Subgroups — #507497

Order 512.

560 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33312	C₂	1	1	yes	maximal, characteristic
2	C₄×#500	C₂	1	1	yes	maximal, characteristic
3	#13452	C₂	1	4	yes	maximal
4	#13282	C₂	1	2	yes	maximal
5	#13177	C₂	1	4	yes	maximal
6	#12906	C₂	1	2	yes	maximal
7	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	—	2	1	no	—
9	C₂×#500	C₂²	1	1	yes	characteristic
10	C₂×#500	C₂²	1	2	yes	—
11	#2868	—	2	1	no	—
12	C₄×C₄⋊Q₈	—	2	1	no	—
13	C₂×#367	C₂²	1	8	yes	—
14	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	#1990	—	2	1	no	—
17	#1942	—	2	1	no	—
18	#1942	C₂²	1	2	yes	—
19	#1845	—	2	1	no	—
20	#1845	—	2	1	no	—
21	#1845	C₂²	1	2	yes	—
22	#1845	C₂²	1	2	yes	—
23	#1845	C₂²	1	1	yes	characteristic
24	C₄×#107	—	2	1	no	—
25	C₄×#107	C₂²	1	2	yes	—
26	C₄²×C₈	C₂²	1	1	yes	characteristic
27	#1380	C₂²	1	8	yes	—
28	#1374	C₂²	1	2	yes	—
29	C₂×C₄⋊Q₈	—	2	1	no	—
30	C₂×C₄⋊Q₈	—	2	2	no	—
31	C₂×C₄⋊Q₈	—	4	1	no	—
32	C₂×C₄⋊Q₈	—	4	1	no	—
33	C₂×C₄×Q₈	—	4	1	no	—
34	C₂²×C₁₆	C₂³	1	4	yes	—
35	#500	—	2	2	no	—
36	#500	—	4	4	no	—
37	#500	—	4	4	no	—
38	#500	Q₈	1	4	yes	complemented
39	C₂×#107	—	2	1	no	—
40	C₂×#107	—	2	2	no	—
41	C₂×#107	—	2	2	no	—
42	C₂×#107	—	2	2	no	—
43	C₂×#107	—	2	1	no	—
44	C₂×#107	C₂³	1	4	yes	—
45	C₂×#107	C₂³	1	2	yes	—
46	C₂×#107	C₂³	1	2	yes	—
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	C₂³	1	1	yes	characteristic
51	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	#391	—	4	1	no	—
54	#389	—	2	2	no	—
55	#389	—	4	1	no	—
56	#384	—	2	1	no	—
57	#384	—	2	2	no	—
58	#384	—	2	2	no	—
59	#384	—	4	1	no	—
60	#384	—	4	1	no	—
61	C₄×C₄⋊C₄	—	2	2	no	—
62	C₄×C₄⋊C₄	—	4	1	no	—
63	C₄³	D₄	1	1	yes	characteristic
64	#367	—	4	8	no	—
65	C₄⋊C₁₆	Q₈	1	8	yes	complemented
66	C₄×C₁₆	—	2	4	no	—
67	#340	—	2	2	no	—
68	#340	—	2	2	no	—
69	C₂²×Q₈	—	4	1	no	—
70	C₂²×Q₈	—	4	2	no	—
71	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₈	C₂⁴	1	1	yes	characteristic
75	C₄⋊Q₈	—	2	4	no	—
76	C₄⋊Q₈	—	4	4	no	—
77	C₄⋊Q₈	—	4	4	no	—
78	C₄⋊Q₈	—	4	2	no	—
79	C₄⋊Q₈	—	8	4	no	—
80	C₄⋊Q₈	—	8	4	no	—
81	C₄×Q₈	—	8	4	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	—	2	4	no	—
84	C₂×C₄⋊C₄	—	2	2	no	—
85	C₂×C₄⋊C₄	—	4	1	no	—
86	C₂×C₄⋊C₄	—	4	1	no	—
87	C₂×C₄⋊C₄	—	4	2	no	—
88	C₂×C₄⋊C₄	—	4	2	no	—
89	C₂×C₄⋊C₄	—	4	2	no	—
90	C₂×C₄⋊C₄	—	4	2	no	—
91	C₂×C₄⋊C₄	—	4	2	no	—
92	C₂×C₄⋊C₄	—	4	2	no	—
93	C₂×C₄⋊C₄	—	4	1	no	—
94	C₂×C₄²	—	4	2	no	—
95	C₂×C₄²	D₈	1	1	yes	characteristic
96	C₂×C₄²	D₈	1	1	yes	characteristic
97	C₂×C₄²	D₈	1	1	yes	characteristic
98	C₂×C₄²	D₈	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
102	C₂×C₁₆	—	2	4	no	—
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	C₂×Q₈	1	8	yes	—
105	#107	—	2	4	no	—
106	#107	—	2	2	no	—
107	#107	—	2	4	no	—
108	#107	—	2	8	no	—
109	#107	—	4	4	no	—
110	#107	—	4	4	no	—
111	#107	—	4	2	no	—
112	#107	—	4	4	no	—
113	#107	C₂×Q₈	1	4	yes	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	Q₁₆	1	4	yes	—
118	C₄×C₈	C₂×Q₈	1	2	yes	—
119	C₄×C₈	C₂×Q₈	1	4	yes	—
120	C₄×C₈	#41	1	2	yes	—
121	#95	—	4	2	no	—
122	#95	—	4	2	no	—
123	C₂×Q₈	—	4	4	no	—
