Subgroups — #325802

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28827	C₂	1	2	yes	maximal
2	#28814	C₂	1	2	yes	maximal
3	#10579	C₂	1	4	yes	maximal
4	#9615	C₂	1	2	yes	maximal
5	#9614	C₂	1	2	yes	maximal
6	#8818	C₂	1	1	yes	maximal, characteristic
7	C₈×#107	C₂	1	1	yes	maximal, characteristic
8	C₈×#107	C₂	1	1	yes	maximal, characteristic
9	C₂×#499	—	2	2	no	—
10	#2552	C₂²	1	4	yes	—
11	#2083	—	2	4	no	—
12	#2059	—	2	4	no	—
13	#1948	C₂²	1	4	yes	—
14	#1947	C₂²	1	4	yes	—
15	#1942	—	2	2	no	—
16	#1942	—	2	2	no	—
17	#1845	—	2	2	no	—
18	#1845	C₂²	1	2	yes	—
19	#1845	C₂²	1	2	yes	—
20	#1845	C₂²	1	2	yes	—
21	C₄×#107	C₂²	1	1	yes	characteristic
22	C₄×#107	C₂²	1	1	yes	characteristic
23	C₈×C₄⋊C₄	C₂²	1	2	yes	—
24	C₈×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×#335	C₂²	1	1	yes	characteristic
26	C₂×#335	C₂²	1	1	yes	characteristic
27	C₂×C₈²	C₂²	1	1	yes	characteristic
28	#1296	C₂²	1	4	yes	—
29	#1279	C₂²	1	2	yes	—
30	#1279	C₂²	1	2	yes	—
31	C₂×#125	—	2	4	no	—
32	#499	—	4	8	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#107	—	2	2	no	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	—	2	4	no	—
38	C₂×#107	—	4	2	no	—
39	C₂×#107	D₄	1	1	yes	characteristic
40	C₂×#107	D₄	1	1	yes	characteristic
41	C₂×#107	D₄	1	1	yes	characteristic
42	C₂×#107	D₄	1	1	yes	characteristic
43	C₂×#106	—	2	4	no	—
44	C₂×C₄⋊C₈	C₂³	1	2	yes	—
45	C₂×C₄⋊C₈	C₂³	1	2	yes	—
46	C₂×C₄×C₈	—	2	2	no	—
47	C₂×C₄×C₈	—	2	2	no	—
48	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
51	#398	—	4	4	no	—
52	#389	—	2	4	no	—
53	#389	—	2	4	no	—
54	#384	—	2	4	no	—
55	#384	C₂³	1	4	yes	—
56	C₄×C₄⋊C₄	C₂³	1	2	yes	—
57	C₄×C₄⋊C₄	C₂³	1	2	yes	—
58	#340	—	2	4	no	—
59	#340	—	2	4	no	—
60	#340	D₄	1	2	yes	—
61	#340	D₄	1	2	yes	—
62	#336	—	2	2	no	—
63	#336	—	2	2	no	—
64	#335	Q₈	1	4	yes	—
65	#335	Q₈	1	4	yes	—
66	C₈²	Q₈	1	4	yes	—
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	2	2	no	—
69	C₂²×C₈	—	2	2	no	—
70	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
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	#125	—	4	8	no	—
75	#125	—	8	4	no	—
76	C₂×C₄⋊C₄	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	4	no	—
78	C₂×C₄⋊C₄	—	2	4	no	—
79	C₂×C₄⋊C₄	—	4	4	no	—
80	C₂×C₄⋊C₄	—	4	4	no	—
81	C₂×C₄⋊C₄	—	4	4	no	—
82	C₂×C₄⋊C₄	C₂×D₄	1	2	yes	—
83	C₂×C₄⋊C₄	C₂×D₄	1	2	yes	—
84	C₂×C₄⋊C₄	C₂×D₄	1	2	yes	—
85	C₂×C₄⋊C₄	C₂×D₄	1	2	yes	—
86	C₂×C₄²	—	2	2	no	—
87	C₂×C₄²	—	2	2	no	—
88	C₂×C₄²	C₂⁴	1	1	yes	characteristic
89	#107	—	2	2	no	—
90	#107	—	2	2	no	—
91	#107	—	4	4	no	—
92	#107	—	4	4	no	—
93	#107	—	8	4	no	—
94	#107	—	8	2	no	—
95	#107	—	8	2	no	—
96	#107	—	8	4	no	—
97	#107	D₈	1	4	yes	—
98	#107	Q₁₆	1	4	yes	—
99	#106	—	4	8	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	C₂×Q₈	1	4	yes	—
103	C₄⋊C₈	C₂×Q₈	1	4	yes	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	4	4	no	—
106	C₄×C₈	—	4	4	no	—
107	C₄×C₈	C₂×Q₈	1	2	yes	—
108	C₄×C₈	C₂×Q₈	1	2	yes	—
109	C₄×C₈	C₂×Q₈	1	2	yes	—
110	C₄×C₈	C₂×Q₈	1	2	yes	—
111	C₄×C₈	C₂×Q₈	1	2	yes	—
112	#95	—	2	2	no	—
113	#95	—	2	2	no	—
