Subgroups — #126644

Order 512.

504 conjugacy classes of subgroups.

The subgroup lattice has 241 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1450	C₂	1	1	yes	maximal, characteristic
4	#7151	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂	—	2	1	no	—
6	C₂³×C₁₆	—	2	1	no	—
7	C₂×#431	C₄	1	1	yes	characteristic
8	C₂×#431	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	#2150	—	2	2	no	—
11	#2150	C₄	1	4	yes	—
12	C₂×C₄⋊C₁₆	—	2	1	no	—
13	C₂×C₄⋊C₁₆	—	2	2	no	—
14	C₂×C₄⋊C₁₆	—	2	1	no	—
15	C₂×C₄⋊C₁₆	C₄	1	4	yes	—
16	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
18	#1450	—	2	2	no	—
19	#1450	—	2	2	no	—
20	C₂×#322	C₄	1	2	yes	—
21	C₂×C₈²	C₂²	1	2	yes	—
22	C₂³×C₈	D₄	1	1	yes	characteristic
23	C₂³×C₈	Q₈	1	1	yes	characteristic
24	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×M₃₂	—	2	4	no	—
26	C₂×M₃₂	—	4	2	no	—
27	C₂×M₃₂	—	4	2	no	—
28	C₂²×C₁₆	—	2	4	no	—
29	C₂²×C₁₆	—	2	2	no	—
30	C₂²×C₁₆	—	2	2	no	—
31	C₂²×C₁₆	—	4	2	no	—
32	C₂²×C₁₆	—	4	2	no	—
33	#431	—	2	2	no	—
34	#431	—	2	2	no	—
35	#431	—	2	2	no	—
36	#431	C₈	1	4	yes	—
37	C₂×C₄⋊C₈	—	2	1	no	—
38	C₂×C₄⋊C₈	C₈	1	2	yes	—
39	C₂×C₂²⋊C₈	—	2	1	no	—
40	C₂×C₂²⋊C₈	C₈	1	2	yes	—
41	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
46	C₂×C₄×C₈	D₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	2	yes	—
48	C₂×C₄×C₈	D₄	1	2	yes	—
49	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	4	no	—
52	C₄⋊C₁₆	—	2	4	no	—
53	C₄⋊C₁₆	—	2	2	no	—
54	C₄⋊C₁₆	—	2	2	no	—
55	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
56	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
57	#322	—	2	4	no	—
58	C₈²	—	2	2	no	—
59	C₈²	—	2	2	no	—
60	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
62	C₂³×C₄	M₁₆	1	1	yes	characteristic
63	C₂²×C₈	—	2	2	no	—
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	—	2	2	no	—
66	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₄⋊C₄	1	2	yes	—
70	C₂²×C₈	C₄⋊C₄	1	2	yes	—
71	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×C₈	1	4	yes	—
73	C₂²×C₈	D₈	1	2	yes	—
74	C₂²×C₈	SD₁₆	1	2	yes	—
75	C₂²×C₈	SD₁₆	1	2	yes	—
76	C₂²×C₈	Q₁₆	1	2	yes	—
77	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₄²	C₄²	1	2	yes	—
80	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
81	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₈	1	2	yes	—
84	C₂×C₄²	M₁₆	1	1	yes	characteristic
85	C₂×C₄²	M₁₆	1	2	yes	—
86	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
87	M₃₂	—	4	8	no	—
88	C₂×C₁₆	—	2	4	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	—	4	8	no	—
91	C₂×C₁₆	—	4	2	no	—
92	C₂×C₁₆	—	4	2	no	—
93	C₂×C₁₆	—	4	2	no	—
94	C₂×C₁₆	—	4	2	no	—
95	C₄⋊C₈	—	2	4	no	—
96	C₄⋊C₈	—	2	4	no	—
97	C₂²⋊C₈	—	2	4	no	—
98	C₂²⋊C₈	—	2	4	no	—
99	C₄×C₈	—	2	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	2	2	no	—
102	C₄×C₈	C₄²	1	2	yes	—
103	C₄×C₈	C₂²⋊C₄	1	2	yes	—
104	C₄×C₈	C₂²⋊C₄	1	2	yes	—
105	C₄×C₈	C₂²⋊C₄	1	2	yes	—
106	C₄×C₈	C₄⋊C₄	1	2	yes	—
107	C₄×C₈	C₄⋊C₄	1	2	yes	—
108	C₄×C₈	C₂×C₈	1	4	yes	—
109	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂×D₄	1	1	yes	characteristic
112	C₄×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
114	C₂²×C₄	—	2	4	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	#95	1	2	yes	—
117	C₂²×C₄	C₄×C₈	1	2	yes	—
118	C₂²×C₄	#97	1	2	yes	—
119	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