124	C₂×Q₈	—	8	4	no	—
125	C₂×Q₈	—	8	4	no	—
126	C₂×Q₈	—	8	2	no	—
127	C₂×Q₈	—	8	4	no	—
128	C₂×Q₈	—	8	2	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	—	4	2	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
134	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
136	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₈	C₄⋊Q₈	1	2	yes	—
145	C₂×C₈	C₂×Q₁₆	1	2	yes	—
146	C₂×C₈	C₂×Q₁₆	1	2	yes	—
147	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₈	C₂²×Q₈	1	2	yes	—
149	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₈	C₂×#41	1	1	yes	characteristic
151	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	8	4	no	—
160	C₄⋊C₄	—	8	4	no	—
161	C₄⋊C₄	—	8	4	no	—
162	C₄⋊C₄	—	8	2	no	—
163	C₄⋊C₄	—	8	4	no	—
164	C₄⋊C₄	—	8	4	no	—
165	C₄⋊C₄	—	8	4	no	—
166	C₄⋊C₄	—	8	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	8	4	no	—
172	C₄²	SD₃₂	1	4	yes	—
173	C₄²	SD₃₂	1	4	yes	—
174	C₄²	C₂²⋊Q₈	1	2	yes	—
175	C₄²	C₄⋊Q₈	1	2	yes	—
176	C₄²	C₂×Q₁₆	1	2	yes	—
177	C₄²	#137	1	2	yes	—
178	C₁₆	—	2	8	no	—
179	C₂³	C₂²×D₈	1	1	yes	characteristic
180	Q₈	—	8	8	no	—
181	Q₈	—	16	8	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	8	4	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	4	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	#474	1	2	yes	—
198	C₂×C₄	#474	1	2	yes	—
199	C₂×C₄	#500	1	2	yes	—
200	C₂×C₄	#500	1	2	yes	—
201	C₂×C₄	C₂×SD₃₂	1	2	yes	—
202	C₂×C₄	C₂×SD₃₂	1	2	yes	—
203	C₂×C₄	C₂×SD₃₂	1	2	yes	—
204	C₂×C₄	C₂×SD₃₂	1	2	yes	—
205	C₂×C₄	C₂×SD₃₂	1	2	yes	—
206	C₂×C₄	C₂×SD₃₂	1	2	yes	—
207	C₂×C₄	#509	1	2	yes	—
208	C₂×C₄	#510	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Q₈	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
211	C₂×C₄	#535	1	1	yes	characteristic
212	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
213	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₂×C₄	C₂×#137	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	#477	1	4	yes	—
218	C₈	#554	1	2	yes	—
219	C₈	Q₈²	1	2	yes	—
220	C₂²	C₂×#474	1	1	yes	characteristic
221	C₂²	C₂×#500	1	1	yes	characteristic
222	C₂²	#3205	1	1	yes	characteristic
223	C₂²	C₂²×SD₃₂	1	1	yes	characteristic
224	C₂²	C₂²×SD₃₂	1	1	yes	characteristic
225	C₂²	C₂×#509	1	1	yes	characteristic
226	C₂²	C₂×#510	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	8	4	no	—
231	C₄	—	16	2	no	—
232	C₄	—	16	4	no	—
233	C₄	#2225	1	4	yes	—
234	C₄	#2226	1	4	yes	—
235	C₄	#2254	1	4	yes	—
236	C₄	#2567	1	1	yes	characteristic
237	C₄	C₂×#477	1	1	yes	characteristic
238	C₄	#3165	1	1	yes	characteristic
239	C₄	#3209	1	1	yes	characteristic
240	C₂	#29208	1	1	yes	characteristic
241	C₂	C₂×#2226	1	1	yes	characteristic
242	C₂	C₂×#2225	1	1	yes	characteristic
243	C₂	C₂×#2254	1	1	yes	characteristic
244	C₂	#33731	1	1	yes	characteristic
245	C₂	#33772	1	1	yes	characteristic
246	C₂	#33774	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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
38. #500	Q₈	semidirect
65. C₄⋊C₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#33312	2
C₄×#500	2
#13452	2
#13282	2
#13177	2
#12906	2
C₄×C₄⋊C₁₆	2