389	1	1	yes	characteristic
173	C₂×C₄	#394	1	1	yes	characteristic
174	C₂×C₄	#397	1	1	yes	characteristic
175	C₂×C₄	#398	1	2	yes	—
176	C₂×C₄	C₄×D₈	1	1	yes	characteristic
177	C₂×C₄	#440	1	2	yes	—
178	C₂×C₄	#459	1	2	yes	—
179	C₂×C₄	#463	1	2	yes	—
180	C₂×C₄	#464	1	2	yes	—
181	C₂×C₄	#466	1	1	yes	characteristic
182	C₂×C₄	#469	1	1	yes	characteristic
183	C₂×C₄	#470	1	1	yes	characteristic
184	C₂×C₄	#480	1	2	yes	—
185	C₂×C₄	#482	1	2	yes	—
186	C₂×C₄	#485	1	2	yes	—
187	C₂×C₄	#499	1	1	yes	characteristic
188	C₂×C₄	#501	1	2	yes	—
189	C₈	—	4	2	no	—
190	C₈	—	8	4	no	—
191	C₈	—	8	4	no	—
192	C₈	#382	1	1	yes	characteristic
193	C₈	#385	1	1	yes	characteristic
194	C₈	#387	1	2	yes	—
195	C₈	#393	1	1	yes	characteristic
196	C₈	#400	1	2	yes	—
197	C₈	#400	1	1	yes	characteristic
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	#1928	1	2	yes	—
204	C₂²	#1942	1	1	yes	characteristic
205	C₂²	#2056	1	1	yes	characteristic
206	C₂²	#2076	1	1	yes	characteristic
207	C₂²	#2083	1	2	yes	—
208	C₄	—	8	2	no	—
209	C₄	#1442	1	1	yes	characteristic
210	C₄	#1925	1	2	yes	—
211	C₄	#1944	1	1	yes	characteristic
212	C₄	#2054	1	1	yes	characteristic
213	C₄	#2075	1	1	yes	characteristic
214	C₄	#2085	1	2	yes	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	2	2	no	—
218	C₂	—	2	2	no	—
219	C₂	#11256	1	1	yes	characteristic
220	C₂	#13173	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1942
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #1381	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#368	2
C₂×#1388	2
C₂×#1381	2
#12545	2
#10873	2