Subgroups — C₂×#3502 #62784

Order 256.

3413 conjugacy classes of subgroups.

The subgroup lattice has 142 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3502	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈²	C₂	1	4	yes	maximal, complemented
3	C₂×#557	C₂	1	8	yes	maximal, complemented
4	C₂×#557	C₂	1	4	yes	maximal, complemented
5	C₂×#544	C₂	1	6	yes	maximal
6	C₂×#533	C₂	1	6	yes	maximal
7	C₂×#519	C₂	1	2	yes	maximal
8	C₂²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈	C₂²	1	1	yes	characteristic
10	Q₈²	C₂²	1	64	yes	complemented
11	#557	C₂²	1	128	yes	complemented
12	#557	C₂²	1	64	yes	complemented
13	#544	C₂²	1	96	yes	—
14	#533	C₂²	1	96	yes	—
15	C₂×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₂×C₄⋊Q₈	C₂²	1	6	yes	—
17	C₂×C₄⋊Q₈	C₂²	1	24	yes	—
18	C₂×#125	C₂²	1	12	yes	—
19	C₂×#125	C₂²	1	24	yes	—
20	C₂×C₂²⋊Q₈	C₂²	1	4	yes	—
21	C₂×C₂²⋊Q₈	C₂²	1	12	yes	—
22	#519	C₂²	1	32	yes	—
23	C₂×C₄×Q₈	C₂²	1	4	yes	—
24	C₂×C₄×Q₈	C₂²	1	4	yes	—
25	C₂×C₄×Q₈	C₂²	1	16	yes	—
26	C₂×C₄×Q₈	C₂²	1	8	yes	—
27	C₂×C₄×Q₈	C₂²	1	8	yes	—
28	C₂×C₄×Q₈	C₂²	1	12	yes	—
29	C₂×C₄×Q₈	C₂²	1	12	yes	—
30	C₂×#117	C₂²	1	12	yes	—
31	C₂²×C₄⋊C₄	C₂²	1	3	yes	—
32	C₂²×C₄²	C₂²	1	3	yes	—
33	C₂²×Q₈	Q₈	1	16	yes	complemented
34	C₂²×Q₈	C₂³	1	4	yes	—
35	C₂²×Q₈	C₂³	1	12	yes	—
36	C₂²×Q₈	C₂³	1	4	yes	—
37	C₂²×Q₈	C₂³	1	8	yes	—
38	C₂³×C₄	C₂³	1	1	yes	characteristic
39	C₂³×C₄	C₂³	1	3	yes	—
40	C₂³×C₄	C₂³	1	3	yes	—
41	C₄⋊Q₈	C₂³	1	48	yes	—
42	C₄⋊Q₈	C₂³	1	48	yes	—
43	C₄⋊Q₈	C₂³	1	192	yes	—
44	#125	C₂³	1	96	yes	—
45	#125	C₂³	1	192	yes	—
46	C₂²⋊Q₈	C₂³	1	32	yes	—
47	C₂²⋊Q₈	C₂³	1	96	yes	—
48	C₄×Q₈	C₂³	1	32	yes	—
49	C₄×Q₈	C₂³	1	32	yes	—
50	C₄×Q₈	C₂³	1	64	yes	—
51	C₄×Q₈	C₂³	1	64	yes	—
52	C₄×Q₈	C₂³	1	96	yes	—
53	C₄×Q₈	C₂³	1	96	yes	—
54	#117	C₂³	1	96	yes	—
55	C₂×C₄⋊C₄	C₂³	1	12	yes	—
56	C₂×C₄⋊C₄	C₂³	1	24	yes	—
57	C₂×C₄⋊C₄	C₂³	1	12	yes	—
58	C₂×C₄⋊C₄	C₂³	1	12	yes	—
59	C₂×C₄⋊C₄	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄	C₂³	1	12	yes	—
62	C₂×C₄⋊C₄	C₂³	1	24	yes	—
63	C₂×C₂²⋊C₄	C₂³	1	8	yes	—
64	C₂×C₄²	C₂³	1	24	yes	—
65	C₂×C₄²	C₂³	1	6	yes	—
66	C₂×C₄²	C₂³	1	6	yes	—
