Subgroups — #149378

Order 512.

434 conjugacy classes of subgroups.

The subgroup lattice has 223 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2161	C₂	1	1	yes	maximal, characteristic
2	C₂×#1385	C₂	1	1	yes	maximal, characteristic
3	C₂×#1381	C₂	1	2	yes	maximal
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	#2161	—	2	2	no	—
12	#2161	C₄	1	4	yes	complemented
13	C₂×#368	—	2	2	no	—
14	C₂×#344	—	2	2	no	—
15	#1738	—	2	2	no	—
16	#1385	—	2	2	no	—
17	#1381	—	2	4	no	—
18	#1374	—	2	2	no	—
19	C₂²×M₁₆	—	2	2	no	—
20	C₂²×M₁₆	—	2	2	no	—
21	C₂²×M₁₆	D₄	1	1	yes	characteristic
22	C₂³×C₈	C₂³	1	1	yes	characteristic
23	C₂×#117	D₄	1	1	yes	characteristic
24	C₂×M₃₂	—	2	2	no	—
25	C₂×M₃₂	—	2	2	no	—
26	C₂×M₃₂	C₂×C₄	1	2	yes	—
27	C₂×M₃₂	D₄	1	2	yes	—
28	C₂²×C₁₆	—	2	1	no	—
29	C₂²×C₁₆	—	2	1	no	—
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	—	2	2	no	—
32	C₂²×C₁₆	D₄	1	2	yes	—
33	C₂²×C₁₆	D₄	1	2	yes	—
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	—	2	2	no	—
44	C₂×#108	C₂×C₄	1	4	yes	—
45	C₂×#108	Q₈	1	4	yes	—
46	#405	—	2	2	no	—
47	#405	—	2	2	no	—
48	C₂×#97	—	2	1	no	—
49	C₂×C₄×C₈	—	2	1	no	—
50	#368	—	2	8	no	—
51	#344	—	4	4	no	—
52	#341	—	4	4	no	—
53	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	2	2	no	—
56	C₂×M₁₆	—	2	2	no	—
57	C₂×M₁₆	—	2	4	no	—
58	C₂×M₁₆	—	2	4	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	—	4	2	no	—
69	C₂×M₁₆	Q₁₆	1	2	yes	—
70	C₂²×C₈	—	2	1	no	—
71	C₂²×C₈	—	4	2	no	—
72	C₂²×C₈	—	4	2	no	—
73	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₈	C₂×Q₈	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	2	yes	—
82	C₂²×C₈	#41	1	1	yes	characteristic
83	#117	—	2	2	no	—
84	#117	—	2	2	no	—
85	C₂×C₄⋊C₄	—	2	1	no	—
86	C₂×C₂²⋊C₄	Q₁₆	1	2	yes	—
87	C₂×C₄²	—	2	1	no	—
88	M₃₂	—	2	4	no	—
89	M₃₂	—	2	4	no	—
90	C₂×C₁₆	—	2	2	no	—
91	C₂×C₁₆	—	2	2	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	2	4	no	—
94	C₂×C₁₆	—	4	1	no	—
95	C₂×C₁₆	—	4	1	no	—
96	C₂×C₁₆	—	4	2	no	—
97	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
98	#108	—	2	4	no	—
99	#108	—	2	4	no	—
100	#108	—	2	4	no	—
101	#108	—	2	4	no	—
102	#108	—	4	2	no	—
103	#108	—	4	2	no	—
104	#108	—	4	2	no	—
105	#108	—	4	2	no	—
106	#97	—	4	2	no	—
107	C₄×C₈	—	4	1	no	—
108	C₄×C₈	—	4	1	no	—
109	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
112	C₂²×C₄	#121	1	1	yes	characteristic
113	C₂²×C₄	C₂²⋊Q₈	1	2	yes	—
114	C₂²×C₄	#123	1	2	yes	—
115	C₂²×C₄	#127	1	1	yes	characteristic
116	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
117	C₂²×C₄	#135	1	1	yes	characteristic
118	C₂²×C₄	#135	1	1	yes	characteristic
119	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
120	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
121	M₁₆	—	2	4	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	4	4	no	—
124	M₁₆	—	4	4	no	—
125	M₁₆	—	4	4	no	—
126	M₁₆	—	4	4	no	—
127	M₁₆	—	8	2	no	—
128	M₁₆	—	8	2	no	—
