Subgroups — C₄×#1368 #129089

Order 512.

300 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	C₄×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#1368	C₂	1	2	yes	maximal
3	C₄×#334	C₂	1	2	yes	maximal
4	#7156	C₂	1	2	yes	maximal
5	C₂×#346	C₄	1	1	yes	characteristic
6	C₂×#346	C₄	1	1	yes	characteristic
7	C₂×#346	C₂²	1	1	yes	characteristic
8	C₂×#346	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₄⋊C₈	—	2	2	no	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	C₂×#334	C₂²	1	4	yes	—
14	#1368	C₄	1	8	yes	direct product factor
15	#1277	—	2	2	no	—
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊C₈	—	2	4	no	—
21	C₂×C₄×C₈	—	4	2	no	—
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
26	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄³	D₄	1	1	yes	characteristic
29	#346	—	2	2	no	—
30	#346	—	2	2	no	—
31	#346	C₈	1	4	yes	complemented
32	#346	C₂×C₄	1	4	yes	—
33	C₄×C₁₆	—	2	2	no	—
34	C₄×C₁₆	C₈	1	4	yes	complemented
35	#336	—	4	2	no	—
36	#334	—	2	4	no	—
37	#334	C₂×C₄	1	8	yes	—
38	C₂²×C₈	—	4	4	no	—
39	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	SD₁₆	1	2	yes	—
46	C₂×C₄²	SD₁₆	1	2	yes	—
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₆	—	2	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	C₄²	1	2	yes	—
55	C₂×C₁₆	C₄²	1	2	yes	—
56	C₂×C₁₆	C₂×C₈	1	2	yes	—
57	C₂×C₁₆	C₂×C₈	1	2	yes	—
58	C₂×C₁₆	C₂×C₈	1	2	yes	—
59	C₂×C₁₆	C₂×C₈	1	2	yes	—
60	C₄⋊C₈	—	2	8	no	—
61	C₄⋊C₈	—	4	4	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	8	4	no	—
66	C₄×C₈	C₂×C₈	1	2	yes	—
67	C₄×C₈	C₂×C₈	1	2	yes	—
68	C₄×C₈	M₁₆	1	2	yes	—
69	C₄×C₈	M₁₆	1	2	yes	—
70	C₄×C₈	C₂²×C₄	1	2	yes	—
71	C₄×C₈	#41	1	2	yes	—
72	C₂²×C₄	#106	1	2	yes	—
73	C₂²×C₄	#106	1	1	yes	characteristic
74	C₂²×C₄	#106	1	1	yes	characteristic
75	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	4	8	no	—
84	C₂×C₈	—	8	4	no	—
85	C₂×C₈	—	8	4	no	—
86	C₂×C₈	C₄⋊C₈	1	2	yes	—
87	C₂×C₈	C₄⋊C₈	1	2	yes	—
88	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₈	#117	1	1	yes	characteristic
90	C₂×C₈	C₄×Q₈	1	2	yes	—
91	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
95	C₄²	—	2	4	no	—
96	C₄²	—	4	2	no	—
97	C₄²	C₄⋊C₈	1	2	yes	—
98	C₄²	C₄⋊C₈	1	2	yes	—
99	C₄²	#108	1	2	yes	—
100	C₄²	#108	1	2	yes	—
101	C₄²	C₄×D₄	1	2	yes	—
102	C₄²	#135	1	2	yes	—
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	2	no	—
105	C₁₆	—	2	2	no	—
106	C₁₆	C₄×C₈	1	4	yes	complemented
107	C₂³	C₂×#106	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	#334	1	2	yes	—
114	C₂×C₄	#334	1	4	yes	—
115	C₂×C₄	#334	1	2	yes	—
116	C₂×C₄	#365	1	4	yes	—
117	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₂×C₄	#427	1	1	yes	characteristic
121	C₂×C₄	C₂×#108	1	1	yes	characteristic
122	C₂×C₄	C₂×#108	1	1	yes	characteristic
123	C₂×C₄	C₄×SD₁₆	1	4	yes	—
124	C₈	—	2	2	no	—
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₈	—	8	8	no	—
128	C₈	C₂×C₄×C₈	1	1	yes	characteristic
129	C₈	C₄×M₁₆	1	1	yes	characteristic
130	C₈	#431	1	1	yes	characteristic
131	C₈	#431	1	1	yes	characteristic
132	C₂²	C₂×#334	1	1	yes	characteristic
133	C₂²	C₂×#334	1	1	yes	characteristic
134	C₂²	C₄×#106	1	1	yes	characteristic
135	C₂²	C₂×#365	1	2	yes	—
136	C₂²	#2155	1	2	yes	—
137	C₄	—	2	4	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	4	no	—
140	C₄	#1368	1	4	yes	direct product factor
141	C₄	#1564	1	1	yes	characteristic
142	C₄	#1564	1	1	yes	characteristic
143	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
144	C₄	C₄×#108	1	1	yes	characteristic
145	C₂	C₄×#334	1	1	yes	characteristic
146	C₂	C₂×#1368	1	1	yes	characteristic
147	C₂	#11999	1	1	yes	characteristic
148	C₂	#12000	1	2	yes	—
149	C₂	C₄×#365	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	#106
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. #1368	C₄	direct (N × H)
31. #346	C₈	semidirect
34. C₄×C₁₆	C₈	semidirect
106. C₁₆	C₄×C₈	semidirect
140. C₄	#1368	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#346	2
C₂×#1368	2
C₄×#334	2
#7156	2

Subgroups — C4×#1368 #129089

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#1368 #129089