Subgroups — #483463

Order 512.

426 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33484	C₂	1	1	yes	maximal, characteristic
2	#17834	C₂	1	1	yes	maximal, characteristic
3	#17826	C₂	1	1	yes	maximal, characteristic
4	#12030	C₂	1	1	yes	maximal, characteristic, complemented
5	#12030	C₂	1	2	yes	maximal, complemented
6	#12029	C₂	1	1	yes	maximal, characteristic, complemented
7	#12028	C₂	1	2	yes	maximal, complemented
8	#12020	C₂	1	1	yes	maximal, characteristic, complemented
9	#12019	C₂	1	2	yes	maximal, complemented
10	#12000	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#1368	C₂	1	2	yes	maximal, complemented
12	#2990	—	2	1	no	—
13	C₈×#41	C₂²	1	1	yes	characteristic
14	#2906	—	2	1	no	—
15	#2183	C₄	1	1	yes	characteristic
16	#2183	C₄	1	1	yes	characteristic
17	#2168	C₄	1	2	yes	—
18	#2168	C₄	1	2	yes	—
19	D₄×C₁₆	C₄	1	1	yes	characteristic
20	D₄×C₁₆	C₄	1	1	yes	characteristic
21	#2163	C₂²	1	2	yes	—
22	#2163	C₂²	1	1	yes	characteristic
23	C₄×M₃₂	C₂²	1	1	yes	characteristic
24	C₂×#346	C₂²	1	2	yes	—
25	#1574	—	2	1	no	—
26	#1573	—	2	2	no	—
27	#1570	—	2	2	no	—
28	#1570	—	2	1	no	—
29	#1568	C₂²	1	2	yes	—
30	#1566	C₂²	1	2	yes	—
31	#1566	C₂²	1	1	yes	characteristic
32	#1566	C₂²	1	1	yes	characteristic
33	#1564	C₂²	1	1	yes	characteristic
34	C₂×#334	C₂²	1	2	yes	—
35	C₂×#334	C₂²	1	1	yes	characteristic
36	C₂×#334	C₂²	1	2	yes	—
37	#1368	C₂²	1	1	yes	characteristic
38	#1368	C₂²	1	2	yes	—
39	#1368	C₂²	1	2	yes	—
40	#1368	C₂²	1	2	yes	—
41	#1368	C₂²	1	1	yes	characteristic
42	#1368	C₂²	1	1	yes	characteristic
43	#1368	C₂²	1	1	yes	characteristic
44	#1368	C₂²	1	2	yes	—
45	#1368	C₂²	1	2	yes	—
46	#1368	C₂²	1	2	yes	—
47	C₂×#131	—	4	1	no	—
48	C₄×#41	D₄	1	1	yes	characteristic
49	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
50	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₁₆	C₂×C₄	1	2	yes	—
52	C₂²×C₁₆	C₂×C₄	1	2	yes	—
53	#446	—	4	1	no	—
54	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
55	C₈×Q₈	Q₈	1	1	yes	characteristic
56	#435	—	4	2	no	—
57	C₈×D₄	—	4	1	no	—
58	C₈×D₄	C₂×C₄	1	1	yes	characteristic
59	C₈×D₄	C₂×C₄	1	2	yes	—
60	C₈×D₄	Q₈	1	2	yes	—
61	C₈×D₄	Q₈	1	1	yes	characteristic
62	#432	—	2	2	no	—
63	#431	—	2	1	no	—
64	#431	D₄	1	2	yes	—
65	#431	D₄	1	1	yes	characteristic
66	#424	—	4	1	no	—
67	#424	—	4	2	no	—
68	#423	—	2	2	no	—
69	#423	—	2	1	no	—
70	#423	—	2	1	no	—
71	C₂×C₄⋊C₈	—	2	2	no	—
72	C₂×C₄⋊C₈	—	2	1	no	—
73	C₂×C₄⋊C₈	—	2	2	no	—
74	C₂×C₄×C₈	C₂³	1	2	yes	—
75	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
76	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
77	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
78	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
79	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
80	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
81	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
82	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
83	#346	C₂×C₄	1	2	yes	—
84	#346	C₂×C₄	1	2	yes	—
85	#346	C₂³	1	1	yes	characteristic
86	#346	C₂³	1	2	yes	—
87	#346	C₂³	1	1	yes	characteristic
88	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
89	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
90	#335	—	2	1	no	—
91	#335	—	2	2	no	—
92	#334	—	2	2	no	—
93	#334	—	2	1	no	—
94	#334	—	2	2	no	—
95	#334	—	2	2	no	—
96	#334	—	2	1	no	—
97	#334	—	2	1	no	—
98	#334	C₂³	1	1	yes	characteristic
99	#334	C₂³	1	1	yes	characteristic
100	#334	C₂³	1	2	yes	—
101	#334	C₂³	1	1	yes	characteristic
102	#334	C₂³	1	2	yes	—
103	#334	C₂³	1	1	yes	characteristic
104	C₂×#41	SD₁₆	1	1	yes	characteristic
105	#131	—	8	4	no	—
106	C₂×M₁₆	—	4	2	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂²×C₈	—	4	2	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	—	4	2	no	—
112	C₂²×C₈	C₂²×C₄	1	2	yes	—
113	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₈	C₂×Q₈	1	2	yes	—
115	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
116	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
117	C₄×Q₈	SD₁₆	1	1	yes	characteristic
118	C₄×D₄	C₄⋊C₄	1	2	yes	—
119	C₄×D₄	C₄⋊C₄	1	1	yes	characteristic
120	C₄×D₄	SD₁₆	1	1	yes	characteristic
121	C₄×D₄	SD₁₆	1	2	yes	—
122	#117	SD₁₆	1	1	yes	characteristic
123	#117	SD₁₆	1	2	yes	—
124	C₂×C₄²	C₂×D₄	1	2	yes	—
125	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