129	M₁₆	—	8	2	no	—
130	M₁₆	—	8	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	4	1	no	—
133	C₂×C₈	—	4	1	no	—
134	C₂×C₈	—	4	4	no	—
135	C₂×C₈	—	4	4	no	—
136	C₂×C₈	—	8	2	no	—
137	C₂×C₈	—	8	2	no	—
138	C₂×C₈	—	8	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₈	#117	1	2	yes	—
142	C₂×C₈	C₄×D₄	1	1	yes	characteristic
143	C₂×C₈	C₄×Q₈	1	2	yes	—
144	C₂×C₈	#121	1	1	yes	characteristic
145	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₈	#123	1	1	yes	characteristic
148	C₂×C₈	#123	1	1	yes	characteristic
149	C₂×C₈	#123	1	2	yes	—
150	C₂×C₈	#123	1	2	yes	—
151	C₂×C₈	#124	1	1	yes	characteristic
152	C₂×C₈	#124	1	1	yes	characteristic
153	C₂×C₈	#125	1	2	yes	—
154	C₂×C₈	#125	1	2	yes	—
155	C₂×C₈	#126	1	2	yes	—
156	C₂×C₈	#126	1	2	yes	—
157	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
158	C₄⋊C₄	—	4	2	no	—
159	C₂²⋊C₄	—	2	4	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	1	no	—
162	C₁₆	—	2	4	no	—
163	C₁₆	—	4	2	no	—
164	C₂³	—	4	2	no	—
165	C₂³	C₄×Q₁₆	1	1	yes	characteristic
166	C₂³	#467	1	1	yes	characteristic
167	C₂³	#477	1	2	yes	—
168	C₂³	#484	1	2	yes	—
169	C₂³	#494	1	1	yes	characteristic
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	#387	1	2	yes	—
173	C₂×C₄	#390	1	1	yes	characteristic
174	C₂×C₄	#397	1	1	yes	characteristic
175	C₂×C₄	#398	1	1	yes	characteristic
176	C₂×C₄	#399	1	2	yes	—
177	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
178	C₂×C₄	#442	1	2	yes	—
179	C₂×C₄	#467	1	1	yes	characteristic
180	C₂×C₄	#469	1	1	yes	characteristic
181	C₂×C₄	#469	1	1	yes	characteristic
182	C₂×C₄	#477	1	2	yes	—
183	C₂×C₄	#478	1	2	yes	—
184	C₂×C₄	#478	1	2	yes	—
185	C₂×C₄	#482	1	2	yes	—
186	C₂×C₄	#482	1	2	yes	—
187	C₂×C₄	#484	1	2	yes	—
188	C₂×C₄	#495	1	1	yes	characteristic
189	C₂×C₄	#496	1	2	yes	—
190	C₈	—	4	2	no	—
191	C₈	—	8	4	no	—
192	C₈	—	8	4	no	—
193	C₈	#381	1	1	yes	characteristic
194	C₈	#387	1	2	yes	—
195	C₈	#391	1	1	yes	characteristic
196	C₈	#397	1	1	yes	characteristic
197	C₈	#398	1	1	yes	characteristic
198	C₈	#400	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	4	1	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	1	no	—
204	C₂²	#1921	1	2	yes	—
205	C₂²	#1969	1	1	yes	characteristic
206	C₂²	#2076	1	1	yes	characteristic
207	C₂²	#2087	1	1	yes	characteristic
208	C₂²	#2095	1	2	yes	—
209	C₄	—	8	2	no	—
210	C₄	#1445	1	1	yes	characteristic
211	C₄	#1921	1	2	yes	—
212	C₄	#1972	1	1	yes	characteristic
213	C₄	#2076	1	1	yes	characteristic
214	C₄	#2087	1	1	yes	characteristic
215	C₄	#2092	1	2	yes	—
216	C₂	—	2	1	no	—
217	C₂	—	2	2	no	—
218	C₂	—	2	2	no	—
219	C₂	—	2	1	no	—
220	C₂	#11377	1	1	yes	characteristic
221	C₂	#13520	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2087
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
12. #2161	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2161	2
C₂×#1385	2
C₂×#1381	2
#12545	2
#10873	2