Subgroups — #483403

Order 512.

520 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#17793	C₂	1	2	yes	maximal
3	#12024	C₂	1	4	yes	maximal, complemented
4	#12023	C₂	1	2	yes	maximal, complemented
5	#12022	C₂	1	2	yes	maximal, complemented
6	#12008	C₂	1	4	yes	maximal, complemented
7	C₂³×C₁₆	C₄	1	2	yes	—
8	#2920	—	2	2	no	—
9	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₄	1	2	yes	—
14	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	#1568	—	2	2	no	—
20	#1568	C₂²	1	4	yes	—
21	#1567	C₂²	1	4	yes	—
22	#1566	—	2	2	no	—
23	#1564	—	2	2	no	—
24	#1564	C₂²	1	4	yes	—
25	#1372	C₂²	1	8	yes	complemented
26	#1371	C₂²	1	8	yes	complemented
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	Q₈	1	1	yes	characteristic
29	C₂²×C₄²	D₄	1	1	yes	characteristic
30	C₂²×C₁₆	—	2	8	no	—
31	C₂²×C₁₆	C₂×C₄	1	2	yes	—
32	C₂²×C₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₁₆	C₂×C₄	1	2	yes	—
34	C₂²×C₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₁₆	C₂×C₄	1	2	yes	—
37	#432	—	2	4	no	—
38	#432	—	4	2	no	—
39	#431	—	2	4	no	—
40	#431	—	4	2	no	—
41	#423	—	2	4	no	—
42	#407	—	4	2	no	—
43	C₄×M₁₆	—	4	2	no	—
44	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×C₈	D₄	1	1	yes	characteristic
48	C₂×C₄×C₈	D₄	1	1	yes	characteristic
49	C₂×C₄×C₈	D₄	1	1	yes	characteristic
50	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
51	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
52	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
53	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
54	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
57	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
58	C₄×C₁₆	C₂×C₄	1	2	yes	—
59	C₄×C₁₆	C₂×C₄	1	2	yes	—
60	C₄×C₁₆	C₂×C₄	1	2	yes	—
61	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₆	C₂×C₄	1	2	yes	—
64	C₄×C₁₆	C₂³	1	2	yes	—
65	C₄×C₁₆	C₂³	1	2	yes	—
66	#335	—	2	4	no	—
67	#335	C₂³	1	8	yes	—
68	#334	—	2	4	no	—
69	#334	—	2	4	no	—
70	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
71	C₂³×C₄	D₈	1	1	yes	characteristic
72	C₂³×C₄	Q₁₆	1	1	yes	characteristic
73	C₂×M₁₆	—	4	4	no	—
74	C₂²×C₈	—	2	4	no	—
75	C₂²×C₈	—	2	4	no	—
76	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄²	D₈	1	1	yes	characteristic
92	C₂×C₄²	D₈	1	1	yes	characteristic
93	C₂×C₄²	D₈	1	1	yes	characteristic
94	C₂×C₄²	Q₁₆	1	1	yes	characteristic
95	C₂×C₄²	Q₁₆	1	1	yes	characteristic
96	C₂×C₄²	Q₁₆	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	C₂×C₁₆	—	2	4	no	—
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	—	2	4	no	—
105	C₂×C₁₆	M₁₆	1	8	yes	complemented
106	C₂×C₁₆	C₂²×C₄	1	2	yes	—
107	C₂×C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×C₁₆	C₂²×C₄	1	2	yes	—
109	C₂×C₁₆	C₂²×C₄	1	2	yes	—
110	C₄⋊C₈	—	2	8	no	—
111	C₄⋊C₈	—	4	4	no	—
112	C₂²⋊C₈	—	4	4	no	—
113	C₂²⋊C₈	—	4	4	no	—
114	#97	—	4	4	no	—
115	C₄×C₈	—	4	4	no	—
116	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
121	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
123	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
124	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₄×C₈	C₂×D₄	1	1	yes	characteristic
126	C₄×C₈	C₂×D₄	1	1	yes	characteristic
127	C₄×C₈	C₂×D₄	1	1	yes	characteristic
128	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
129	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
130	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
131	C₄×C₈	C₂⁴	1	1	yes	characteristic