120	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
121	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
122	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
123	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊C₈	1	2	yes	—
125	C₂²×C₄	#106	1	2	yes	—
126	C₂²×C₄	#107	1	2	yes	—
127	C₂²×C₄	#108	1	2	yes	—
128	C₂²×C₄	#108	1	2	yes	—
129	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
131	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	8	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	#95	1	2	yes	—
143	C₂×C₈	#95	1	2	yes	—
144	C₂×C₈	#102	1	4	yes	—
145	C₂×C₈	#102	1	4	yes	—
146	C₂×C₈	#103	1	4	yes	—
147	C₂×C₈	#103	1	4	yes	—
148	C₂×C₈	#106	1	2	yes	—
149	C₂×C₈	#107	1	2	yes	—
150	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₈	C₂×D₈	1	1	yes	characteristic
155	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
158	C₄²	#95	1	2	yes	—
159	C₄²	C₄×C₈	1	2	yes	—
160	C₄²	#97	1	2	yes	—
161	C₄²	C₂²⋊C₈	1	2	yes	—
162	C₄²	C₂²⋊C₈	1	2	yes	—
163	C₄²	C₄⋊C₈	1	2	yes	—
164	C₄²	C₂×C₄²	1	1	yes	characteristic
165	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₄²	C₂²×C₈	1	1	yes	characteristic
167	C₄²	C₂×M₁₆	1	1	yes	characteristic
168	C₁₆	—	4	4	no	—
169	C₁₆	—	4	4	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	#334	1	2	yes	—
173	C₂³	#335	1	2	yes	—
174	C₂³	#336	1	2	yes	—
175	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	#334	1	4	yes	—
184	C₂×C₄	#334	1	2	yes	—
185	C₂×C₄	#335	1	4	yes	—
186	C₂×C₄	#335	1	2	yes	—
187	C₂×C₄	#336	1	2	yes	—
188	C₂×C₄	#336	1	2	yes	—
189	C₂×C₄	#336	1	2	yes	—
190	C₂×C₄	#340	1	4	yes	—
191	C₂×C₄	#341	1	4	yes	—
192	C₂×C₄	C₂×#95	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
194	C₂×C₄	C₂×#97	1	1	yes	characteristic
195	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
196	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
197	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
198	C₂×C₄	C₂×#106	1	1	yes	characteristic
199	C₂×C₄	C₂×#107	1	1	yes	characteristic
200	C₂×C₄	C₂×#108	1	1	yes	characteristic
201	C₂×C₄	C₂×#108	1	1	yes	characteristic
202	C₂×C₄	#504	1	4	yes	—
203	C₈	—	2	2	no	—
204	C₈	—	2	1	no	—
205	C₈	—	2	1	no	—
206	C₈	—	4	4	no	—
207	C₈	—	4	4	no	—
208	C₈	#340	1	4	yes	—
209	C₈	C₂×#102	1	1	yes	characteristic
210	C₈	C₂×#102	1	1	yes	characteristic
211	C₈	C₂×#103	1	1	yes	characteristic
212	C₈	C₂×#103	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	#1277	1	4	yes	—
218	C₂²	C₂×#334	1	1	yes	characteristic
219	C₂²	C₂×#335	1	1	yes	characteristic
220	C₂²	C₂×#336	1	1	yes	characteristic
221	C₂²	#2151	1	2	yes	—
222	C₂²	#2153	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	#1277	1	4	yes	—
228	C₄	C₂×#334	1	1	yes	characteristic
229	C₄	C₂×#335	1	1	yes	characteristic
230	C₄	C₂×#340	1	1	yes	characteristic
231	C₄	C₂×#341	1	1	yes	characteristic
232	C₄	#2108	1	2	yes	—
233	C₄	#2113	1	2	yes	—
234	C₂	—	2	4	no	—
235	C₂	C₂×#1277	1	1	yes	characteristic
236	C₂	#11997	1	2	yes	—
237	C₂	#12005	1	2	yes	—
238	C₂	#12545	1	1	yes	characteristic
239	C₂	#12549	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×C₄⋊C₈
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×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
4. #7151	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2150	2
C₂²×C₄⋊C₁₆	2
C₂×#1450	2
#7151	2