Subgroups — #134667

Order 512.

470 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈²	C₂	1	1	yes	maximal, characteristic
2	#12783	C₂	1	2	yes	maximal
3	C₂×#1370	C₂	1	2	yes	maximal, complemented
4	#7151	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
9	C₂×C₂²⋊C₁₆	—	2	2	no	—
10	C₂×C₈²	—	2	4	no	—
11	C₂×C₈²	C₄	1	1	yes	characteristic
12	C₂×C₈²	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1374	—	2	2	no	—
16	#1370	—	2	4	no	—
17	C₂³×C₈	—	2	1	no	—
18	C₂³×C₈	C₈	1	1	yes	characteristic
19	C₂³×C₈	C₈	1	1	yes	characteristic
20	C₂³×C₈	D₄	1	1	yes	characteristic
21	C₂³×C₈	Q₈	1	1	yes	characteristic
22	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₆	—	2	4	no	—
24	C₂×C₄×C₈	—	2	1	no	—
25	C₂×C₄×C₈	—	2	2	no	—
26	C₂×C₄×C₈	—	2	2	no	—
27	C₂×C₄×C₈	—	2	2	no	—
28	C₂×C₄×C₈	—	2	2	no	—
29	C₂×C₄×C₈	—	2	2	no	—
30	C₂×C₄×C₈	—	2	2	no	—
31	C₂×C₄×C₈	C₈	1	1	yes	characteristic
32	C₂×C₄×C₈	C₈	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	4	no	—
42	C₄⋊C₁₆	—	2	4	no	—
43	C₂²⋊C₁₆	—	4	4	no	—
44	C₈²	—	2	2	no	—
45	C₈²	—	2	2	no	—
46	C₈²	—	2	2	no	—
47	C₈²	D₄	1	4	yes	—
48	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
50	C₂³×C₄	M₁₆	1	1	yes	characteristic
51	C₂²×C₈	—	2	4	no	—
52	C₂²×C₈	—	2	4	no	—
53	C₂²×C₈	—	2	4	no	—
54	C₂²×C₈	—	2	2	no	—
55	C₂²×C₈	—	2	2	no	—
56	C₂²×C₈	—	2	1	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	—	2	2	no	—
60	C₂²×C₈	—	2	2	no	—
61	C₂²×C₈	—	2	2	no	—
62	C₂²×C₈	—	2	2	no	—
63	C₂²×C₈	—	2	2	no	—
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	—	2	2	no	—
66	C₂²×C₈	C₁₆	1	4	yes	complemented
67	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₈	D₈	1	2	yes	—
74	C₂²×C₈	Q₁₆	1	2	yes	—
75	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₄²	—	2	1	no	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	—	2	1	no	—
80	C₂×C₄²	—	2	1	no	—
81	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄²	M₁₆	1	1	yes	characteristic
84	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₁₆	—	4	4	no	—
86	C₂×C₁₆	—	4	4	no	—
87	C₂×C₁₆	—	4	4	no	—
88	C₄×C₈	—	2	4	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	2	1	no	—
94	C₄×C₈	—	2	1	no	—
95	C₄×C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	C₂²⋊C₄	1	2	yes	—
100	C₄×C₈	C₂²⋊C₄	1	2	yes	—
101	C₄×C₈	M₁₆	1	2	yes	—
102	C₄×C₈	M₁₆	1	2	yes	—
103	C₄×C₈	C₂×D₄	1	2	yes	—
104	C₄×C₈	#41	1	2	yes	—
105	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	1	no	—
116	C₂²×C₄	—	2	1	no	—
117	C₂²×C₄	—	2	1	no	—
118	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
121	C₂²×C₄	#107	1	2	yes	—
122	C₂²×C₄	#108	1	2	yes	—
123	C₂²×C₄	C₂×C₁₆	1	2	yes	—
124	C₂²×C₄	M₃₂	1	2	yes	—
125	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
127	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
128	C₂×C₈	—	2	16	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	—	2	2	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	8	no	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	2	1	no	—
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	2	1	no	—
148	C₂×C₈	—	2	1	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	C₂²⋊C₈	1	2	yes	—
153	C₂×C₈	C₂²⋊C₈	1	2	yes	—
154	C₂×C₈	#107	1	2	yes	—
155	C₂×C₈	#108	1	2	yes	—
156	C₂×C₈	C₂×C₁₆	1	2	yes	—
157	C₂×C₈	C₂×C₁₆	1	2	yes	—
158	C₂×C₈	M₃₂	1	2	yes	—
159	C₂×C₈	M₃₂	1	2	yes	—
160	C₂×C₈	#121	1	2	yes	—
161	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
162	C₂×C₈	C₂×D₈	1	1	yes	characteristic
163	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
164	C₂×C₈	#135	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₄²	#117	1	1	yes	characteristic
173	C₄²	C₂×M₁₆	1	1	yes	characteristic
174	C₄²	C₂×M₁₆	1	1	yes	characteristic
175	C₁₆	—	8	4	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	#335	1	2	yes	—
181	C₂³	C₄⋊C₁₆	1	2	yes	—
182	C₂³	#364	1	2	yes	—
183	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	#335	1	2	yes	—
203	C₂×C₄	C₄⋊C₁₆	1	2	yes	—
204	C₂×C₄	#364	1	2	yes	—
205	C₂×C₄	#380	1	1	yes	characteristic
206	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
207	C₂×C₄	#407	1	1	yes	characteristic
208	C₂×C₄	#423	1	1	yes	characteristic
209	C₂×C₄	C₂×#107	1	1	yes	characteristic
210	C₂×C₄	#427	1	1	yes	characteristic
211	C₂×C₄	C₂×#108	1	1	yes	characteristic
212	C₂×C₄	C₂×#108	1	1	yes	characteristic
213	C₂×C₄	#431	1	1	yes	characteristic
214	C₂×C₄	#432	1	1	yes	characteristic
215	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
216	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
217	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
218	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
219	C₈	—	2	4	no	—
220	C₈	—	2	4	no	—
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	C₂²⋊C₁₆	1	4	yes	complemented
228	C₈	#466	1	2	yes	—
229	C₈	#467	1	2	yes	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	#1370	1	4	yes	complemented
237	C₂²	C₂×#335	1	1	yes	characteristic
238	C₂²	#1564	1	1	yes	characteristic
239	C₂²	#1797	1	1	yes	characteristic
240	C₂²	C₂×C₄⋊C₁₆	1	1	yes	characteristic
241	C₂²	#2150	1	1	yes	characteristic
242	C₂²	C₂×#364	1	1	yes	characteristic
243	C₂²	C₂×#364	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	#1567	1	1	yes	characteristic
250	C₄	#1568	1	1	yes	characteristic
251	C₄	#1808	1	1	yes	characteristic
252	C₄	#1810	1	1	yes	characteristic
253	C₄	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
254	C₄	#2112	1	1	yes	characteristic
255	C₄	#2162	1	1	yes	characteristic
256	C₄	#2163	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	#8596	1	1	yes	characteristic
259	C₂	C₂×#1370	1	1	yes	characteristic
260	C₂	#12004	1	1	yes	characteristic
261	C₂	#12012	1	1	yes	characteristic
262	C₂	#12013	1	1	yes	characteristic
263	C₂	#12783	1	1	yes	characteristic
264	C₂	#12788	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
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
3. C₂×#1370	C₂	semidirect
4. #7151	C₂	semidirect
66. C₂²×C₈	C₁₆	semidirect
227. C₈	C₂²⋊C₁₆	semidirect
236. C₂²	#1370	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈²	2
#12783	2
C₂×#1370	2
#7151	2