Subgroups — #506431

Order 512.

486 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33474	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
3	#28297	C₂	1	2	yes	maximal
4	#13471	C₂	1	4	yes	maximal
5	#13244	C₂	1	2	yes	maximal
6	#13171	C₂	1	2	yes	maximal
7	#12909	C₂	1	1	yes	maximal, characteristic
8	#12819	C₂	1	2	yes	maximal
9	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
10	#3078	—	2	2	no	—
11	#2960	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₁₆	C₂²	1	2	yes	—
13	#2183	—	2	1	no	—
14	#2183	—	2	1	no	—
15	Q₈×C₁₆	—	2	1	no	—
16	Q₈×C₁₆	—	2	1	no	—
17	C₄×Q₃₂	—	2	1	no	—
18	C₄×Q₃₂	—	2	1	no	—
19	C₄×Q₃₂	—	2	1	no	—
20	C₄×Q₃₂	—	2	1	no	—
21	C₂×#368	C₂²	1	2	yes	—
22	C₂×#368	C₂²	1	2	yes	—
23	C₂×#366	C₂²	1	1	yes	characteristic
24	#2151	C₂²	1	2	yes	—
25	#2150	C₂²	1	1	yes	characteristic
26	C₂×#362	C₂²	1	4	yes	—
27	C₂×#358	C₂²	1	2	yes	—
28	#2108	C₂²	1	2	yes	—
29	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
30	#2064	—	2	4	no	—
31	#1979	C₂²	1	4	yes	—
32	#1952	C₂²	1	4	yes	—
33	#1848	C₂²	1	2	yes	—
34	#1801	—	2	2	no	—
35	#1387	C₂²	1	4	yes	—
36	C₂²×Q₁₆	C₂³	1	2	yes	—
37	#533	D₄	1	1	yes	characteristic
38	C₂×C₄×Q₈	—	2	2	no	—
39	C₂×Q₃₂	—	2	1	no	—
40	C₂×Q₃₂	—	2	1	no	—
41	C₂×Q₃₂	—	2	2	no	—
42	C₂×Q₃₂	Q₈	1	4	yes	complemented
43	C₂×M₃₂	C₂³	1	2	yes	—
44	C₂²×C₁₆	C₂³	1	1	yes	characteristic
45	C₂²×C₁₆	C₂³	1	1	yes	characteristic
46	#479	—	4	4	no	—
47	#477	—	4	2	no	—
48	#475	—	4	2	no	—
49	#446	—	2	1	no	—
50	#446	—	2	1	no	—
51	C₈×Q₈	—	2	1	no	—
52	C₈×Q₈	—	2	1	no	—
53	C₄×Q₁₆	—	2	2	no	—
54	C₄×Q₁₆	—	2	2	no	—
55	C₄×Q₁₆	—	2	2	no	—
56	C₄×Q₁₆	—	2	2	no	—
57	C₂×#108	—	2	2	no	—
58	C₂×#108	C₂³	1	4	yes	—
59	#428	—	2	4	no	—
60	C₂×#107	C₂³	1	2	yes	—
61	#424	D₄	1	2	yes	—
62	#423	—	2	2	no	—
63	#423	D₄	1	1	yes	characteristic
64	#419	—	2	4	no	—
65	C₂×#103	—	2	2	no	—
66	C₂×#101	—	2	4	no	—
67	#405	C₂³	1	2	yes	—
68	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
69	#368	—	2	4	no	—
70	#368	—	4	2	no	—
71	#366	—	2	1	no	—
72	#366	—	2	1	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	C₄⋊C₁₆	—	2	2	no	—
75	C₄⋊C₁₆	—	2	1	no	—
76	C₄⋊C₁₆	—	2	1	no	—
77	#362	—	4	4	no	—
78	#358	—	2	2	no	—
79	#358	—	2	2	no	—
80	#346	—	2	2	no	—
81	C₄×C₁₆	—	2	2	no	—
82	C₄×C₁₆	D₄	1	1	yes	characteristic
83	C₄×C₁₆	D₄	1	1	yes	characteristic
84	C₄×C₁₆	D₄	1	1	yes	characteristic
85	C₄×C₁₆	D₄	1	1	yes	characteristic
86	#343	—	2	4	no	—
87	C₂²×Q₈	—	2	2	no	—
88	C₂×Q₁₆	—	2	2	no	—
89	C₂×Q₁₆	—	2	2	no	—
90	C₂×Q₁₆	—	2	2	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	2	2	no	—
93	C₂×Q₁₆	C₂×Q₈	1	4	yes	—
94	C₂×M₁₆	—	2	4	no	—
95	C₂×M₁₆	C₂×D₄	1	2	yes	—
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂⁴	1	1	yes	characteristic
98	C₄⋊Q₈	D₈	1	1	yes	characteristic
99	C₄⋊Q₈	D₈	1	1	yes	characteristic
100	#125	D₈	1	2	yes	—
101	C₂²⋊Q₈	—	2	1	no	—
102	C₂²⋊Q₈	—	2	1	no	—
103	C₄×Q₈	—	2	1	no	—
104	C₄×Q₈	—	2	1	no	—
105	C₄×Q₈	—	4	2	no	—
106	C₄×Q₈	—	4	2	no	—
107	C₄×Q₈	—	4	2	no	—
108	C₄×Q₈	—	4	2	no	—
109	#117	C₂×D₄	1	2	yes	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	4	2	no	—
112	C₂×C₄²	—	4	2	no	—
113	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
114	Q₃₂	—	2	4	no	—
115	Q₃₂	—	2	4	no	—
116	M₃₂	—	2	2	no	—
117	M₃₂	—	2	2	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	2	no	—
121	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
122	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
123	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
124	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
