Subgroups — #483752

Order 512.

418 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33508	C₂	1	2	yes	maximal, complemented
2	#17959	C₂	1	1	yes	maximal, characteristic
3	#12057	C₂	1	2	yes	maximal, complemented
4	#12056	C₂	1	2	yes	maximal, complemented
5	#12054	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#12039	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#1370	C₂	1	1	yes	maximal, characteristic, complemented
10	#11622	C₂	1	2	yes	maximal, complemented
11	#11410	C₂	1	1	yes	maximal, characteristic, complemented
12	#3168	C₄	1	1	yes	characteristic
13	#2966	C₂²	1	2	yes	—
14	#2183	C₂²	1	2	yes	—
15	Q₈×C₁₆	C₂²	1	2	yes	—
16	#2169	C₂²	1	2	yes	—
17	#2168	—	2	2	no	—
18	D₄×C₁₆	—	2	2	no	—
19	D₄×C₁₆	C₂²	1	2	yes	—
20	#2164	—	2	2	no	—
21	#2162	C₂²	1	1	yes	characteristic
22	#2162	C₂²	1	1	yes	characteristic
23	#2151	—	2	2	no	—
24	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
25	#1591	C₄	1	1	yes	characteristic
26	#1590	C₄	1	2	yes	—
27	#1589	C₄	1	1	yes	characteristic
28	C₈×Q₁₆	C₂²	1	1	yes	characteristic
29	C₈×SD₁₆	C₂²	1	2	yes	—
30	C₈×D₈	C₂²	1	1	yes	characteristic
31	#1568	C₄	1	1	yes	characteristic
32	#1494	C₄	1	2	yes	—
33	C₂×C₈²	C₂²	1	1	yes	characteristic
34	#1370	C₂²	1	1	yes	characteristic
35	#1370	C₂²	1	1	yes	characteristic
36	#1370	C₂²	1	2	yes	—
37	Q₈⋊C₁₆	C₂²	1	2	yes	—
38	Q₈⋊C₁₆	C₂²	1	2	yes	—
39	D₄⋊C₁₆	C₂²	1	2	yes	—
40	D₄⋊C₁₆	C₂²	1	2	yes	—
41	#1312	C₂²	1	1	yes	characteristic
42	#1312	C₂²	1	1	yes	characteristic
43	C₈×C₁₆	C₂²	1	1	yes	characteristic
44	C₈×C₁₆	C₂²	1	1	yes	characteristic
45	#529	C₂×C₄	1	2	yes	—
46	C₂×M₃₂	—	2	2	no	—
47	C₂²×C₁₆	—	2	2	no	—
48	C₂²×C₁₆	—	2	2	no	—
49	#499	C₈	1	1	yes	characteristic
50	#495	C₈	1	1	yes	characteristic
51	#488	C₈	1	2	yes	—
52	#467	C₈	1	1	yes	characteristic
53	#466	C₈	1	1	yes	characteristic
54	#465	C₈	1	2	yes	—
55	#446	C₂×C₄	1	2	yes	—
56	C₈×Q₈	C₂³	1	2	yes	—
57	C₄×Q₁₆	C₂×C₄	1	1	yes	characteristic
58	C₄×SD₁₆	C₂×C₄	1	2	yes	—
59	C₄×D₈	C₂×C₄	1	1	yes	characteristic
60	#436	C₂×C₄	1	2	yes	—
61	#435	—	2	2	no	—
62	C₈×D₄	—	2	2	no	—
63	C₈×D₄	C₂³	1	2	yes	—
64	#433	—	2	2	no	—
65	#432	C₂×C₄	1	2	yes	—
66	#405	—	2	2	no	—
67	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
70	C₄⋊C₁₆	—	2	2	no	—
71	C₄⋊C₁₆	—	2	2	no	—
72	C₄⋊C₁₆	—	2	2	no	—
73	C₄⋊C₁₆	—	2	1	no	—
74	C₄⋊C₁₆	D₄	1	1	yes	characteristic
75	C₄⋊C₁₆	D₄	1	1	yes	characteristic
76	C₄⋊C₁₆	C₂³	1	2	yes	—
77	C₄⋊C₁₆	C₂³	1	2	yes	—
78	C₂²⋊C₁₆	—	2	1	no	—
79	C₂²⋊C₁₆	—	2	2	no	—
80	C₂²⋊C₁₆	—	2	2	no	—
81	C₂²⋊C₁₆	D₄	1	1	yes	characteristic
82	C₂²⋊C₁₆	D₄	1	1	yes	characteristic
83	#346	—	2	2	no	—
84	C₄×C₁₆	—	2	2	no	—
85	C₄×C₁₆	C₂³	1	1	yes	characteristic
86	C₄×C₁₆	C₂³	1	1	yes	characteristic
87	C₄×C₁₆	C₂³	1	1	yes	characteristic
88	C₄×C₁₆	C₂³	1	1	yes	characteristic
89	#335	C₂×C₄	1	1	yes	characteristic
90	#335	C₂×C₄	1	1	yes	characteristic
91	#334	C₂×C₄	1	2	yes	—
92	Q₈⋊C₈	C₂×C₄	1	2	yes	—
93	Q₈⋊C₈	C₂×C₄	1	2	yes	—
94	D₄⋊C₈	C₂×C₄	1	2	yes	—
95	D₄⋊C₈	C₂×C₄	1	2	yes	—
96	C₈²	C₂×C₄	1	1	yes	characteristic
97	C₈²	C₂×C₄	1	1	yes	characteristic
98	C₈²	C₂³	1	1	yes	characteristic
99	C₈²	C₂³	1	1	yes	characteristic
100	C₂×Q₁₆	C₂×C₈	1	1	yes	characteristic
101	C₂×SD₁₆	C₂×C₈	1	2	yes	—
102	C₂×D₈	C₂×C₈	1	1	yes	characteristic
103	C₂×M₁₆	—	2	2	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	2	no	—
106	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
107	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	#41	1	1	yes	characteristic
110	#126	—	2	2	no	—
111	#125	C₂×C₈	1	2	yes	—
112	#124	C₂×C₈	1	2	yes	—
113	#123	—	2	2	no	—
114	C₂²⋊Q₈	C₂×C₈	1	2	yes	—
115	#121	C₂×C₈	1	2	yes	—
116	C₄×Q₈	C₂²×C₄	1	2	yes	—
117	C₄×D₄	—	2	2	no	—
118	C₄×D₄	C₂²×C₄	1	2	yes	—
119	#117	—	2	2	no	—
120	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
121	M₃₂	—	8	2	no	—
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	1	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	8	2	no	—
127	C₂×C₁₆	—	8	2	no	—
