Subgroups — #506098

Order 512.

444 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28015	C₂	1	2	yes	maximal
2	#13481	C₂	1	2	yes	maximal
3	#13475	C₂	1	2	yes	maximal
4	#13236	C₂	1	1	yes	maximal, characteristic
5	#13236	C₂	1	1	yes	maximal, characteristic
6	#13236	C₂	1	1	yes	maximal, characteristic
7	#13165	C₂	1	1	yes	maximal, characteristic
8	#13133	C₂	1	2	yes	maximal
9	#12905	C₂	1	1	yes	maximal, characteristic
10	C₄×#366	C₂	1	1	yes	maximal, characteristic
11	C₁₆×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
13	#2511	—	2	2	no	—
14	C₂×#366	C₂²	1	1	yes	characteristic
15	C₂×#366	C₂²	1	1	yes	characteristic
16	C₂×#366	C₂²	1	1	yes	characteristic
17	C₂×#366	C₂²	1	2	yes	—
18	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×#358	C₂²	1	2	yes	—
20	C₂×#358	C₂²	1	2	yes	—
21	C₂×#358	C₂²	1	2	yes	—
22	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#2072	—	2	2	no	—
26	#2068	—	2	2	no	—
27	#1969	C₂²	1	2	yes	—
28	#1969	C₂²	1	2	yes	—
29	#1969	C₂²	1	2	yes	—
30	#1942	C₂²	1	2	yes	—
31	#1928	—	2	2	no	—
32	#1845	C₂²	1	2	yes	—
33	C₄×#107	C₂²	1	2	yes	—
34	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	#1380	C₂²	1	2	yes	—
36	#1380	C₂²	1	2	yes	—
37	#1374	C₂²	1	1	yes	characteristic
38	#1374	C₂²	1	1	yes	characteristic
39	C₂²×Q₁₆	C₂³	1	2	yes	—
40	C₂×Q₃₂	—	2	2	no	—
41	C₂×Q₃₂	—	2	2	no	—
42	C₂×Q₃₂	—	2	2	no	—
43	C₂²×C₁₆	C₂³	1	1	yes	characteristic
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	1	yes	characteristic
46	C₂²×C₁₆	C₂³	1	1	yes	characteristic
47	C₂×#107	—	2	2	no	—
48	C₂×#107	C₂³	1	2	yes	—
49	C₂×#107	C₂³	1	2	yes	—
50	C₂×#107	C₂³	1	2	yes	—
51	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
52	C₂×#103	—	2	2	no	—
53	C₂×#103	—	2	2	no	—
54	C₂×#103	—	2	2	no	—
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
57	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
58	#400	—	4	2	no	—
59	#400	—	4	2	no	—
60	#391	—	2	2	no	—
61	#391	—	2	2	no	—
62	#391	—	2	2	no	—
63	#389	—	2	2	no	—
64	#387	—	4	2	no	—
65	#387	—	4	2	no	—
66	#384	—	2	2	no	—
67	C₄×C₄⋊C₄	—	2	2	no	—
68	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
69	#366	—	2	2	no	—
70	#366	—	2	2	no	—
71	#366	—	2	2	no	—
72	#366	—	4	2	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	#358	—	2	4	no	—
75	#358	—	4	2	no	—
76	#358	—	4	2	no	—
77	C₄×C₁₆	—	2	2	no	—
78	C₄×C₁₆	—	2	2	no	—
79	C₄×C₁₆	—	2	2	no	—
80	#340	—	2	2	no	—
81	#340	—	2	2	no	—
82	#336	D₄	1	1	yes	characteristic
83	#336	D₄	1	1	yes	characteristic
84	C₂²×Q₈	—	2	2	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	—	2	2	no	—
87	C₂×Q₁₆	—	2	2	no	—
88	C₂×Q₁₆	—	2	2	no	—
89	C₂×Q₁₆	—	2	2	no	—
90	C₂×Q₁₆	—	2	2	no	—
91	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂⁴	1	1	yes	characteristic
95	C₂×C₄⋊C₄	—	2	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₄⋊C₄	—	4	2	no	—
100	C₂×C₄⋊C₄	—	4	2	no	—
101	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
102	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
103	C₂×C₄²	—	4	2	no	—
104	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
107	Q₃₂	—	4	4	no	—
108	C₂×C₁₆	—	2	1	no	—
109	C₂×C₁₆	—	2	1	no	—
110	C₂×C₁₆	—	2	1	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	2	1	no	—
116	C₂×C₁₆	#41	1	2	yes	—
117	C₂×C₁₆	#41	1	2	yes	—
118	C₂×C₁₆	#41	1	2	yes	—
119	C₂×C₁₆	#41	1	2	yes	—
120	#107	—	2	2	no	—