125	C₂×C₁₆	C₂×Q₈	1	2	yes	—
126	C₂×C₁₆	#41	1	2	yes	—
127	#108	—	2	4	no	—
128	#108	—	2	4	no	—
129	#108	—	8	2	no	—
130	#107	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	—	4	4	no	—
133	#106	—	4	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	2	1	no	—
138	C₄⋊C₈	—	4	2	no	—
139	C₄⋊C₈	—	4	2	no	—
140	#103	—	4	2	no	—
141	#103	—	4	2	no	—
142	#103	—	4	4	no	—
143	#103	—	4	4	no	—
144	#101	—	8	4	no	—
145	#97	—	2	2	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	C₂×D₄	1	1	yes	characteristic
148	C₄×C₈	C₂×D₄	1	1	yes	characteristic
149	C₄×C₈	#41	1	1	yes	characteristic
150	C₄×C₈	#41	1	1	yes	characteristic
151	C₂×Q₈	—	2	1	no	—
152	C₂×Q₈	—	2	1	no	—
153	C₂×Q₈	—	2	4	no	—
154	C₂×Q₈	—	4	2	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂×Q₈	—	4	2	no	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	—	4	2	no	—
161	C₂²×C₄	#135	1	1	yes	characteristic
162	C₂²×C₄	#135	1	1	yes	characteristic
163	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
164	Q₁₆	—	2	4	no	—
165	Q₁₆	—	2	4	no	—
166	Q₁₆	—	4	2	no	—
167	Q₁₆	—	4	2	no	—
168	M₁₆	—	2	2	no	—
169	M₁₆	—	2	2	no	—
170	M₁₆	—	4	4	no	—
171	M₁₆	—	4	4	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	4	4	no	—
176	C₂×C₈	#121	1	1	yes	characteristic
177	C₂×C₈	#121	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
179	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
180	C₂×C₈	C₂×#41	1	1	yes	characteristic
181	C₂×C₈	C₂×#41	1	1	yes	characteristic
182	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	8	2	no	—
191	C₄⋊C₄	—	8	2	no	—
192	C₄⋊C₄	C₂×D₈	1	2	yes	—
193	C₄⋊C₄	C₂×D₈	1	2	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	8	2	no	—
197	C₄²	—	8	2	no	—
198	C₄²	#121	1	1	yes	characteristic
199	C₄²	#121	1	1	yes	characteristic
200	C₄²	C₂×D₈	1	1	yes	characteristic
201	C₄²	C₂×D₈	1	1	yes	characteristic
202	C₁₆	—	2	1	no	—
203	C₁₆	—	2	1	no	—
204	C₁₆	—	4	2	no	—
205	C₁₆	C₂²⋊Q₈	1	2	yes	—
206	C₁₆	C₂²⋊Q₈	1	2	yes	—
207	C₂³	C₂×#135	1	1	yes	characteristic
208	Q₈	—	4	2	no	—
209	Q₈	—	4	2	no	—
210	Q₈	—	4	4	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	8	2	no	—
213	Q₈	—	8	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	2	no	—
222	C₂×C₄	—	16	2	no	—
223	C₂×C₄	#466	1	1	yes	characteristic
224	C₂×C₄	#466	1	1	yes	characteristic
225	C₂×C₄	#466	1	1	yes	characteristic
226	C₂×C₄	#466	1	1	yes	characteristic
227	C₂×C₄	C₂×#121	1	1	yes	characteristic
228	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
229	C₂×C₄	#550	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
231	C₂×C₄	#576	1	1	yes	characteristic
232	C₂×C₄	#578	1	1	yes	characteristic
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	—	8	4	no	—
237	C₈	C₂×C₂²⋊Q₈	1	1	yes	characteristic
238	C₈	#537	1	1	yes	characteristic
239	C₈	#554	1	1	yes	characteristic
240	C₈	#556	1	1	yes	characteristic
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	C₂×#466	1	1	yes	characteristic
244	C₂²	#3380	1	1	yes	characteristic
245	C₂²	#3393	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	8	4	no	—
252	C₄	—	16	2	no	—
253	C₄	#2222	1	2	yes	—
254	C₄	#2538	1	1	yes	characteristic
255	C₄	#3061	1	1	yes	characteristic
256	C₄	Q₈×D₈	1	1	yes	characteristic
257	C₄	#3391	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	#28263	1	1	yes	characteristic
261	C₂	C₂×#2222	1	1	yes	characteristic
262	C₂	#33617	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
42. C₂×Q₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33474	2
C₂×C₄×Q₃₂	2
#28297	2
#13471	2
#13244	2
#13171	2
#12909	2
#12819	2