128	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
129	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
130	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
131	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
132	#107	C₂×C₈	1	1	yes	characteristic
133	#107	C₂×C₈	1	1	yes	characteristic
134	#106	C₂×C₈	1	2	yes	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	C₂²×C₄	1	2	yes	—
138	C₄⋊C₈	C₂²×C₄	1	2	yes	—
139	#103	C₂×C₈	1	2	yes	—
140	#103	C₂×C₈	1	2	yes	—
141	#102	C₂×C₈	1	2	yes	—
142	#102	C₂×C₈	1	2	yes	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	—	2	2	no	—
145	C₂²⋊C₈	—	2	2	no	—
146	#97	—	2	2	no	—
147	#97	—	2	2	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	2	1	no	—
150	C₄×C₈	C₂×C₈	1	1	yes	characteristic
151	C₄×C₈	C₂×C₈	1	1	yes	characteristic
152	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
153	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
154	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
155	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
157	C₄×C₈	C₂×D₄	1	1	yes	characteristic
158	C₄×C₈	#41	1	1	yes	characteristic
159	C₄×C₈	C₂⁴	1	1	yes	characteristic
160	C₂×Q₈	C₂²×C₈	1	2	yes	—
161	C₂×D₄	—	2	2	no	—
162	C₂×D₄	C₂²×C₈	1	2	yes	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
166	C₂²×C₄	#131	1	1	yes	characteristic
167	Q₁₆	—	2	2	no	—
168	SD₁₆	—	4	2	no	—
169	D₈	—	2	2	no	—
170	M₁₆	—	8	2	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	C₄×D₄	1	1	yes	characteristic
184	C₂×C₈	C₄×D₄	1	1	yes	characteristic
185	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
186	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
187	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
188	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
189	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₈	C₂×#41	1	1	yes	characteristic
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₂²×C₈	1	2	yes	—
196	C₄⋊C₄	C₂²×C₈	1	2	yes	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	C₄×D₄	1	1	yes	characteristic
204	C₄²	C₂²×C₈	1	1	yes	characteristic
205	C₄²	#131	1	1	yes	characteristic
206	C₄²	C₂³×C₄	1	1	yes	characteristic
207	C₁₆	—	2	1	no	—
208	C₁₆	—	2	1	no	—
209	C₁₆	—	2	1	no	—
210	C₁₆	—	2	1	no	—
211	C₁₆	—	4	1	no	—
212	C₁₆	—	4	1	no	—
213	C₁₆	—	4	2	no	—
214	C₁₆	—	4	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	C₈×D₄	1	1	yes	characteristic
217	Q₈	—	2	2	no	—
218	Q₈	—	2	2	no	—
219	D₄	—	2	2	no	—
220	D₄	—	2	2	no	—
221	D₄	—	4	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₈×D₄	1	1	yes	characteristic
233	C₂×C₄	C₈×D₄	1	1	yes	characteristic
234	C₂×C₄	C₈×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
237	C₂×C₄	C₂×#131	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	2	2	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	4	2	no	—
248	C₈	#504	1	2	yes	—
249	C₈	#576	1	1	yes	characteristic
250	C₈	#578	1	1	yes	characteristic
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	#2170	1	2	yes	—
257	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	#2951	1	1	yes	characteristic
266	C₄	#2957	1	1	yes	characteristic
267	C₄	C₂×#504	1	1	yes	characteristic
268	C₄	#3407	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	8	2	no	—
271	C₂	#17857	1	1	yes	characteristic
272	C₂	#33438	1	1	yes	characteristic
273	C₂	C₂×#2170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#466	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ 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
1. #33508	C₂	semidirect
3. #12057	C₂	semidirect
4. #12056	C₂	semidirect
5. #12054	C₂	semidirect
6. C₁₆×Q₁₆	C₂	semidirect
7. #12039	C₂	semidirect
8. C₁₆×D₈	C₂	semidirect
9. C₂×#1370	C₂	semidirect
10. #11622	C₂	semidirect
11. #11410	C₂	semidirect

Maximal subgroups

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

subgroup	index
#33508	2
#17959	2
#12057	2
#12056	2
#12054	2
C₁₆×Q₁₆	2
#12039	2
C₁₆×D₈	2
C₂×#1370	2
#11622	2
#11410	2