Subgroups — C₂×#1387 #12568

Order 256.

307 conjugacy classes of subgroups.

The subgroup lattice has 165 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#108	C₂	1	1	yes	maximal, characteristic
3	C₂²×#107	C₂	1	1	yes	maximal, characteristic
4	#1387	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₁₆	—	2	1	no	—
6	C₂³×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄	—	2	1	no	—
8	C₂×M₃₂	—	2	2	no	—
9	C₂×M₃₂	—	2	1	no	—
10	C₂×M₃₂	C₄	1	4	yes	complemented
11	C₂×M₃₂	C₂²	1	2	yes	—
12	C₂²×C₁₆	—	2	1	no	—
13	C₂×#108	—	2	1	no	—
14	C₂×#108	—	2	2	no	—
15	C₂×#108	C₄	1	4	yes	complemented
16	C₂×#108	C₂²	1	2	yes	—
17	C₂×#107	—	2	1	no	—
18	C₂×#107	—	2	1	no	—
19	C₂×#107	—	2	1	no	—
20	C₂×#107	C₄	1	2	yes	—
21	C₂×#107	C₄	1	2	yes	—
22	C₂×#107	C₂²	1	2	yes	—
23	C₂³×C₄	—	4	1	no	—
24	C₂³×C₄	D₄	1	1	yes	characteristic
25	C₂×M₁₆	—	2	4	no	—
26	C₂×M₁₆	—	2	2	no	—
27	C₂×M₁₆	—	4	1	no	—
28	C₂×M₁₆	—	4	2	no	—
29	C₂²×C₈	—	4	1	no	—
30	C₂²×C₈	C₂×C₄	1	2	yes	—
31	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₈	D₄	1	1	yes	characteristic
35	C₂²×C₈	D₄	1	1	yes	characteristic
36	C₂²×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₈	D₄	1	2	yes	—
38	C₂²×C₈	Q₈	1	2	yes	—
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊C₄	—	2	2	no	—
41	C₂×C₄⋊C₄	—	2	2	no	—
42	C₂×C₄⋊C₄	—	2	2	no	—
43	C₂×C₄⋊C₄	—	4	1	no	—
44	C₂×C₄⋊C₄	—	4	1	no	—
45	C₂×C₄⋊C₄	—	4	1	no	—
46	M₃₂	—	2	4	no	—
47	M₃₂	—	2	2	no	—
48	M₃₂	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆	—	2	4	no	—
50	C₂×C₁₆	—	2	2	no	—
51	#108	—	2	4	no	—
52	#108	—	2	2	no	—
53	#108	C₂×C₄	1	4	yes	complemented
54	#107	—	2	2	no	—
55	#107	—	2	2	no	—
56	#107	—	2	2	no	—
57	#107	C₂×C₄	1	4	yes	—
58	C₂⁴	SD₁₆	1	1	yes	characteristic
59	C₂²×C₄	—	4	2	no	—
60	C₂²×C₄	—	4	2	no	—
61	C₂²×C₄	—	4	2	no	—
62	C₂²×C₄	—	8	4	no	—
63	C₂²×C₄	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄	D₈	1	1	yes	characteristic
68	C₂²×C₄	D₈	1	1	yes	characteristic
69	C₂²×C₄	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄	SD₁₆	1	2	yes	—
71	C₂²×C₄	Q₁₆	1	2	yes	—
72	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
73	M₁₆	—	2	4	no	—
74	M₁₆	—	4	4	no	—
75	M₁₆	—	4	2	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	C₄²	1	2	yes	—
79	C₂×C₈	C₄²	1	2	yes	—
80	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈	C₂²⋊C₄	1	2	yes	—
85	C₂×C₈	C₄⋊C₄	1	2	yes	—
86	C₂×C₈	C₄⋊C₄	1	2	yes	—
87	C₂×C₈	C₄⋊C₄	1	2	yes	—
88	C₂×C₈	C₄⋊C₄	1	2	yes	—
89	C₂×C₈	C₄⋊C₄	1	2	yes	—
90	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₁₆	—	2	4	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	2	2	no	—
103	C₂³	#102	1	1	yes	characteristic
104	C₂³	#102	1	1	yes	characteristic
105	C₂³	#103	1	2	yes	—
106	C₂³	#106	1	2	yes	—
107	C₂³	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	—	8	4	no	—
111	C₂×C₄	—	8	4	no	—
112	C₂×C₄	#95	1	2	yes	—
113	C₂×C₄	#95	1	2	yes	—
114	C₂×C₄	#102	1	1	yes	characteristic
115	C₂×C₄	#102	1	1	yes	characteristic
116	C₂×C₄	#102	1	2	yes	—
117	C₂×C₄	#102	1	2	yes	—
118	C₂×C₄	#103	1	2	yes	—
119	C₂×C₄	#103	1	2	yes	—
120	C₂×C₄	#103	1	2	yes	—
121	C₂×C₄	#106	1	2	yes	—
122	C₂×C₄	#107	1	2	yes	—
123	C₂×C₄	#107	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
130	C₈	—	4	4	no	—
131	C₈	#95	1	2	yes	—
132	C₈	#95	1	2	yes	—
133	C₈	C₂×C₄²	1	1	yes	characteristic
134	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	4	no	—
142	C₂²	—	2	4	no	—
143	C₂²	#340	1	2	yes	—
144	C₂²	#340	1	2	yes	—
145	C₂²	#361	1	2	yes	—
146	C₂²	#362	1	2	yes	—
147	C₂²	C₂×#102	1	1	yes	characteristic
148	C₂²	C₂×#103	1	1	yes	characteristic
149	C₂²	C₂×#106	1	1	yes	characteristic
150	C₄	—	8	4	no	—
151	C₄	#340	1	2	yes	—
152	C₄	#340	1	2	yes	—
153	C₄	C₂×#95	1	1	yes	characteristic
154	C₄	C₂×#102	1	1	yes	characteristic
155	C₄	C₂×#103	1	1	yes	characteristic
156	C₄	C₂×#107	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#1387	1	4	yes	direct product factor
161	C₂	C₂×#340	1	1	yes	characteristic
162	C₂	C₂×#361	1	1	yes	characteristic
163	C₂	C₂×#362	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#102
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂⁴	SD₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #1387	C₂	direct (N × H)
10. C₂×M₃₂	C₄	semidirect
15. C₂×#108	C₄	semidirect
48. M₃₂	C₂×C₄	semidirect
53. #108	C₂×C₄	semidirect
160. C₂	#1387	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×M₃₂	2
C₂²×#108	2
C₂²×#107	2
#1387	2

Subgroups — C2×#1387 #12568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1387 #12568