Subgroups — #483472

Order 512.

417 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₆×#41	C₂	1	1	yes	maximal, characteristic
2	#17836	C₂	1	2	yes	maximal
3	#12036	C₂	1	2	yes	maximal, complemented
4	#12035	C₂	1	4	yes	maximal, complemented
5	#12024	C₂	1	2	yes	maximal, complemented
6	#12022	C₂	1	1	yes	maximal, characteristic, complemented
7	#12008	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#1371	C₂	1	2	yes	maximal, complemented
9	#2990	—	2	2	no	—
10	C₈×#41	C₂²	1	1	yes	characteristic
11	Q₈×C₁₆	C₄	1	1	yes	characteristic
12	Q₈×C₁₆	C₄	1	1	yes	characteristic
13	D₄×C₁₆	C₄	1	2	yes	—
14	D₄×C₁₆	C₄	1	1	yes	characteristic
15	D₄×C₁₆	C₄	1	2	yes	—
16	D₄×C₁₆	C₄	1	1	yes	characteristic
17	#2162	C₂²	1	2	yes	—
18	#2162	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	2	yes	—
20	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
21	#1574	—	2	2	no	—
22	#1573	—	2	4	no	—
23	#1568	C₂²	1	4	yes	—
24	#1567	C₂²	1	2	yes	—
25	#1564	C₂²	1	2	yes	—
26	C₂×#335	C₂²	1	4	yes	—
27	#1372	C₂²	1	2	yes	—
28	#1372	C₂²	1	4	yes	—
29	#1371	C₂²	1	2	yes	—
30	#1371	C₂²	1	4	yes	—
31	#1371	C₂²	1	1	yes	characteristic
32	#1371	C₂²	1	2	yes	—
33	#1371	C₂²	1	1	yes	characteristic
34	C₄×#41	D₄	1	1	yes	characteristic
35	C₂²×C₁₆	C₂×C₄	1	2	yes	—
36	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₁₆	C₂×C₄	1	2	yes	—
38	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
39	#446	—	4	2	no	—
40	C₈×Q₈	C₂×C₄	1	1	yes	characteristic
41	C₈×Q₈	Q₈	1	1	yes	characteristic
42	#435	—	4	4	no	—
43	C₈×D₄	—	4	2	no	—
44	C₈×D₄	C₂×C₄	1	2	yes	—
45	C₈×D₄	C₂×C₄	1	1	yes	characteristic
46	C₈×D₄	Q₈	1	2	yes	—
47	C₈×D₄	Q₈	1	1	yes	characteristic
48	#432	—	2	4	no	—
49	#431	—	2	2	no	—
50	#431	D₄	1	2	yes	—
51	#431	D₄	1	1	yes	characteristic
52	#423	—	2	2	no	—
53	C₂×C₄⋊C₈	—	2	4	no	—
54	C₂×C₄×C₈	C₂³	1	2	yes	—
55	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
56	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
57	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
58	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
59	C₄⋊C₁₆	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
61	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
62	C₂²⋊C₁₆	C₂×C₄	1	2	yes	—
63	C₄×C₁₆	C₂×C₄	1	2	yes	—
64	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
65	C₄×C₁₆	C₂×C₄	1	2	yes	—
66	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
67	C₄×C₁₆	C₂³	1	1	yes	characteristic
68	C₄×C₁₆	C₂³	1	2	yes	—
69	C₄×C₁₆	C₂³	1	1	yes	characteristic
70	#335	—	2	4	no	—
71	#335	—	2	2	no	—
72	#335	C₂³	1	2	yes	—
73	#335	C₂³	1	4	yes	—
74	#335	C₂³	1	2	yes	—
75	#334	—	2	2	no	—
76	#334	—	2	4	no	—
77	C₂×#41	Q₁₆	1	1	yes	characteristic
78	C₂×M₁₆	—	4	2	no	—
79	C₂²×C₈	—	4	4	no	—
80	C₂²×C₈	C₂²×C₄	1	2	yes	—
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂×Q₈	1	2	yes	—
83	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
84	C₄×Q₈	C₄⋊C₄	1	1	yes	characteristic
85	C₄×Q₈	D₈	1	1	yes	characteristic
86	C₄×D₄	C₄⋊C₄	1	2	yes	—
87	C₄×D₄	C₄⋊C₄	1	1	yes	characteristic
88	C₄×D₄	D₈	1	1	yes	characteristic
89	C₄×D₄	D₈	1	2	yes	—
90	#117	Q₁₆	1	1	yes	characteristic
91	#117	Q₁₆	1	2	yes	—
92	C₂×C₄²	C₂×D₄	1	2	yes	—
93	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₂×C₁₆	—	2	4	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	4	2	no	—
97	C₂×C₁₆	—	4	1	no	—
98	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₁₆	C₂²×C₄	1	2	yes	—
100	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₁₆	C₂²×C₄	1	2	yes	—
