Subgroups — #134677

Order 512.

330 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#504	C₂	1	1	yes	maximal, characteristic
2	#12008	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic, complemented
4	#8605	C₂	1	2	yes	maximal
5	#7159	C₂	1	2	yes	maximal, complemented
6	C₂×#504	C₄	1	1	yes	characteristic
7	C₂×#504	C₄	1	1	yes	characteristic
8	C₄×#131	C₂²	1	1	yes	characteristic
9	#2108	—	2	1	no	—
10	#2108	—	2	1	no	—
11	#2108	C₄	1	1	yes	characteristic
12	#2108	C₄	1	1	yes	characteristic
13	C₄×M₃₂	—	2	1	no	—
14	C₄×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	—	2	1	no	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1564	C₂²	1	2	yes	—
18	C₂×#335	C₂²	1	2	yes	—
19	#1477	—	2	2	no	—
20	#1372	—	2	1	no	—
21	#1371	—	2	1	no	—
22	#1371	—	2	1	no	—
23	#1371	—	2	1	no	—
24	#1283	—	2	2	no	—
25	C₂×#131	C₂×C₄	1	1	yes	characteristic
26	C₂×#131	Q₈	1	1	yes	characteristic
27	C₄×#41	D₄	1	1	yes	characteristic
28	#504	—	2	2	no	—
29	#504	—	2	2	no	—
30	#504	—	2	2	no	—
31	#504	C₈	1	4	yes	complemented
32	C₂×M₃₂	—	2	1	no	—
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
35	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₁₆	—	2	1	no	—
37	C₂²×C₁₆	—	2	1	no	—
38	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
40	#431	—	2	2	no	—
41	C₂×C₄⋊C₈	—	2	2	no	—
42	#405	—	2	1	no	—
43	#405	—	2	1	no	—
44	#405	C₂×C₄	1	1	yes	characteristic
45	#405	Q₈	1	1	yes	characteristic
46	C₄×M₁₆	—	2	1	no	—
47	C₄×M₁₆	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	—	2	1	no	—
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	#346	—	2	1	no	—
51	#346	—	2	1	no	—
52	#346	—	2	1	no	—
53	C₄×C₁₆	—	2	1	no	—
54	C₄×C₁₆	—	2	1	no	—
55	C₄×C₁₆	—	2	1	no	—
56	C₄×C₁₆	D₄	1	1	yes	characteristic
57	C₄×C₁₆	D₄	1	1	yes	characteristic
58	C₄×C₁₆	D₄	1	1	yes	characteristic
59	C₄×C₁₆	D₄	1	1	yes	characteristic
60	#335	—	2	2	no	—
61	#335	—	2	2	no	—
62	#335	—	2	2	no	—
63	#334	—	2	2	no	—
64	Q₈⋊C₈	—	4	2	no	—
65	D₄⋊C₈	—	4	2	no	—
66	C₂×#41	C₄⋊C₄	1	1	yes	characteristic
67	#131	—	2	2	no	—
68	#131	—	2	2	no	—
69	#131	—	2	1	no	—
70	#131	—	2	1	no	—
71	#131	C₂×C₈	1	2	yes	—
72	#131	M₁₆	1	2	yes	—
73	C₂×M₁₆	—	2	1	no	—
74	C₂×M₁₆	—	2	1	no	—
75	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
76	C₂×M₁₆	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	—	2	1	no	—
78	C₂²×C₈	—	2	1	no	—
79	C₂²×C₈	—	4	2	no	—
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₄×Q₈	D₈	1	1	yes	characteristic
83	C₄×Q₈	Q₁₆	1	1	yes	characteristic
84	C₄×D₄	—	2	1	no	—
85	C₄×D₄	—	2	1	no	—
86	C₄×D₄	D₈	1	1	yes	characteristic
87	C₄×D₄	Q₁₆	1	1	yes	characteristic
88	#117	—	2	1	no	—
89	#117	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
92	M₃₂	—	2	4	no	—
93	M₃₂	—	2	1	no	—
94	M₃₂	—	2	1	no	—
95	M₃₂	—	2	4	no	—
96	M₃₂	C₂×C₈	1	2	yes	—
97	M₃₂	M₁₆	1	2	yes	—
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	—	2	1	no	—
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₁₆	C₂×C₈	1	2	yes	—
109	C₂×C₁₆	M₁₆	1	2	yes	—
110	C₄⋊C₈	—	4	2	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	4	2	no	—
113	C₂²⋊C₈	—	4	2	no	—
114	#97	—	2	1	no	—
115	#97	—	2	1	no	—
116	#97	—	2	1	no	—
117	C₄×C₈	—	2	1	no	—
118	C₄×C₈	—	2	1	no	—
119	C₄×C₈	—	2	1	no	—
120	C₄×C₈	—	4	2	no	—
