Subgroups — #146994

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 217 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1848	C₂	1	2	yes	maximal
3	C₂×#1385	C₂	1	4	yes	maximal
4	C₂²×M₃₂	C₂²	1	2	yes	—
5	C₂²×#108	—	2	2	no	—
6	C₂²×#108	C₂²	1	4	yes	—
7	C₂×#423	—	2	2	no	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄×M₃₂	—	2	2	no	—
10	C₄×M₃₂	—	2	2	no	—
11	C₂×#346	C₄	1	2	yes	—
12	C₂×C₄×C₁₆	C₄	1	2	yes	—
13	#1848	—	2	2	no	—
14	#1848	—	2	2	no	—
15	#1848	C₄	1	4	yes	—
16	#1848	C₄	1	4	yes	—
17	#1385	—	2	8	no	—
18	C₂²×M₁₆	—	2	4	no	—
19	C₂³×C₈	D₄	1	1	yes	characteristic
20	C₂³×C₈	C₂³	1	1	yes	characteristic
21	C₂²×C₄²	D₄	1	1	yes	characteristic
22	C₂×M₃₂	—	2	4	no	—
23	C₂×M₃₂	—	2	4	no	—
24	C₂×M₃₂	D₄	1	4	yes	—
25	C₂×M₃₂	Q₈	1	4	yes	—
26	C₂²×C₁₆	C₂×C₄	1	4	yes	—
27	C₂×#108	—	2	8	no	—
28	C₂×#108	—	2	8	no	—
29	C₂×#108	—	2	4	no	—
30	C₂×#108	—	2	4	no	—
31	C₂×#108	—	2	4	no	—
32	C₂×#108	—	4	4	no	—
33	C₂×#108	—	4	4	no	—
34	C₂×#108	C₂×C₄	1	8	yes	—
35	#423	—	2	4	no	—
36	#423	—	2	4	no	—
37	#423	—	4	4	no	—
38	C₂×C₄⋊C₈	—	4	4	no	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₈	D₄	1	2	yes	—
44	C₂×C₄×C₈	D₄	1	2	yes	—
45	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
46	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
47	#346	—	2	4	no	—
48	C₄×C₁₆	—	2	2	no	—
49	C₄×C₁₆	—	2	2	no	—
50	C₂³×C₄	Q₁₆	1	2	yes	—
51	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
52	C₂×M₁₆	—	2	8	no	—
53	C₂×M₁₆	—	4	4	no	—
54	C₂×M₁₆	—	4	4	no	—
55	C₂×M₁₆	—	4	4	no	—
56	C₂×M₁₆	—	4	4	no	—
57	C₂×M₁₆	—	4	4	no	—
58	C₂²×C₈	—	4	4	no	—
59	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	D₈	1	2	yes	—
64	C₂²×C₈	SD₁₆	1	4	yes	—
65	C₂²×C₈	Q₁₆	1	2	yes	—
66	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₈	C₂²×C₄	1	2	yes	—
68	C₂²×C₈	C₂×D₄	1	2	yes	—
69	C₂²×C₈	C₂×D₄	1	2	yes	—
70	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₈	C₂×Q₈	1	2	yes	—
72	C₂²×C₈	#41	1	4	yes	—
73	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
74	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄²	D₈	1	2	yes	—
78	C₂×C₄²	SD₁₆	1	4	yes	—
79	M₃₂	—	2	8	no	—
80	M₃₂	—	2	8	no	—
81	C₂×C₁₆	—	2	4	no	—
82	C₂×C₁₆	—	2	4	no	—
83	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
84	#108	—	2	16	no	—
85	#108	—	4	8	no	—
86	#108	—	4	8	no	—
87	#108	—	4	8	no	—
88	C₄⋊C₈	—	4	8	no	—
89	C₄⋊C₈	—	4	8	no	—
90	C₄×C₈	C₄²	1	1	yes	characteristic
91	C₄×C₈	C₄²	1	1	yes	characteristic
92	C₄×C₈	C₄²	1	2	yes	—
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₈	C₂²⋊C₄	1	2	yes	—
96	C₄×C₈	C₄⋊C₄	1	2	yes	—
97	C₄×C₈	C₄⋊C₄	1	2	yes	—
98	C₄×C₈	C₄⋊C₄	1	2	yes	—
99	C₄×C₈	C₄⋊C₄	1	2	yes	—
100	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₄	—	4	4	no	—
102	C₂²×C₄	#103	1	4	yes	—
103	C₂²×C₄	#103	1	4	yes	—
104	C₂²×C₄	#107	1	2	yes	—
105	C₂²×C₄	#107	1	2	yes	—
106	C₂²×C₄	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	#124	1	2	yes	—
109	C₂²×C₄	#127	1	1	yes	characteristic
110	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
