Subgroups — #497891

Order 512.

610 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2178	C₂	1	1	yes	maximal, characteristic
2	#33460	C₂	1	2	yes	maximal
3	#22692	C₂	1	2	yes	maximal, complemented
4	#13172	C₂	1	2	yes	maximal, complemented
5	#12946	C₂	1	4	yes	maximal, complemented
6	#12906	C₂	1	1	yes	maximal, characteristic, complemented
7	#12821	C₂	1	2	yes	maximal, complemented
8	#12664	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#508	C₂²	1	1	yes	characteristic
10	C₂×#443	C₂²	1	2	yes	—
11	#2951	C₂²	1	4	yes	—
12	#2913	—	2	2	no	—
13	#2178	C₄	1	8	yes	complemented
14	#2177	—	2	2	no	—
15	#2176	—	2	2	no	—
16	C₄×SD₃₂	—	2	4	no	—
17	C₂×#368	C₂²	1	1	yes	characteristic
18	C₂×#367	C₂²	1	2	yes	—
19	C₂×#367	C₂²	1	2	yes	—
20	C₂×#364	C₂²	1	1	yes	characteristic
21	C₂×#359	C₂²	1	2	yes	—
22	#2139	C₂²	1	4	yes	—
23	#2108	C₂²	1	2	yes	—
24	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
25	#1952	C₂²	1	4	yes	—
26	#1859	—	2	4	no	—
27	#1845	C₂²	1	2	yes	—
28	#1816	—	2	2	no	—
29	#1772	C₂²	1	1	yes	characteristic
30	#1387	C₂²	1	4	yes	—
31	#1379	C₂²	1	2	yes	—
32	C₂×#135	C₂³	1	2	yes	—
33	C₂×#131	—	2	2	no	—
34	#520	—	2	4	no	—
35	#508	C₂×C₄	1	4	yes	—
36	#508	Q₈	1	4	yes	—
37	C₂×Q₃₂	D₄	1	1	yes	characteristic
38	C₂×SD₃₂	D₄	1	2	yes	—
39	C₂×D₁₆	D₄	1	1	yes	characteristic
40	C₂×M₃₂	C₂³	1	2	yes	—
41	C₂²×C₁₆	C₂³	1	1	yes	characteristic
42	C₂²×C₁₆	C₂³	1	1	yes	characteristic
43	#443	—	2	4	no	—
44	#443	C₂×C₄	1	8	yes	complemented
45	#440	—	4	4	no	—
46	C₄×Q₁₆	—	2	4	no	—
47	C₄×D₈	—	2	4	no	—
48	C₂×#108	D₄	1	1	yes	characteristic
49	C₂×#108	C₂³	1	2	yes	—
50	#428	—	2	4	no	—
51	#428	—	4	4	no	—
52	#427	—	4	2	no	—
53	#427	D₄	1	1	yes	characteristic
54	C₂×#107	—	2	2	no	—
55	C₂×#107	—	4	2	no	—
56	C₂×#107	C₂³	1	4	yes	—
57	C₂×#104	—	2	2	no	—
58	#417	—	2	4	no	—
59	#405	C₂³	1	2	yes	—
60	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
61	#384	—	2	2	no	—
62	#368	C₂×C₄	1	4	yes	complemented
63	#367	—	2	2	no	—
64	#367	—	2	2	no	—
65	#367	—	4	2	no	—
66	#364	—	2	2	no	—
67	#359	C₂×C₄	1	8	yes	complemented
68	#358	—	2	4	no	—
69	#357	—	2	4	no	—
70	#346	—	2	2	no	—
71	C₄×C₁₆	—	2	2	no	—
72	C₄×C₁₆	C₂×C₄	1	4	yes	complemented
73	#343	—	2	4	no	—
74	#339	D₄	1	2	yes	—
75	C₂×#41	—	2	2	no	—
76	#135	—	2	2	no	—
77	#135	—	2	2	no	—
78	#135	C₂²×C₄	1	4	yes	—
79	#135	C₂×Q₈	1	4	yes	—
80	C₂×Q₁₆	C₂×D₄	1	2	yes	—
81	C₂×SD₁₆	—	2	2	no	—
82	C₂×D₈	C₂×D₄	1	2	yes	—
83	#131	—	2	8	no	—
84	#131	—	4	4	no	—
85	C₂×M₁₆	—	2	2	no	—
86	C₂×M₁₆	—	4	2	no	—
87	C₂×M₁₆	C₂×D₄	1	2	yes	—
88	C₂²×C₈	—	4	2	no	—
89	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₈	C₂⁴	1	1	yes	characteristic
91	C₄×Q₈	—	4	4	no	—
92	C₄×D₄	—	4	4	no	—
93	C₄×D₄	—	8	4	no	—
94	#117	—	4	4	no	—
95	#117	C₂×D₄	1	2	yes	—
96	C₂×C₄⋊C₄	—	2	4	no	—
97	C₂×C₄⋊C₄	—	4	4	no	—
98	C₂×C₄⋊C₄	—	8	2	no	—
99	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
100	Q₃₂	C₄⋊C₄	1	4	yes	complemented
101	SD₃₂	C₄⋊C₄	1	8	yes	complemented
102	D₁₆	C₄⋊C₄	1	4	yes	complemented
103	M₃₂	—	2	2	no	—
104	M₃₂	—	2	2	no	—
105	C₂×C₁₆	—	2	1	no	—
106	C₂×C₁₆	—	2	1	no	—
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	C₂²×C₄	1	2	yes	—
109	C₂×C₁₆	C₂²×C₄	1	2	yes	—
110	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
111	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
112	C₂×C₁₆	C₂×Q₈	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	2	2	no	—
115	#108	C₂²×C₄	1	4	yes	—
116	#107	—	2	4	no	—
117	#107	—	2	4	no	—
118	#107	—	4	2	no	—
119	#107	—	4	2	no	—