121	C₄×C₈	C₂×D₄	1	1	yes	characteristic
122	C₄×C₈	C₂×D₄	1	1	yes	characteristic
123	C₄×C₈	#41	1	1	yes	characteristic
124	C₄×C₈	#41	1	1	yes	characteristic
125	#41	—	2	1	no	—
126	#41	—	2	1	no	—
127	#41	C₄⋊C₈	1	2	yes	—
128	#41	#108	1	2	yes	—
129	C₂×Q₈	#107	1	1	yes	characteristic
130	C₂×D₄	—	2	1	no	—
131	C₂×D₄	#107	1	1	yes	characteristic
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄	#135	1	1	yes	characteristic
137	C₂²×C₄	#135	1	1	yes	characteristic
138	M₁₆	—	2	4	no	—
139	M₁₆	—	2	1	no	—
140	M₁₆	—	2	1	no	—
141	M₁₆	—	2	2	no	—
142	M₁₆	—	2	2	no	—
143	M₁₆	C₄⋊C₈	1	2	yes	—
144	M₁₆	#108	1	2	yes	—
145	C₂×C₈	—	2	4	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	8	2	no	—
153	C₂×C₈	—	8	2	no	—
154	C₂×C₈	—	16	2	no	—
155	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₈	#117	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
161	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
162	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	#107	1	1	yes	characteristic
166	C₄⋊C₄	#107	1	1	yes	characteristic
167	C₂²⋊C₄	—	2	1	no	—
168	C₂²⋊C₄	—	2	1	no	—
169	C₂²⋊C₄	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	2	1	no	—
172	C₄²	—	2	1	no	—
173	C₄²	#121	1	1	yes	characteristic
174	C₄²	#121	1	1	yes	characteristic
175	C₄²	C₂×D₈	1	1	yes	characteristic
176	C₄²	C₂×Q₁₆	1	1	yes	characteristic
177	C₁₆	—	2	2	no	—
178	C₁₆	—	2	1	no	—
179	C₁₆	—	2	2	no	—
180	C₁₆	—	2	1	no	—
181	C₁₆	C₂²⋊C₈	1	2	yes	—
182	C₁₆	C₂²⋊C₈	1	2	yes	—
183	C₂³	—	2	1	no	—
184	C₂³	#427	1	1	yes	characteristic
185	Q₈	—	2	1	no	—
186	Q₈	#335	1	2	yes	—
187	D₄	—	2	2	no	—
188	D₄	—	2	2	no	—
189	D₄	—	2	1	no	—
190	D₄	#335	1	2	yes	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	#380	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
204	C₂×C₄	#423	1	1	yes	characteristic
205	C₂×C₄	C₂×#107	1	1	yes	characteristic
206	C₂×C₄	C₂×#108	1	1	yes	characteristic
207	C₂×C₄	C₂×#108	1	1	yes	characteristic
208	C₂×C₄	#466	1	1	yes	characteristic
209	C₂×C₄	#466	1	1	yes	characteristic
210	C₂×C₄	#467	1	1	yes	characteristic
211	C₂×C₄	#467	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	2	no	—
215	C₈	—	2	1	no	—
216	C₈	—	16	2	no	—
217	C₈	—	16	2	no	—
218	C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
219	C₈	#407	1	1	yes	characteristic
220	C₈	#431	1	1	yes	characteristic
221	C₈	#432	1	1	yes	characteristic
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	#1564	1	1	yes	characteristic
228	C₂²	#1568	1	1	yes	characteristic
229	C₂²	#1808	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	C₂×#335	1	1	yes	characteristic
237	C₄	#1567	1	1	yes	characteristic
238	C₄	#1797	1	1	yes	characteristic
239	C₄	#1810	1	1	yes	characteristic
240	C₄	#2221	1	1	yes	characteristic
241	C₄	#2222	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	#8596	1	1	yes	characteristic
246	C₂	#12841	1	1	yes	characteristic
247	C₂	#12842	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1808
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131 ◁ 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
2. #12008	C₂	semidirect
3. C₂×#1371	C₂	semidirect
5. #7159	C₂	semidirect
31. #504	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#504	2
#12008	2
C₂×#1371	2
#8605	2
#7159	2