121	#107	—	2	2	no	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	8	2	no	—
127	C₄⋊C₈	—	2	2	no	—
128	#103	—	4	4	no	—
129	#103	—	8	2	no	—
130	#103	—	8	2	no	—
131	C₄×C₈	—	2	1	no	—
132	C₄×C₈	—	2	1	no	—
133	C₄×C₈	—	2	1	no	—
134	C₄×C₈	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	#95	—	4	2	no	—
138	#95	—	4	2	no	—
139	#95	—	4	2	no	—
140	#95	—	4	2	no	—
141	#95	—	4	2	no	—
142	#95	—	4	2	no	—
143	#95	—	4	2	no	—
144	#95	—	4	2	no	—
145	#95	D₈	1	1	yes	characteristic
146	#95	D₈	1	1	yes	characteristic
147	C₂×Q₈	—	4	2	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	4	2	no	—
152	C₂×Q₈	—	4	2	no	—
153	C₂²×C₄	—	4	2	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
159	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
160	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
162	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
164	Q₁₆	—	4	2	no	—
165	Q₁₆	—	4	2	no	—
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	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	#135	1	2	yes	—
177	C₂×C₈	C₂×#41	1	1	yes	characteristic
178	C₂×C₈	C₂×#41	1	1	yes	characteristic
179	C₂×C₈	C₂×#41	1	1	yes	characteristic
180	C₂×C₈	C₂×#41	1	1	yes	characteristic
181	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	8	4	no	—
191	C₄⋊C₄	—	16	2	no	—
192	C₄⋊C₄	—	16	2	no	—
193	C₄⋊C₄	Q₃₂	1	4	yes	complemented
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	16	2	no	—
201	C₁₆	—	2	2	no	—
202	C₁₆	—	4	1	no	—
203	C₁₆	—	4	1	no	—
204	C₁₆	#126	1	4	yes	—
205	C₂³	C₂²×D₈	1	1	yes	characteristic
206	Q₈	—	8	2	no	—
207	Q₈	—	8	2	no	—
208	Q₈	—	8	2	no	—
209	Q₈	—	8	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	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	8	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	—	8	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×Q₃₂	1	2	yes	—
237	C₂×C₄	C₂×Q₃₂	1	2	yes	—
238	C₂×C₄	C₂×Q₃₂	1	2	yes	—
239	C₂×C₄	#508	1	2	yes	—
240	C₂×C₄	#532	1	1	yes	characteristic
241	C₂×C₄	#547	1	1	yes	characteristic
242	C₂×C₄	#550	1	1	yes	characteristic
243	C₂×C₄	C₂×#135	1	1	yes	characteristic
244	C₂×C₄	#576	1	1	yes	characteristic
245	C₂×C₄	#578	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	—	4	1	no	—
248	C₈	—	4	1	no	—
249	C₈	C₂×#126	1	1	yes	characteristic
250	C₈	#542	1	1	yes	characteristic
251	C₈	#553	1	1	yes	characteristic
252	C₈	#556	1	1	yes	characteristic
253	C₂²	#3169	1	1	yes	characteristic
254	C₂²	#3334	1	1	yes	characteristic
255	C₂²	#3391	1	1	yes	characteristic
256	C₂²	C₂²×Q₃₂	1	1	yes	characteristic
257	C₂²	C₂×#508	1	1	yes	characteristic
258	C₂²	#3415	1	1	yes	characteristic
259	C₂²	#3417	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	1	no	—
266	C₄	—	16	2	no	—
267	C₄	—	16	2	no	—
268	C₄	—	16	2	no	—
269	C₄	—	16	2	no	—
270	C₄	#2512	1	1	yes	characteristic
271	C₄	#3175	1	1	yes	characteristic
272	C₄	#3339	1	1	yes	characteristic
273	C₄	#3393	1	1	yes	characteristic
274	C₂	#28014	1	1	yes	characteristic
275	C₂	#33734	1	1	yes	characteristic
276	C₂	#33739	1	1	yes	characteristic
277	C₂	#33903	1	1	yes	characteristic
278	C₂	#33905	1	1	yes	characteristic
279	C₂	#33962	1	1	yes	characteristic
280	C₂	#33963	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
193. C₄⋊C₄	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#28015	2
#13481	2
#13475	2
#13236	2
#13165	2
#13133	2
#12905	2
C₄×#366	2
C₁₆×C₄⋊C₄	2