120	#107	—	4	4	no	—
121	#107	D₈	1	2	yes	—
122	#106	—	4	4	no	—
123	#106	D₈	1	2	yes	—
124	#104	—	2	8	no	—
125	#103	—	4	4	no	—
126	#102	—	4	4	no	—
127	#97	—	2	2	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	C₂²×C₄	1	2	yes	—
130	C₄×C₈	#41	1	2	yes	—
131	#41	—	2	4	no	—
132	#41	—	2	4	no	—
133	#41	—	4	2	no	—
134	#41	—	4	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×D₄	—	2	2	no	—
137	C₂×D₄	—	4	2	no	—
138	C₂²×C₄	—	2	1	no	—
139	C₂²×C₄	—	4	2	no	—
140	C₂²×C₄	—	4	4	no	—
141	C₂²×C₄	—	8	4	no	—
142	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
143	Q₁₆	—	2	2	no	—
144	Q₁₆	C₂×C₄⋊C₄	1	4	yes	—
145	SD₁₆	—	2	4	no	—
146	SD₁₆	—	2	4	no	—
147	D₈	—	2	2	no	—
148	D₈	C₂×C₄⋊C₄	1	4	yes	—
149	M₁₆	—	2	4	no	—
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	8	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	8	no	—
157	C₂×C₈	C₄×D₄	1	2	yes	—
158	C₂×C₈	C₂×D₈	1	1	yes	characteristic
159	C₂×C₈	C₂×D₈	1	1	yes	characteristic
160	C₂×C₈	#135	1	2	yes	—
161	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₈	C₂×#41	1	1	yes	characteristic
165	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₈	2₋¹⁺⁴	1	1	yes	characteristic
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	4	4	no	—
169	C₄⋊C₄	—	8	4	no	—
170	C₄⋊C₄	—	16	4	no	—
171	C₄⋊C₄	C₂×D₈	1	4	yes	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	8	4	no	—
174	C₄²	—	2	2	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	8	4	no	—
177	C₄²	C₄×D₄	1	2	yes	—
178	C₁₆	—	2	2	no	—
179	C₁₆	—	4	2	no	—
180	C₁₆	C₂×C₄⋊C₄	1	2	yes	—
181	C₁₆	C₂×C₄⋊C₄	1	2	yes	—
182	C₂³	—	4	2	no	—
183	C₂³	#576	1	1	yes	characteristic
184	Q₈	—	2	4	no	—
185	Q₈	—	4	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	4	no	—
188	D₄	—	4	4	no	—
189	D₄	—	4	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	8	4	no	—
197	C₂×C₄	—	16	4	no	—
198	C₂×C₄	—	32	4	no	—
199	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
200	C₂×C₄	D₄²	1	1	yes	characteristic
201	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
202	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
203	C₂×C₄	C₂×#135	1	1	yes	characteristic
204	C₂×C₄	#578	1	1	yes	characteristic
205	C₈	—	4	4	no	—
206	C₈	—	4	2	no	—
207	C₈	C₄×D₈	1	2	yes	—
208	C₈	C₄×D₈	1	2	yes	—
209	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₈	#520	1	1	yes	characteristic
211	C₈	#547	1	1	yes	characteristic
212	C₈	#554	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	4	4	no	—
215	C₂²	—	8	2	no	—
216	C₂²	—	8	2	no	—
217	C₂²	#2951	1	1	yes	characteristic
218	C₂²	D₄×D₈	1	1	yes	characteristic
219	C₂²	#3380	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	4	4	no	—
222	C₄	—	4	4	no	—
223	C₄	—	16	4	no	—
224	C₄	D₄×C₄⋊C₄	1	1	yes	characteristic
225	C₄	C₂×C₄×D₈	1	1	yes	characteristic
226	C₄	#3283	1	1	yes	characteristic
227	C₄	Q₈×D₈	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	8	4	no	—
231	C₂	C₄⋊C₄×D₈	1	1	yes	characteristic
232	C₂	#33856	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #22692	C₂	semidirect
4. #13172	C₂	semidirect
5. #12946	C₂	semidirect
6. #12906	C₂	semidirect
7. #12821	C₂	semidirect
8. #12664	C₂	semidirect
13. #2178	C₄	semidirect
44. #443	C₂×C₄	semidirect
62. #368	C₂×C₄	semidirect
67. #359	C₂×C₄	semidirect
72. C₄×C₁₆	C₂×C₄	semidirect
100. Q₃₂	C₄⋊C₄	semidirect
101. SD₃₂	C₄⋊C₄	semidirect
102. D₁₆	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2178	2
#33460	2
#22692	2
#13172	2
#12946	2
#12906	2
#12821	2
#12664	2