Subgroups — #494091

Order 512.

624 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	C₂×#2108	C₂	1	1	yes	maximal, characteristic
2	#20807	C₂	1	2	yes	maximal
3	#12892	C₂	1	4	yes	maximal, complemented
4	#12891	C₂	1	4	yes	maximal, complemented
5	#12559	C₂	1	2	yes	maximal
6	C₂×#1381	C₂	1	2	yes	maximal
7	C₂²×M₃₂	C₂²	1	1	yes	characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#430	—	2	2	no	—
10	C₂²×#108	C₂²	1	4	yes	—
11	#2905	—	2	2	no	—
12	C₂×#405	C₂²	1	1	yes	characteristic
13	#2108	C₄	1	8	yes	—
14	C₂×#346	C₂²	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	#1849	—	2	4	no	—
17	#1848	C₂²	1	8	yes	—
18	#1381	C₂²	1	4	yes	—
19	#1381	C₂²	1	4	yes	—
20	#1381	C₂²	1	8	yes	—
21	C₂²×M₁₆	—	2	4	no	—
22	C₂²×M₁₆	D₄	1	1	yes	characteristic
23	C₂³×C₈	C₂³	1	1	yes	characteristic
24	C₂×#117	D₄	1	1	yes	characteristic
25	C₂×M₃₂	—	2	4	no	—
26	C₂×M₃₂	C₂×C₄	1	4	yes	—
27	C₂²×C₁₆	—	2	4	no	—
28	C₂²×C₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₁₆	C₂³	1	1	yes	characteristic
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	#430	—	4	8	no	—
33	C₂×#108	—	2	8	no	—
34	C₂×#108	—	2	8	no	—
35	C₂×#108	—	2	8	no	—
36	C₂×#108	—	2	4	no	—
37	C₂×#108	C₂³	1	8	yes	—
38	#424	—	4	8	no	—
39	#423	—	2	8	no	—
40	#405	C₂×C₄	1	4	yes	—
41	#405	Q₈	1	4	yes	—
42	C₂×#97	D₄	1	2	yes	—
43	C₂×C₄×C₈	C₂³	1	2	yes	—
44	#346	C₂×C₄	1	8	yes	—
45	C₄×C₁₆	C₂×C₄	1	8	yes	—
46	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×M₁₆	—	2	8	no	—
48	C₂×M₁₆	—	4	4	no	—
49	C₂×M₁₆	—	4	8	no	—
50	C₂×M₁₆	—	4	8	no	—
51	C₂×M₁₆	C₄⋊C₄	1	4	yes	—
52	C₂×M₁₆	SD₁₆	1	8	yes	—
53	C₂²×C₈	—	4	4	no	—
54	C₂²×C₈	C₂²×C₄	1	2	yes	—
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₈	C₂×D₄	1	2	yes	—
57	C₂²×C₈	C₂×Q₈	1	2	yes	—
58	C₂²×C₈	#41	1	4	yes	—
59	C₂²×C₈	#41	1	4	yes	—
60	C₂²×C₈	C₂⁴	1	1	yes	characteristic
61	#117	C₄⋊C₄	1	4	yes	—
62	#117	D₈	1	4	yes	—
63	C₂×C₄⋊C₄	Q₁₆	1	2	yes	—
64	C₂×C₂²⋊C₄	Q₁₆	1	2	yes	—
65	C₂×C₄²	C₂×D₄	1	2	yes	—
66	M₃₂	—	2	8	no	—
67	C₂×C₁₆	—	2	8	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂²×C₄	1	2	yes	—
72	C₂×C₁₆	C₂²×C₄	1	4	yes	—
73	C₂×C₁₆	C₂²×C₄	1	2	yes	—
74	#108	—	2	8	no	—
75	#108	—	2	8	no	—
76	#108	—	4	8	no	—
77	#108	—	4	16	no	—
78	C₄⋊C₈	—	4	16	no	—
79	#97	C₄⋊C₄	1	8	yes	—
80	C₄×C₈	C₂²×C₄	1	4	yes	—
81	C₄×C₈	C₂×Q₈	1	4	yes	—
82	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
84	C₂²×C₄	#124	1	4	yes	—
85	C₂²×C₄	C₂×D₈	1	2	yes	—
86	C₂²×C₄	C₂×SD₁₆	1	4	yes	—
87	C₂²×C₄	C₂×Q₁₆	1	4	yes	—
88	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
90	M₁₆	—	4	8	no	—
91	M₁₆	—	4	8	no	—
92	M₁₆	—	8	8	no	—
93	M₁₆	#106	1	16	yes	—
94	C₂×C₈	—	4	8	no	—
95	C₂×C₈	—	8	8	no	—
96	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₈	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₈	#117	1	4	yes	—
99	C₂×C₈	#117	1	4	yes	—
100	C₂×C₈	#124	1	4	yes	—
101	C₂×C₈	#125	1	8	yes	—
102	C₂×C₈	C₂×SD₁₆	1	8	yes	—
103	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₈	C₂×#41	1	2	yes	—
106	C₂×C₈	C₂×#41	1	2	yes	—
107	C₄⋊C₄	#107	1	8	yes	—
108	C₂²⋊C₄	#107	1	8	yes	—
109	C₄²	C₂×C₄⋊C₄	1	4	yes	—
110	C₄²	C₂×D₈	1	4	yes	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	2	no	—
113	C₁₆	—	2	4	no	—
114	C₂³	—	4	2	no	—
115	C₂³	C₂×#107	1	2	yes	—
116	C₂³	#487	1	4	yes	—
117	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	#382	1	4	yes	—
120	C₂×C₄	C₂×#106	1	4	yes	—
121	C₂×C₄	C₂×#107	1	8	yes	—
122	C₂×C₄	C₂×#107	1	2	yes	—
123	C₂×C₄	#486	1	8	yes	—
124	C₂×C₄	#487	1	4	yes	—
125	C₂×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₂×#124	1	2	yes	—
127	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
128	C₂×C₄	C₂²×SD₁₆	1	2	yes	—
129	C₈	—	8	8	no	—
130	C₈	#382	1	4	yes	—
131	C₈	C₂×#106	1	8	yes	—
132	C₈	C₂×#117	1	1	yes	characteristic
133	C₈	C₂×#117	1	1	yes	characteristic
134	C₈	C₂×#125	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	4	no	—
137	C₂²	—	4	4	no	—
138	C₂²	#1834	1	4	yes	—
139	C₂²	C₂²×#107	1	1	yes	characteristic
140	C₂²	C₂×#487	1	2	yes	—
141	C₄	—	2	8	no	—
142	C₄	#1834	1	4	yes	—
143	C₄	C₂×#382	1	1	yes	characteristic
144	C₄	C₂²×#106	1	1	yes	characteristic
145	C₄	C₂×#486	1	2	yes	—
146	C₂	—	2	4	no	—
147	C₂	—	2	2	no	—
148	C₂	C₂×#1834	1	1	yes	characteristic
149	C₂	#33414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ 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. #12892	C₂	semidirect
4. #12891	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2108	2
#20807	2
#12892	2
#12891	2
#12559	2
C₂×#1381	2