Subgroups — #225760

Order 512.

521 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22733	C₂	1	2	yes	maximal
2	#18042	C₂	1	2	yes	maximal
3	C₂×#1591	C₂	1	1	yes	maximal, characteristic
4	#9625	C₂	1	2	yes	maximal
5	#8909	C₂	1	4	yes	maximal
6	#8822	C₂	1	1	yes	maximal, characteristic
7	#8663	C₂	1	2	yes	maximal
8	#8293	C₂	1	1	yes	maximal, characteristic
9	Q₈×M₁₆	C₂²	1	4	yes	—
10	C₂×#446	C₂²	1	2	yes	—
11	C₂×C₄×Q₁₆	C₂²	1	1	yes	characteristic
12	#1950	—	2	4	no	—
13	#1950	C₂²	1	4	yes	—
14	#1849	—	2	2	no	—
15	#1848	C₂²	1	2	yes	—
16	#1778	—	2	2	no	—
17	C₄×#108	C₂²	1	1	yes	characteristic
18	#1593	—	2	2	no	—
19	#1591	C₄	1	2	yes	—
20	#1591	C₄	1	2	yes	—
21	#1591	C₄	1	2	yes	—
22	#1591	C₄	1	2	yes	—
23	#1587	—	2	4	no	—
24	#1579	—	2	2	no	—
25	#1566	C₂²	1	2	yes	—
26	#1566	C₂²	1	2	yes	—
27	#1564	C₂²	1	1	yes	characteristic
28	C₂×#335	C₂²	1	1	yes	characteristic
29	#1487	C₂²	1	4	yes	—
30	C₂×Q₈⋊C₈	C₂²	1	2	yes	—
31	#1451	C₂²	1	2	yes	—
32	C₂×C₈²	C₂²	1	1	yes	characteristic
33	#1301	C₂²	1	4	yes	—
34	#1278	C₂²	1	2	yes	—
35	C₂²×Q₁₆	Q₈	1	1	yes	characteristic
36	C₂×C₄×Q₈	C₂³	1	2	yes	—
37	#446	—	2	4	no	—
38	#446	—	2	2	no	—
39	#446	—	2	2	no	—
40	#446	—	4	4	no	—
41	#446	C₂×C₄	1	4	yes	—
42	#446	C₂×C₄	1	4	yes	—
43	C₈×Q₈	—	2	4	no	—
44	C₄×Q₁₆	C₂×C₄	1	2	yes	—
45	C₄×Q₁₆	C₂×C₄	1	2	yes	—
46	C₄×Q₁₆	D₄	1	2	yes	—
47	C₄×Q₁₆	D₄	1	2	yes	—
48	#431	C₂³	1	2	yes	—
49	C₂×#108	—	2	2	no	—
50	C₂×#108	—	2	4	no	—
51	C₂×#108	—	2	2	no	—
52	C₂×#108	D₄	1	2	yes	—
53	C₂×#107	Q₈	1	1	yes	characteristic
54	#423	—	2	4	no	—
55	#423	C₂³	1	4	yes	—
56	C₂×C₄⋊C₈	—	2	2	no	—
57	C₂×C₄⋊C₈	C₂³	1	2	yes	—
58	C₂×#103	Q₈	1	2	yes	—
59	C₄×M₁₆	—	2	4	no	—
60	C₄×M₁₆	C₂³	1	2	yes	—
61	C₂×#97	—	2	2	no	—
62	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
64	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
65	#341	—	2	4	no	—
66	#341	D₄	1	2	yes	—
67	#335	—	2	1	no	—
68	#335	C₂×C₄	1	2	yes	—
69	#335	C₂×C₄	1	2	yes	—
70	#334	—	2	2	no	—
71	#334	—	2	2	no	—
72	#334	—	2	1	no	—
73	#334	—	4	2	no	—
74	Q₈⋊C₈	—	2	4	no	—
75	Q₈⋊C₈	—	2	4	no	—
76	Q₈⋊C₈	C₂×C₄	1	4	yes	—
77	Q₈⋊C₈	C₂×C₄	1	4	yes	—
78	#322	—	2	2	no	—
79	#322	—	2	2	no	—
80	C₈²	C₂×C₄	1	1	yes	characteristic
81	C₈²	C₂×C₄	1	1	yes	characteristic
82	C₈²	C₂×C₄	1	1	yes	characteristic
83	C₈²	C₂×C₄	1	1	yes	characteristic
84	C₂²×Q₈	C₂×Q₈	1	2	yes	—
85	C₂×Q₁₆	—	4	2	no	—
86	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
87	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
88	C₂×M₁₆	—	2	4	no	—
89	C₂×M₁₆	C₂×D₄	1	4	yes	—
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₄×Q₈	—	2	2	no	—
97	C₄×Q₈	—	2	2	no	—
98	C₄×Q₈	C₂²×C₄	1	4	yes	—
99	C₄×Q₈	C₂×D₄	1	4	yes	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	C₂×Q₈	1	2	yes	—
102	C₂×C₄²	—	2	2	no	—
103	C₂×C₄²	C₂⁴	1	1	yes	characteristic
104	#108	—	2	4	no	—
105	#108	—	4	4	no	—
106	#108	—	8	2	no	—
107	#108	—	8	4	no	—
108	#107	C₄⋊C₄	1	2	yes	—
109	#107	C₄⋊C₄	1	2	yes	—
110	C₄⋊C₈	—	2	4	no	—
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	4	no	—