126	M₃₂	—	2	2	no	—
127	M₃₂	—	4	1	no	—
128	C₂×C₁₆	—	2	4	no	—
129	C₂×C₁₆	—	4	2	no	—
130	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₁₆	C₂²×C₄	1	2	yes	—
134	C₂×C₁₆	C₂²×C₄	1	2	yes	—
135	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	2	1	no	—
139	C₄⋊C₈	—	2	1	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	1	no	—
142	C₄⋊C₈	—	4	2	no	—
143	C₄⋊C₈	—	4	1	no	—
144	C₄⋊C₈	—	4	2	no	—
145	C₄⋊C₈	—	4	2	no	—
146	C₄⋊C₈	—	4	1	no	—
147	C₄⋊C₈	—	4	1	no	—
148	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
149	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
150	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
151	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
152	C₂²⋊C₈	—	4	1	no	—
153	C₂²⋊C₈	—	4	2	no	—
154	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
155	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
156	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
157	#97	—	4	2	no	—
158	C₄×C₈	—	4	1	no	—
159	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₄	1	2	yes	—
161	C₄×C₈	C₂×D₄	1	2	yes	—
162	C₄×C₈	C₂×D₄	1	1	yes	characteristic
163	C₄×C₈	C₂×Q₈	1	2	yes	—
164	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
165	C₄×C₈	C₂⁴	1	1	yes	characteristic
166	#41	—	2	4	no	—
167	C₂×Q₈	#106	1	1	yes	characteristic
168	C₂×D₄	#106	1	2	yes	—
169	C₂×D₄	#106	1	1	yes	characteristic
170	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
172	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
173	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
174	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
175	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
176	M₁₆	—	8	4	no	—
177	M₁₆	—	8	2	no	—
178	M₁₆	—	16	1	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	8	4	no	—
192	C₂×C₈	—	8	2	no	—
193	C₂×C₈	—	16	2	no	—
194	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
196	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
197	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
200	C₄⋊C₄	#106	1	2	yes	—
201	C₄⋊C₄	#106	1	1	yes	characteristic
202	C₄⋊C₄	#106	1	2	yes	—
203	C₄⋊C₄	#106	1	1	yes	characteristic
204	C₂²⋊C₄	#106	1	2	yes	—
205	C₂²⋊C₄	#106	1	2	yes	—
206	C₂²⋊C₄	#106	1	2	yes	—
207	C₄²	C₂×C₄⋊C₄	1	2	yes	—
208	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₄²	C₂×SD₁₆	1	1	yes	characteristic
210	C₄²	C₂×SD₁₆	1	2	yes	—
211	C₄²	C₂×SD₁₆	1	2	yes	—
212	C₄²	C₂×SD₁₆	1	1	yes	characteristic
213	C₄²	C₂²×D₄	1	1	yes	characteristic
214	C₁₆	—	2	2	no	—
215	C₁₆	—	2	1	no	—
216	C₁₆	—	2	1	no	—
217	C₁₆	—	2	2	no	—
218	C₁₆	—	2	1	no	—
219	C₁₆	#131	1	2	yes	complemented
220	C₂³	C₂×#106	1	1	yes	characteristic
221	C₂³	C₂×#106	1	2	yes	—
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	#365	1	4	yes	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	C₂×#106	1	2	yes	—
234	C₂×C₄	C₂×#106	1	1	yes	characteristic
235	C₂×C₄	C₂×#106	1	2	yes	—
236	C₂×C₄	C₂×#106	1	1	yes	characteristic
237	C₂×C₄	C₂×#106	1	2	yes	—
238	C₂×C₄	C₂×#106	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
241	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	2	no	—
247	C₈	—	8	1	no	—
248	C₈	—	8	1	no	—
249	C₈	—	8	2	no	—
250	C₈	—	8	1	no	—
251	C₈	—	8	1	no	—
252	C₈	—	8	2	no	—
253	C₈	C₂×#131	1	1	yes	characteristic
254	C₈	#568	1	1	yes	characteristic
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	C₂×#365	1	4	yes	—
261	C₂²	C₂²×#106	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	1	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	C₂×#365	1	2	yes	—
270	C₄	#2906	1	1	yes	characteristic
271	C₄	#2916	1	1	yes	characteristic
272	C₂	—	2	4	no	—
273	C₂	—	4	1	no	—
274	C₂	#17826	1	1	yes	characteristic
275	C₂	C₂²×#365	1	1	yes	characteristic
276	C₂	#33404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ 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
4. #12030	C₂	semidirect
5. #12030	C₂	semidirect
6. #12029	C₂	semidirect
7. #12028	C₂	semidirect
8. #12020	C₂	semidirect
9. #12019	C₂	semidirect
10. #12000	C₂	semidirect
11. C₂×#1368	C₂	semidirect
219. C₁₆	#131	semidirect

Maximal subgroups

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

subgroup	index
#33484	2
#17834	2
#17826	2
#12030	2
#12029	2
#12028	2
#12020	2
#12019	2
#12000	2
C₂×#1368	2