102	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₄⋊C₈	—	4	2	no	—
109	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
110	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
111	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
112	C₄⋊C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₂²⋊C₈	—	4	2	no	—
114	C₂²⋊C₈	—	4	4	no	—
115	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
116	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
117	C₂²⋊C₈	C₄⋊C₄	1	2	yes	—
118	#97	—	4	4	no	—
119	C₄×C₈	—	4	2	no	—
120	C₄×C₈	C₂²×C₄	1	2	yes	—
121	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
122	C₄×C₈	C₂×D₄	1	1	yes	characteristic
123	C₄×C₈	C₂×D₄	1	2	yes	—
124	C₄×C₈	C₂×Q₈	1	2	yes	—
125	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
126	C₄×C₈	C₂⁴	1	1	yes	characteristic
127	#41	—	2	4	no	—
128	C₂×Q₈	#107	1	1	yes	characteristic
129	C₂×D₄	#107	1	2	yes	—
130	C₂×D₄	#107	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
133	C₂²×C₄	C₂×D₈	1	2	yes	—
134	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
135	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
136	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
137	M₁₆	—	16	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	4	1	no	—
144	C₂×C₈	—	4	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	16	4	no	—
148	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
152	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
154	C₄⋊C₄	#107	1	2	yes	—
155	C₄⋊C₄	#107	1	1	yes	characteristic
156	C₄⋊C₄	#107	1	2	yes	—
157	C₄⋊C₄	#107	1	1	yes	characteristic
158	C₂²⋊C₄	#107	1	2	yes	—
159	C₂²⋊C₄	#107	1	2	yes	—
160	C₂²⋊C₄	#107	1	2	yes	—
161	C₄²	C₂×C₄⋊C₄	1	2	yes	—
162	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₄²	C₂×D₈	1	1	yes	characteristic
164	C₄²	C₂×D₈	1	2	yes	—
165	C₄²	C₂×Q₁₆	1	2	yes	—
166	C₄²	C₂×Q₁₆	1	1	yes	characteristic
167	C₄²	C₂²×D₄	1	1	yes	characteristic
168	C₁₆	—	2	2	no	—
169	C₁₆	—	2	1	no	—
170	C₁₆	—	2	1	no	—
171	C₁₆	—	2	2	no	—
172	C₁₆	—	2	1	no	—
173	C₁₆	#131	1	2	yes	—
174	C₂³	C₂×#107	1	1	yes	characteristic
175	C₂³	C₂×#107	1	2	yes	—
176	Q₈	—	2	2	no	—
177	D₄	—	2	4	no	—
178	D₄	#368	1	4	yes	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	C₂×#107	1	2	yes	—
190	C₂×C₄	C₂×#107	1	2	yes	—
191	C₂×C₄	C₂×#107	1	1	yes	characteristic
192	C₂×C₄	C₂×#107	1	1	yes	characteristic
193	C₂×C₄	C₂×#107	1	2	yes	—
194	C₂×C₄	C₂×#107	1	1	yes	characteristic
195	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
197	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	1	no	—
200	C₈	—	2	1	no	—
201	C₈	—	2	2	no	—
202	C₈	—	2	1	no	—
203	C₈	—	8	2	no	—
204	C₈	—	8	4	no	—
205	C₈	—	8	2	no	—
206	C₈	C₂×#131	1	1	yes	characteristic
207	C₈	#568	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂×#368	1	4	yes	—
214	C₂²	C₂²×#107	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	C₂×#368	1	2	yes	—
222	C₄	#2906	1	1	yes	characteristic
223	C₄	#2916	1	1	yes	characteristic
224	C₄	#3415	1	1	yes	characteristic
225	C₄	#3417	1	1	yes	characteristic
226	C₂	—	2	4	no	—
227	C₂	—	4	1	no	—
228	C₂	#17828	1	1	yes	characteristic
229	C₂	#33410	1	1	yes	characteristic
230	C₂	C₂²×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
3. #12036	C₂	semidirect
4. #12035	C₂	semidirect
5. #12024	C₂	semidirect
6. #12022	C₂	semidirect
7. #12008	C₂	semidirect
8. C₂×#1371	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₆×#41	2
#17836	2
#12036	2
#12035	2
#12024	2
#12022	2
#12008	2
C₂×#1371	2