113	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
114	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
115	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
116	M₁₆	—	4	8	no	—
117	M₁₆	—	4	8	no	—
118	M₁₆	—	8	4	no	—
119	M₁₆	—	8	4	no	—
120	C₂×C₈	—	4	8	no	—
121	C₂×C₈	—	8	4	no	—
122	C₂×C₈	—	8	4	no	—
123	C₂×C₈	#95	1	1	yes	characteristic
124	C₂×C₈	#95	1	1	yes	characteristic
125	C₂×C₈	#95	1	2	yes	—
126	C₂×C₈	#102	1	4	yes	—
127	C₂×C₈	#102	1	4	yes	—
128	C₂×C₈	#103	1	4	yes	—
129	C₂×C₈	#103	1	4	yes	—
130	C₂×C₈	#106	1	2	yes	—
131	C₂×C₈	#106	1	2	yes	—
132	C₂×C₈	#107	1	2	yes	—
133	C₂×C₈	#107	1	2	yes	—
134	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₈	#117	1	2	yes	—
139	C₂×C₈	C₄×D₄	1	4	yes	—
140	C₂×C₈	C₄×Q₈	1	4	yes	—
141	C₂×C₈	#121	1	4	yes	—
142	C₂×C₈	C₂²⋊Q₈	1	4	yes	—
143	C₂×C₈	#125	1	2	yes	—
144	C₂×C₈	C₄⋊Q₈	1	2	yes	—
145	C₄²	#95	1	1	yes	characteristic
146	C₄²	#95	1	2	yes	—
147	C₄²	#95	1	1	yes	characteristic
148	C₄²	#102	1	4	yes	—
149	C₄²	#102	1	4	yes	—
150	C₄²	#106	1	2	yes	—
151	C₄²	#106	1	2	yes	—
152	C₁₆	—	2	8	no	—
153	C₂³	—	4	2	no	—
154	C₂³	C₂×#103	1	2	yes	—
155	C₂³	C₂×#107	1	1	yes	characteristic
156	C₂³	#487	1	2	yes	—
157	C₂³	#494	1	1	yes	characteristic
158	C₂³	#500	1	1	yes	characteristic
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	#340	1	4	yes	—
164	C₂×C₄	#340	1	2	yes	—
165	C₂×C₄	#340	1	2	yes	—
166	C₂×C₄	C₂×#95	1	1	yes	characteristic
167	C₂×C₄	#382	1	1	yes	characteristic
168	C₂×C₄	#384	1	1	yes	characteristic
169	C₂×C₄	#390	1	2	yes	—
170	C₂×C₄	#391	1	2	yes	—
171	C₂×C₄	C₂×#102	1	2	yes	—
172	C₂×C₄	C₂×#102	1	2	yes	—
173	C₂×C₄	C₂×#103	1	2	yes	—
174	C₂×C₄	C₂×#106	1	1	yes	characteristic
175	C₂×C₄	C₂×#106	1	1	yes	characteristic
176	C₂×C₄	C₂×#107	1	1	yes	characteristic
177	C₂×C₄	#486	1	2	yes	—
178	C₂×C₄	#486	1	2	yes	—
179	C₂×C₄	#487	1	2	yes	—
180	C₂×C₄	#492	1	2	yes	—
181	C₂×C₄	#493	1	1	yes	characteristic
182	C₂×C₄	#498	1	2	yes	—
183	C₂×C₄	#500	1	1	yes	characteristic
184	C₈	—	8	8	no	—
185	C₈	#340	1	2	yes	—
186	C₈	#340	1	2	yes	—
187	C₈	#340	1	4	yes	—
188	C₈	C₄×C₄⋊C₄	1	2	yes	—
189	C₈	#380	1	2	yes	—
190	C₈	#389	1	4	yes	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	1	no	—
195	C₂²	—	4	1	no	—
196	C₂²	C₂×#340	1	1	yes	characteristic
197	C₂²	#1834	1	1	yes	characteristic
198	C₂²	#1845	1	1	yes	characteristic
199	C₂²	#1956	1	2	yes	—
200	C₂²	#1990	1	2	yes	—
201	C₄	—	2	4	no	—
202	C₄	—	2	2	no	—
203	C₄	#1385	1	4	yes	complemented
204	C₄	C₄⋊#95	1	1	yes	characteristic
205	C₄	C₂×#340	1	1	yes	characteristic
206	C₄	#1834	1	1	yes	characteristic
207	C₄	#1844	1	1	yes	characteristic
208	C₄	#1957	1	2	yes	—
209	C₄	#1990	1	2	yes	—
210	C₂	—	2	1	no	—
211	C₂	—	2	1	no	—
212	C₂	—	2	2	no	—
213	C₂	—	2	2	no	—
214	C₂	#10841	1	1	yes	characteristic
215	C₂	C₂×#1385	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
203. C₄	#1385	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×#1848	2
C₂×#1385	2