Subgroups — #7623221

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 203 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#60666	C₂	1	2	yes	maximal
2	#33732	C₂	1	6	yes	maximal, complemented
3	#33730	C₂	1	6	yes	maximal, complemented
4	#33729	C₂	1	6	yes	maximal, complemented
5	#33726	C₂	1	3	yes	maximal, complemented
6	C₂×#2253	C₂	1	3	yes	maximal, complemented
7	#33411	C₂	1	2	yes	maximal
8	#33410	C₂	1	2	yes	maximal
9	C₄×#504	C₂	1	1	yes	maximal, characteristic
10	#3504	—	2	2	no	—
11	C₂×#504	C₂²	1	1	yes	characteristic
12	C₂×#504	C₂²	1	1	yes	characteristic
13	#3166	—	2	6	no	—
14	#3164	C₂²	1	12	yes	—
15	#3163	—	2	6	no	—
16	#3160	C₂²	1	6	yes	—
17	C₂×#499	C₂²	1	6	yes	—
18	#2913	C₂²	1	4	yes	—
19	#2912	—	2	2	no	—
20	C₄×#131	C₂²	1	1	yes	characteristic
21	#2255	C₂²	1	24	yes	—
22	#2254	C₂²	1	6	yes	—
23	#2253	C₂²	1	3	yes	—
24	#2253	C₂²	1	6	yes	—
25	#2253	C₂²	1	6	yes	—
26	#2253	C₂²	1	6	yes	—
27	#2253	C₂²	1	1	yes	characteristic
28	#2253	C₂²	1	6	yes	—
29	#2252	C₂²	1	6	yes	—
30	#2159	C₂²	1	12	yes	—
31	#2159	C₂²	1	12	yes	—
32	#2158	C₂²	1	6	yes	—
33	#2158	C₂²	1	6	yes	—
34	C₂×#367	C₂²	1	6	yes	—
35	C₂×#366	C₂²	1	6	yes	—
36	#2108	C₂²	1	3	yes	—
37	#2108	C₂²	1	3	yes	—
38	C₄×M₃₂	C₂²	1	3	yes	—
39	C₂×C₄×C₁₆	C₂²	1	3	yes	—
40	C₂×#131	D₄	1	1	yes	characteristic
41	C₂×#131	C₂³	1	1	yes	characteristic
42	#557	—	4	4	no	—
43	#554	—	4	12	no	—
44	#544	—	2	12	no	—
45	#533	—	2	6	no	—
46	C₂×#125	—	2	6	no	—
47	#520	—	2	4	no	—
48	C₄×#41	D₄	1	1	yes	characteristic
49	#504	Q₈	1	8	yes	complemented
50	#504	Q₈	1	8	yes	complemented
51	C₂×M₃₂	C₂³	1	3	yes	—
52	C₂×M₃₂	C₂³	1	3	yes	—
53	C₂²×C₁₆	C₂³	1	3	yes	—
54	C₂²×C₁₆	C₂³	1	3	yes	—
55	#501	—	2	24	no	—
56	#500	C₂³	1	12	yes	—
57	#499	—	2	6	no	—
58	#499	—	2	6	no	—
59	#499	—	2	6	no	—
60	#499	C₂³	1	6	yes	—
61	#499	C₂³	1	12	yes	—
62	#499	C₂³	1	2	yes	—
63	#498	—	2	6	no	—
64	#428	—	2	12	no	—
65	#428	—	2	12	no	—
66	#427	—	2	6	no	—
67	#427	C₂³	1	12	yes	—
68	C₂×#107	C₂³	1	12	yes	—
69	C₂×#106	—	2	6	no	—
70	#405	D₄	1	3	yes	—
71	#405	C₂³	1	3	yes	—
72	C₄×M₁₆	D₄	1	3	yes	—
73	C₂×C₄×C₈	C₂³	1	3	yes	—
74	#367	C₂³	1	6	yes	—
75	#367	C₂³	1	12	yes	—
76	#367	C₂³	1	2	yes	—
77	#367	C₂³	1	12	yes	—
78	#366	C₂³	1	6	yes	—
79	#366	C₂³	1	12	yes	—
80	#366	C₂³	1	2	yes	—
81	#366	C₂³	1	12	yes	—
82	#346	C₂³	1	6	yes	—
83	C₄×C₁₆	C₂³	1	1	yes	characteristic
84	C₄×C₁₆	C₂³	1	3	yes	—
85	C₄×C₁₆	C₂³	1	3	yes	—
86	C₄×C₁₆	C₂³	1	3	yes	—
87	C₂×#41	C₂×D₄	1	1	yes	characteristic
88	#131	Q₁₆	1	8	yes	—
89	#131	C₂×Q₈	1	4	yes	—
90	#131	C₂×Q₈	1	4	yes	—
91	C₂×M₁₆	C₂×D₄	1	3	yes	—
92	C₂×M₁₆	C₂×D₄	1	3	yes	—
93	C₂²×C₈	C₂×D₄	1	3	yes	—
94	C₂²×C₈	C₂⁴	1	3	yes	—
95	C₄⋊Q₈	—	2	12	no	—
96	C₄⋊Q₈	—	4	12	no	—
97	#125	—	2	6	no	—
98	#125	—	2	12	no	—
99	#125	—	2	2	no	—
100	#125	—	4	12	no	—
101	#125	—	4	12	no	—
102	C₂²⋊Q₈	—	4	12	no	—
103	C₂²⋊Q₈	—	4	24	no	—
104	C₄×Q₈	—	4	4	no	—
105	C₄×Q₈	—	4	12	no	—
106	C₄×Q₈	D₈	1	2	yes	—
107	C₄×D₄	—	4	12	no	—
108	C₄×D₄	D₈	1	6	yes	—