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	2	4	no	—
116	C₄⋊C₈	—	4	4	no	—
117	C₄⋊C₈	—	4	4	no	—
118	C₄⋊C₈	C₂²×C₄	1	4	yes	—
119	#103	C₄⋊C₄	1	4	yes	—
120	#103	C₄⋊C₄	1	4	yes	—
121	C₂²⋊C₈	—	2	2	no	—
122	#97	—	2	4	no	—
123	#97	—	2	4	no	—
124	#97	—	2	2	no	—
125	#97	—	4	4	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	4	4	no	—
130	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₄×C₈	C₂×D₄	1	1	yes	characteristic
137	C₄×C₈	C₂×D₄	1	1	yes	characteristic
138	C₄×C₈	#41	1	1	yes	characteristic
139	C₄×C₈	#41	1	1	yes	characteristic
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	4	2	no	—
142	C₂×Q₈	—	4	2	no	—
143	C₂×Q₈	C₂×C₄⋊C₄	1	4	yes	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₂²×Q₈	1	1	yes	characteristic
148	C₂²×C₄	2₋¹⁺⁴	1	1	yes	characteristic
149	Q₁₆	—	2	4	no	—
150	Q₁₆	—	4	2	no	—
151	M₁₆	—	2	4	no	—
152	M₁₆	—	2	4	no	—
153	M₁₆	—	4	4	no	—
154	M₁₆	—	4	4	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	4	1	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	4	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₄×D₄	1	1	yes	characteristic
172	C₂×C₈	C₄×D₄	1	1	yes	characteristic
173	C₂×C₈	C₄×D₄	1	1	yes	characteristic
174	C₂×C₈	C₄×D₄	1	1	yes	characteristic
175	C₂×C₈	#137	1	4	yes	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	C₂×C₄⋊C₄	1	4	yes	—
180	C₄²	—	2	4	no	—
181	C₄²	—	2	4	no	—
182	C₄²	C₂³×C₄	1	1	yes	characteristic
183	C₄²	C₂²×D₄	1	1	yes	characteristic
184	C₄²	C₂×#41	1	1	yes	characteristic
185	C₄²	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂³	D₄×Q₈	1	1	yes	characteristic
187	Q₈	—	2	4	no	—
188	Q₈	—	2	4	no	—
189	Q₈	—	8	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
199	C₂×C₄	#520	1	1	yes	characteristic
200	C₂×C₄	D₄²	1	1	yes	characteristic
201	C₂×C₄	#547	1	1	yes	characteristic
202	C₂×C₄	#554	1	1	yes	characteristic
203	C₂×C₄	C₂×#137	1	2	yes	—
204	C₂×C₄	#577	1	2	yes	—
205	C₈	—	2	2	no	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	2	1	no	—
210	C₈	—	2	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	4	4	no	—
213	C₈	—	4	4	no	—
214	C₈	—	8	2	no	—
215	C₈	—	8	4	no	—
216	C₈	#430	1	2	yes	—
217	C₈	#441	1	2	yes	—
218	C₈	#441	1	2	yes	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	D₄×C₄⋊C₄	1	1	yes	characteristic
222	C₂²	#3387	1	2	yes	—
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	4	4	no	—
227	C₄	—	4	4	no	—
228	C₄	C₂×#430	1	1	yes	characteristic
229	C₄	#2916	1	1	yes	characteristic
230	C₄	C₂×#441	1	1	yes	characteristic
231	C₄	#2952	1	1	yes	characteristic
232	C₄	#3278	1	2	yes	—
233	C₂	—	2	2	no	—
234	C₂	#22669	1	1	yes	characteristic
235	C₂	#22674	1	1	yes	characteristic
236	C₂	#22694	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂³	D₄×Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

Maximal subgroups

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

subgroup	index
#22733	2
#18042	2
C₂×#1591	2
#9625	2
#8909	2
#8822	2
#8663	2
#8293	2