Subgroups — C₄×#1385 #146334

Order 512.

644 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#108	C₂	1	2	yes	maximal
3	C₂×#1385	C₂	1	4	yes	maximal
4	C₂²×M₃₂	C₂²	1	2	yes	—
5	C₂²×#108	C₂²	1	4	yes	—
6	C₂×C₄×M₁₆	—	2	2	no	—
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₄×M₃₂	—	2	4	no	—
9	C₂×#346	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	C₄	1	2	yes	—
11	C₄×#108	—	2	4	no	—
12	C₄×#108	C₄	1	8	yes	—
13	C₂×#341	—	2	2	no	—
14	#1385	C₄	1	16	yes	direct product factor
15	C₂²×M₁₆	—	2	4	no	—
16	C₂³×C₈	D₄	1	1	yes	characteristic
17	C₂³×C₈	C₂³	1	1	yes	characteristic
18	C₂²×C₄²	D₄	1	1	yes	characteristic
19	C₂×M₃₂	—	2	4	no	—
20	C₂×M₃₂	—	2	4	no	—
21	C₂×M₃₂	C₂×C₄	1	8	yes	—
22	C₂²×C₁₆	C₂×C₄	1	4	yes	—
23	C₂×#108	—	2	4	no	—
24	C₂×#108	—	2	8	no	—
25	C₂×#108	C₂×C₄	1	16	yes	—
26	C₂×#108	C₂×C₄	1	8	yes	—
27	C₄×M₁₆	—	2	8	no	—
28	C₄×M₁₆	—	4	4	no	—
29	C₂×#97	—	4	2	no	—
30	C₂×C₄×C₈	—	4	2	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
34	C₂×C₄×C₈	D₄	1	4	yes	—
35	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
36	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
37	#346	—	2	4	no	—
38	C₄×C₁₆	—	2	4	no	—
39	#341	—	2	8	no	—
40	C₂³×C₄	Q₁₆	1	2	yes	—
41	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂×M₁₆	—	2	8	no	—
43	C₂×M₁₆	—	2	16	no	—
44	C₂×M₁₆	—	4	4	no	—
45	C₂×M₁₆	—	4	8	no	—
46	C₂²×C₈	—	2	4	no	—
47	C₂²×C₈	—	4	4	no	—
48	C₂²×C₈	C₂²⋊C₄	1	4	yes	—
49	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²×C₄	1	2	yes	—
53	C₂²×C₈	C₂²×C₄	1	4	yes	—
54	C₂²×C₈	C₂×D₄	1	2	yes	—
55	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₈	#41	1	4	yes	—
57	C₂×C₄²	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄²	D₈	1	2	yes	—
61	C₂×C₄²	SD₁₆	1	2	yes	—
62	C₂×C₄²	SD₁₆	1	2	yes	—
63	M₃₂	—	2	16	no	—
64	C₂×C₁₆	—	2	4	no	—
65	C₂×C₁₆	—	2	4	no	—
66	C₂×C₁₆	C₄²	1	8	yes	—
67	#108	—	2	8	no	—
68	#108	—	2	16	no	—
69	#108	C₄²	1	16	yes	—
70	#97	—	4	8	no	—
71	C₄×C₈	—	4	8	no	—
72	C₄×C₈	C₄²	1	2	yes	—
73	C₄×C₈	C₄²	1	2	yes	—
74	C₄×C₈	C₂²⋊C₄	1	2	yes	—
75	C₄×C₈	C₂²⋊C₄	1	2	yes	—
76	C₄×C₈	C₄⋊C₄	1	4	yes	—
77	C₄×C₈	C₄⋊C₄	1	4	yes	—
78	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
79	C₂²×C₄	—	4	4	no	—
80	C₂²×C₄	#103	1	4	yes	—
81	C₂²×C₄	#103	1	4	yes	—
82	C₂²×C₄	#107	1	2	yes	—
83	C₂²×C₄	#107	1	2	yes	—
84	C₂²×C₄	C₂×C₂²⋊C₄	1	2	yes	—
85	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄	C₄×D₄	1	4	yes	—
87	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
88	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
89	C₂²×C₄	#135	1	4	yes	—
90	M₁₆	—	2	16	no	—
91	M₁₆	—	4	8	no	—
92	M₁₆	—	4	16	no	—
93	C₂×C₈	—	2	8	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	8	no	—
97	C₂×C₈	—	4	16	no	—
98	C₂×C₈	#95	1	2	yes	—
99	C₂×C₈	#95	1	2	yes	—
100	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₈	C₂×C₄²	1	2	yes	—
102	C₂×C₈	C₂×C₄²	1	4	yes	—
103	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₈	#117	1	2	yes	—
106	C₂×C₈	#117	1	4	yes	—
107	C₂×C₈	C₄×D₄	1	8	yes	—
108	C₂×C₈	C₄×Q₈	1	4	yes	—
109	C₄²	#95	1	2	yes	—
110	C₄²	#95	1	2	yes	—
111	C₄²	#102	1	4	yes	—
112	C₄²	#102	1	4	yes	—
113	C₄²	#106	1	2	yes	—
114	C₄²	#106	1	1	yes	characteristic
115	C₄²	#106	1	1	yes	characteristic
116	C₁₆	—	2	8	no	—
117	C₂³	—	4	2	no	—
118	C₂³	C₂×#103	1	2	yes	—
119	C₂³	C₂×#107	1	1	yes	characteristic
120	C₂³	C₄×Q₁₆	1	4	yes	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	#340	1	4	yes	—
126	C₂×C₄	#340	1	2	yes	—
127	C₂×C₄	#340	1	2	yes	—
128	C₂×C₄	C₂×#95	1	1	yes	characteristic
129	C₂×C₄	C₄×C₂²⋊C₄	1	4	yes	—
130	C₂×C₄	C₄×C₄⋊C₄	1	2	yes	—
131	C₂×C₄	C₂×#102	1	2	yes	—
132	C₂×C₄	#416	1	4	yes	—
133	C₂×C₄	C₂×#106	1	1	yes	characteristic
134	C₂×C₄	#427	1	2	yes	—
135	C₂×C₄	C₄×D₈	1	4	yes	—
136	C₂×C₄	C₄×SD₁₆	1	8	yes	—
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	—	4	16	no	—
140	C₈	C₄³	1	1	yes	characteristic
141	C₈	#376	1	1	yes	characteristic
142	C₈	C₄×C₂²⋊C₄	1	2	yes	—
143	C₈	C₄×C₄⋊C₄	1	4	yes	—
144	C₂²	—	2	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	4	no	—
147	C₂²	C₂×#340	1	1	yes	characteristic
148	C₂²	C₄×#103	1	4	yes	—
149	C₂²	C₄×#107	1	2	yes	—
150	C₄	—	2	4	no	—
151	C₄	—	2	2	no	—
152	C₄	#1385	1	4	yes	direct product factor
153	C₄	C₄×#95	1	1	yes	characteristic
154	C₄	#1735	1	1	yes	characteristic
155	C₄	C₄×#102	1	4	yes	—
156	C₄	C₄×#106	1	2	yes	—
157	C₂	—	2	4	no	—
158	C₂	—	2	2	no	—
159	C₂	C₄×#340	1	1	yes	characteristic
160	C₂	C₂×#1385	1	2	yes	—

	subgroup	G/H
center	C₄²	#106
commutator	C₈	C₄³
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₂×#340
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. #1385	C₄	direct (N × H)
152. C₄	#1385	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×M₃₂	2
C₂×C₄×#108	2
C₂×#1385	2

Subgroups — C4×#1385 #146334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#1385 #146334