109	#117	—	2	12	no	—
110	#117	C₂×D₄	1	3	yes	—
111	C₂×C₄⋊C₄	—	2	12	no	—
112	C₂×C₄⋊C₄	—	4	12	no	—
113	C₂×C₄²	C₂×D₄	1	3	yes	—
114	M₃₂	C₂×Q₈	1	12	yes	—
115	M₃₂	C₂×Q₈	1	12	yes	—
116	C₂×C₁₆	C₂×Q₈	1	12	yes	—
117	C₂×C₁₆	C₂×Q₈	1	12	yes	—
118	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
119	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
120	C₂×C₁₆	C₂⁴	1	3	yes	—
121	C₂×C₁₆	C₂⁴	1	3	yes	—
122	#107	—	2	12	no	—
123	#107	—	2	12	no	—
124	#107	C₂⁴	1	12	yes	—
125	#107	C₂⁴	1	4	yes	—
126	#106	—	2	6	no	—
127	#106	—	2	12	no	—
128	#106	—	2	2	no	—
129	#106	—	2	12	no	—
130	#97	C₂×D₄	1	6	yes	—
131	C₄×C₈	C₂×D₄	1	3	yes	—
132	C₄×C₈	C₂×D₄	1	3	yes	—
133	C₄×C₈	C₂⁴	1	1	yes	characteristic
134	C₄×C₈	C₂⁴	1	3	yes	—
135	#41	C₄⋊Q₈	1	4	yes	—
136	#41	C₂×Q₁₆	1	4	yes	—
137	C₂×Q₈	—	4	12	no	—
138	C₂×Q₈	C₂×D₈	1	1	yes	characteristic
139	C₂×D₄	C₂×D₈	1	3	yes	—
140	C₂²×C₄	—	4	12	no	—
141	C₂²×C₄	C₂×D₈	1	6	yes	—
142	C₂²×C₄	C₂²×D₄	1	3	yes	—
143	M₁₆	C₄⋊Q₈	1	12	yes	—
144	M₁₆	C₂×Q₁₆	1	12	yes	—
145	C₂×C₈	C₂×Q₁₆	1	12	yes	—
146	C₂×C₈	C₂²×D₄	1	3	yes	—
147	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₈	C₂²×D₄	1	3	yes	—
149	C₂×C₈	C₂²×Q₈	1	6	yes	—
150	C₂×C₈	C₂²×Q₈	1	6	yes	—
151	C₂×C₈	C₂⁵	1	1	yes	characteristic
152	C₄⋊C₄	—	2	12	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	12	no	—
155	C₄⋊C₄	—	4	12	no	—
156	C₄⋊C₄	—	4	24	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	12	no	—
159	C₄⋊C₄	C₂×D₈	1	6	yes	—
160	C₂²⋊C₄	—	4	12	no	—
161	C₂²⋊C₄	C₂×D₈	1	6	yes	—
162	C₄²	—	4	12	no	—
163	C₄²	C₂×D₈	1	6	yes	—
164	C₄²	C₂²×D₄	1	1	yes	characteristic
165	C₄²	C₂²×D₄	1	3	yes	—
166	C₁₆	C₂²×Q₈	1	2	yes	—
167	C₁₆	C₂²×Q₈	1	2	yes	—
168	C₁₆	C₂²×Q₈	1	6	yes	—
169	C₁₆	C₂²×Q₈	1	6	yes	—
170	C₂³	C₂²×D₈	1	3	yes	—
171	Q₈	—	16	12	no	—
172	Q₈	#500	1	4	yes	—
173	D₄	#500	1	12	yes	—
174	C₂×C₄	—	4	6	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	12	no	—
177	C₂×C₄	—	16	12	no	—
178	C₂×C₄	C₂×C₄⋊Q₈	1	6	yes	—
179	C₂×C₄	C₂²×D₈	1	2	yes	—
180	C₂×C₄	C₂²×D₈	1	6	yes	—
181	C₂×C₄	C₂²×D₈	1	3	yes	—
182	C₂×C₄	C₂²×Q₁₆	1	6	yes	—
183	C₂×C₄	C₂³×D₄	1	1	yes	characteristic
184	C₈	C₂×C₄⋊Q₈	1	6	yes	—
185	C₈	C₂²×Q₁₆	1	2	yes	—
186	C₈	C₂²×Q₁₆	1	6	yes	—
187	C₈	C₂³×Q₈	1	1	yes	characteristic
188	C₈	C₂³×Q₈	1	1	yes	characteristic
189	C₂²	—	2	6	no	—
190	C₂²	C₂×#500	1	6	yes	—
191	C₂²	C₂³×D₈	1	1	yes	characteristic
192	C₄	—	2	2	no	—
193	C₄	—	2	6	no	—
194	C₄	—	8	12	no	—
195	C₄	—	8	4	no	—
196	C₄	C₂×#500	1	6	yes	—
197	C₄	C₂²×C₄⋊Q₈	1	1	yes	characteristic
198	C₄	C₂³×Q₁₆	1	1	yes	characteristic
199	C₂	—	2	6	no	—
200	C₂	C₂²×#500	1	1	yes	characteristic
201	C₂	#62635	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #33732	C₂	semidirect
3. #33730	C₂	semidirect
4. #33729	C₂	semidirect
5. #33726	C₂	semidirect
6. C₂×#2253	C₂	semidirect
49. #504	Q₈	semidirect
50. #504	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#60666	2
#33732	2
#33730	2
#33729	2
#33726	2
C₂×#2253	2
#33411	2
#33410	2
C₄×#504	2