Subgroups — #353044

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 151 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33291	C₂	1	2	yes	maximal
2	#33258	C₂	1	2	yes	maximal
3	#33256	C₂	1	2	yes	maximal
4	#8817	C₂	1	2	yes	maximal
5	#8816	C₂	1	2	yes	maximal
6	#8815	C₂	1	2	yes	maximal
7	#8814	C₂	1	2	yes	maximal
8	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
9	#2859	C₂²	1	4	yes	—
10	#1991	—	2	8	no	—
11	#1942	—	2	8	no	—
12	#1941	—	2	8	no	—
13	#1845	C₂²	1	4	yes	—
14	#1844	C₂²	1	4	yes	—
15	#1834	C₂²	1	4	yes	—
16	#1834	C₂²	1	4	yes	—
17	#1834	C₂²	1	4	yes	—
18	#1834	C₂²	1	4	yes	—
19	C₄²×C₈	C₂²	1	1	yes	characteristic
20	C₄²×C₈	C₂²	1	1	yes	characteristic
21	C₄²×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₈²	C₂²	1	1	yes	characteristic
23	C₂×C₈²	C₂²	1	1	yes	characteristic
24	C₂×C₈²	C₂²	1	1	yes	characteristic
25	C₂×C₈²	C₂²	1	1	yes	characteristic
26	C₂×#125	—	2	4	no	—
27	C₂×#107	—	2	8	no	—
28	C₂×#106	—	2	8	no	—
29	C₂×C₄×C₈	D₄	1	4	yes	—
30	C₂×C₄×C₈	D₄	1	4	yes	—
31	C₂×C₄×C₈	D₄	1	4	yes	—
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#389	—	2	8	no	—
39	#389	—	2	8	no	—
40	#384	C₂³	1	8	yes	—
41	#382	—	2	4	no	—
42	C₄³	C₂³	1	1	yes	characteristic
43	#340	—	2	8	no	—
44	#340	—	2	8	no	—
45	#340	—	2	8	no	—
46	#340	—	2	8	no	—
47	C₈²	Q₈	1	4	yes	—
48	C₈²	Q₈	1	4	yes	—
49	C₈²	Q₈	1	4	yes	—
50	C₈²	Q₈	1	4	yes	—
51	C₂²×C₈	C₂×D₄	1	2	yes	—
52	C₂²×C₈	C₂×D₄	1	2	yes	—
53	C₂²×C₈	C₂×D₄	1	2	yes	—
54	C₂²×C₈	C₂×D₄	1	2	yes	—
55	C₂²×C₈	C₂×D₄	1	2	yes	—
56	C₂²×C₈	C₂×D₄	1	2	yes	—
57	#125	—	4	16	no	—
58	C₂×C₄⋊C₄	—	2	8	no	—
59	C₂×C₄⋊C₄	—	2	8	no	—
60	C₂×C₄⋊C₄	—	2	8	no	—
61	C₂×C₄⋊C₄	—	4	8	no	—
62	C₂×C₄²	C₂×D₄	1	2	yes	—
63	C₂×C₄²	C₂×D₄	1	2	yes	—
64	C₂×C₄²	C₂×D₄	1	2	yes	—
65	C₂×C₄²	C₂⁴	1	1	yes	characteristic
66	#107	—	4	16	no	—
67	#106	—	4	16	no	—
68	C₄×C₈	—	2	8	no	—
69	C₄×C₈	—	2	8	no	—
70	C₄×C₈	—	2	8	no	—
71	C₄×C₈	C₂×Q₈	1	2	yes	—
72	C₄×C₈	C₂×Q₈	1	2	yes	—
73	C₄×C₈	C₂×Q₈	1	2	yes	—
74	C₄×C₈	C₂×Q₈	1	2	yes	—
75	C₄×C₈	C₂×Q₈	1	2	yes	—
76	C₄×C₈	C₂×Q₈	1	2	yes	—
77	C₄×C₈	C₂×Q₈	1	2	yes	—
78	C₄×C₈	C₂×Q₈	1	2	yes	—
79	C₄×C₈	C₂×Q₈	1	2	yes	—
80	C₄×C₈	C₂×Q₈	1	2	yes	—
81	C₄×C₈	C₂×Q₈	1	2	yes	—
82	C₄×C₈	C₂×Q₈	1	2	yes	—
83	#95	—	4	8	no	—
84	C₂²×C₄	—	4	8	no	—
85	C₂²×C₄	#127	1	4	yes	—
86	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	C₄⋊Q₈	1	4	yes	—
94	C₂×C₈	C₄⋊Q₈	1	4	yes	—
95	C₂×C₈	C₄⋊Q₈	1	4	yes	—
96	C₂×C₈	C₄⋊Q₈	1	4	yes	—
97	C₂×C₈	C₄⋊Q₈	1	4	yes	—
98	C₂×C₈	C₄⋊Q₈	1	4	yes	—
99	C₂×C₈	#135	1	8	yes	—
100	C₂×C₈	#135	1	8	yes	—
101	C₄⋊C₄	—	4	16	no	—
102	C₄⋊C₄	—	4	16	no	—
103	C₄⋊C₄	—	4	16	no	—
104	C₄⋊C₄	—	8	16	no	—
105	C₄²	—	2	4	no	—
106	C₄²	—	2	4	no	—
107	C₄²	#135	1	8	yes	—
108	C₄²	C₂²×Q₈	1	1	yes	characteristic
109	C₄²	C₂²×Q₈	1	1	yes	characteristic
110	C₄²	C₂²×Q₈	1	1	yes	characteristic
111	C₄²	C₂²×Q₈	1	1	yes	characteristic
112	C₂³	C₂×#127	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	8	8	no	—
116	C₂×C₄	—	8	8	no	—
117	C₂×C₄	—	8	8	no	—
118	C₂×C₄	#495	1	8	yes	—
119	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₄	C₂×#135	1	4	yes	—
126	C₂×C₄	C₂×#135	1	4	yes	—
127	C₂×C₄	C₂×#135	1	4	yes	—
128	C₈	—	2	4	no	—
129	C₈	—	2	4	no	—
130	C₈	—	2	4	no	—
131	C₈	—	2	4	no	—
132	C₈	#499	1	8	yes	—
133	C₈	#499	1	8	yes	—
134	C₂²	#2868	1	1	yes	characteristic
135	C₂²	C₂×#495	1	2	yes	—
136	C₂²	#3147	1	2	yes	—
137	C₂²	#3147	1	2	yes	—
138	C₄	—	2	8	no	—
139	C₄	—	16	8	no	—
140	C₄	C₂×#499	1	2	yes	—
141	C₄	C₂×#499	1	2	yes	—
142	C₄	#3160	1	2	yes	—
143	C₄	#3160	1	2	yes	—
144	C₄	#3160	1	2	yes	—
145	C₄	#3160	1	2	yes	—
146	C₂	#19600	1	4	yes	—
147	C₂	#33311	1	1	yes	characteristic
148	C₂	#33311	1	1	yes	characteristic
149	C₂	#33313	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ 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
#33291	2
#33258	2
#33256	2
#8817	2
#8816	2
#8815	2
#8814	2
C₄×C₈²	2