132	C₂⁴	#107	1	1	yes	characteristic
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	#107	1	1	yes	characteristic
138	C₂²×C₄	#107	1	1	yes	characteristic
139	C₂²×C₄	#107	1	1	yes	characteristic
140	C₂²×C₄	#107	1	1	yes	characteristic
141	C₂²×C₄	#107	1	1	yes	characteristic
142	C₂²×C₄	#107	1	1	yes	characteristic
143	C₂²×C₄	#107	1	1	yes	characteristic
144	C₂²×C₄	#107	1	1	yes	characteristic
145	C₂²×C₄	#107	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
152	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
153	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
154	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
155	M₁₆	—	8	8	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	4	8	no	—
167	C₂×C₈	#108	1	4	yes	—
168	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₈	C₂×M₁₆	1	4	yes	—
175	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
177	C₄²	#107	1	1	yes	characteristic
178	C₄²	#107	1	1	yes	characteristic
179	C₄²	#107	1	1	yes	characteristic
180	C₄²	#107	1	1	yes	characteristic
181	C₄²	#107	1	1	yes	characteristic
182	C₄²	#107	1	1	yes	characteristic
183	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
185	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₄²	C₂×D₈	1	1	yes	characteristic
187	C₄²	C₂×D₈	1	1	yes	characteristic
188	C₄²	C₂×D₈	1	1	yes	characteristic
189	C₄²	C₂×Q₁₆	1	1	yes	characteristic
190	C₄²	C₂×Q₁₆	1	1	yes	characteristic
191	C₄²	C₂×Q₁₆	1	1	yes	characteristic
192	C₄²	C₂²×D₄	1	1	yes	characteristic
193	C₁₆	—	2	2	no	—
194	C₁₆	—	2	2	no	—
195	C₁₆	C₂×M₁₆	1	4	yes	—
196	C₁₆	C₂×M₁₆	1	4	yes	—
197	C₂³	—	2	2	no	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	C₂×#107	1	1	yes	characteristic
201	C₂³	C₂×#107	1	1	yes	characteristic
202	C₂³	C₂×#107	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	#368	1	4	yes	—
221	C₂×C₄	#423	1	2	yes	—
222	C₂×C₄	C₂×#107	1	1	yes	characteristic
223	C₂×C₄	C₂×#107	1	1	yes	characteristic
224	C₂×C₄	C₂×#107	1	1	yes	characteristic
225	C₂×C₄	C₂×#107	1	1	yes	characteristic
226	C₂×C₄	C₂×#107	1	1	yes	characteristic
227	C₂×C₄	C₂×#107	1	1	yes	characteristic
228	C₂×C₄	C₂×#107	1	1	yes	characteristic
229	C₂×C₄	C₂×#107	1	1	yes	characteristic
230	C₂×C₄	C₂×#107	1	1	yes	characteristic
231	C₂×C₄	C₂×#108	1	2	yes	—
232	C₂×C₄	#508	1	4	yes	—
233	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
235	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	8	8	no	—
241	C₈	#423	1	2	yes	—
242	C₈	C₂×#108	1	2	yes	—
243	C₈	C₂×#108	1	2	yes	—
244	C₈	C₂²×M₁₆	1	2	yes	—
245	C₂²	—	2	8	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	#1567	1	2	yes	—
253	C₂²	#2158	1	2	yes	—
254	C₂²	C₂×#368	1	2	yes	—
255	C₂²	C₂²×#107	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	#1567	1	2	yes	—
263	C₄	#2158	1	2	yes	—
264	C₄	C₂×#368	1	2	yes	—
265	C₄	C₂×#368	1	2	yes	—
266	C₄	C₂×#423	1	1	yes	characteristic
267	C₄	C₂²×#108	1	1	yes	characteristic
268	C₄	C₂×#508	1	2	yes	—
269	C₂	—	2	2	no	—
270	C₂	—	2	2	no	—
271	C₂	—	2	2	no	—
272	C₂	C₂×#1567	1	1	yes	characteristic
273	C₂	C₂×#2158	1	1	yes	characteristic
274	C₂	C₂²×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #12024	C₂	semidirect
4. #12023	C₂	semidirect
5. #12022	C₂	semidirect
6. #12008	C₂	semidirect
25. #1372	C₂²	semidirect
26. #1371	C₂²	semidirect
105. C₂×C₁₆	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₆	2
#17793	2
#12024	2
#12023	2
#12022	2
#12008	2