114	#95	—	4	4	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	2	no	—
119	C₂²×C₄	—	4	4	no	—
120	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
122	C₂²×C₄	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	C₄⋊Q₈	1	2	yes	—
137	C₂×C₈	C₄⋊Q₈	1	2	yes	—
138	C₂×C₈	C₄⋊Q₈	1	2	yes	—
139	C₂×C₈	C₄⋊Q₈	1	2	yes	—
140	C₂×C₈	C₂×D₈	1	2	yes	—
141	C₂×C₈	C₂×Q₁₆	1	2	yes	—
142	C₂×C₈	#135	1	2	yes	—
143	C₂×C₈	#135	1	2	yes	—
144	C₄⋊C₄	—	2	2	no	—
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₄⋊C₄	—	4	4	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	4	no	—
154	C₄⋊C₄	—	8	8	no	—
155	C₄⋊C₄	—	8	8	no	—
156	C₄⋊C₄	—	8	4	no	—
157	C₄⋊C₄	—	16	4	no	—
158	C₄⋊C₄	C₂×D₈	1	4	yes	—
159	C₄⋊C₄	C₂×Q₁₆	1	4	yes	—
160	C₄²	—	4	2	no	—
161	C₄²	—	4	2	no	—
162	C₄²	—	4	2	no	—
163	C₄²	—	4	2	no	—
164	C₄²	C₂²×Q₈	1	1	yes	characteristic
165	C₄²	C₂²×Q₈	1	1	yes	characteristic
166	C₄²	C₂²×Q₈	1	1	yes	characteristic
167	C₄²	2₋¹⁺⁴	1	1	yes	characteristic
168	C₂³	D₄²	1	1	yes	characteristic
169	C₂×C₄	—	2	1	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	4	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	8	4	no	—
184	C₂×C₄	—	8	4	no	—
185	C₂×C₄	—	8	4	no	—
186	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
188	C₂×C₄	#544	1	1	yes	characteristic
189	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
190	C₂×C₄	D₄×Q₈	1	1	yes	characteristic
191	C₂×C₄	Q₈²	1	1	yes	characteristic
192	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
193	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
194	C₂×C₄	C₂×#135	1	1	yes	characteristic
195	C₂×C₄	C₂×#135	1	1	yes	characteristic
196	C₂×C₄	#576	1	1	yes	characteristic
197	C₂×C₄	#576	1	1	yes	characteristic
198	C₂×C₄	#578	1	1	yes	characteristic
199	C₂×C₄	#578	1	1	yes	characteristic
200	C₈	—	2	4	no	—
201	C₈	—	2	2	no	—
202	C₈	—	2	2	no	—
203	C₈	—	4	4	no	—
204	C₈	—	4	4	no	—
205	C₈	#499	1	4	yes	complemented
206	C₈	#500	1	4	yes	—
207	C₂²	#2551	1	1	yes	characteristic
208	C₂²	D₄×D₈	1	1	yes	characteristic
209	C₂²	#3283	1	1	yes	characteristic
210	C₂²	D₄×Q₁₆	1	1	yes	characteristic
211	C₂²	#3287	1	1	yes	characteristic
212	C₂²	#3292	1	1	yes	characteristic
213	C₂²	#3293	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	—	16	4	no	—
221	C₄	C₂×#499	1	1	yes	characteristic
222	C₄	C₂×#500	1	1	yes	characteristic
223	C₄	#3164	1	1	yes	characteristic
224	C₄	#3164	1	1	yes	characteristic
225	C₄	Q₈×D₈	1	1	yes	characteristic
226	C₄	#3380	1	1	yes	characteristic
227	C₄	Q₈×Q₁₆	1	1	yes	characteristic
228	C₄	#3383	1	1	yes	characteristic
229	C₂	#19971	1	1	yes	characteristic
230	C₂	#19975	1	1	yes	characteristic
231	C₂	#20006	1	1	yes	characteristic
232	C₂	#20010	1	1	yes	characteristic
233	C₂	#28813	1	1	yes	characteristic
234	C₂	#28814	1	1	yes	characteristic
235	C₂	#28825	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
205. C₈	#499	semidirect

Maximal subgroups

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

subgroup	index
#28827	2
#28814	2
#10579	2
#9615	2
#9614	2
#8818	2
C₈×#107	2