67	C₂×C₄²	C₂³	1	24	yes	—
68	C₂⁴	C₂⁴	1	1	yes	characteristic
69	C₂×Q₈	—	2	16	no	—
70	C₂×Q₈	—	4	24	no	—
71	C₂×Q₈	C₂×Q₈	1	32	yes	—
72	C₂×Q₈	C₂×Q₈	1	64	yes	complemented
73	C₂×Q₈	C₂⁴	1	16	yes	—
74	C₂×Q₈	C₂⁴	1	48	yes	—
75	C₂×Q₈	C₂⁴	1	16	yes	—
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	4	6	no	—
78	C₂²×C₄	C₂×Q₈	1	24	yes	—
79	C₂²×C₄	C₂⁴	1	4	yes	—
80	C₂²×C₄	C₂⁴	1	1	yes	characteristic
81	C₂²×C₄	C₂⁴	1	8	yes	—
82	C₂²×C₄	C₂⁴	1	8	yes	—
83	C₂²×C₄	C₂⁴	1	6	yes	—
84	C₂²×C₄	C₂⁴	1	12	yes	—
85	C₂²×C₄	C₂⁴	1	12	yes	—
86	C₂²×C₄	C₂⁴	1	6	yes	—
87	C₂²×C₄	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄	C₂⁴	1	48	yes	—
89	C₄⋊C₄	C₂⁴	1	48	yes	—
90	C₄⋊C₄	C₂⁴	1	48	yes	—
91	C₄⋊C₄	C₂⁴	1	32	yes	—
92	C₄⋊C₄	C₂⁴	1	16	yes	—
93	C₄⋊C₄	C₂⁴	1	48	yes	—
94	C₄⋊C₄	C₂⁴	1	96	yes	—
95	C₂²⋊C₄	C₂⁴	1	32	yes	—
96	C₄²	C₂⁴	1	24	yes	—
97	C₄²	C₂⁴	1	24	yes	—
98	C₄²	C₂⁴	1	96	yes	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	C₂²×Q₈	1	4	yes	—
102	C₂³	C₂⁵	1	2	yes	—
103	C₂³	C₂⁵	1	1	yes	characteristic
104	Q₈	—	2	32	no	—
105	Q₈	—	4	48	no	—
106	Q₈	C₂²×Q₈	1	64	yes	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	16	no	—
109	C₂×C₄	—	2	12	no	—
110	C₂×C₄	—	2	16	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	4	12	no	—
113	C₂×C₄	C₂²×Q₈	1	24	yes	—
114	C₂×C₄	C₂²×Q₈	1	48	yes	—
115	C₂×C₄	2₋¹⁺⁴	1	4	yes	—
116	C₂×C₄	C₂⁵	1	2	yes	—
117	C₂×C₄	C₂⁵	1	16	yes	—
118	C₂×C₄	C₂⁵	1	16	yes	—
119	C₂×C₄	C₂⁵	1	12	yes	—
120	C₂×C₄	C₂⁵	1	12	yes	—
121	C₂×C₄	C₂⁵	1	2	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	8	no	—
125	C₂²	C₂³×Q₈	1	4	yes	—
126	C₂²	C₂³×Q₈	1	2	yes	—
127	C₂²	C₂×2₋¹⁺⁴	1	2	yes	—
128	C₂²	#585	1	2	yes	—
129	C₂²	C₂⁶	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	2	16	no	—
132	C₄	—	2	12	no	—
133	C₄	—	2	16	no	—
134	C₄	C₂³×Q₈	1	24	yes	—
135	C₄	C₂×2₋¹⁺⁴	1	4	yes	—
136	C₂	—	2	4	no	—
137	C₂	#3502	1	4	yes	direct product factor
138	C₂	C₂⁴×Q₈	1	1	yes	characteristic
139	C₂	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
140	C₂	C₂×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶
prime radical	C₂⁴	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3502	C₂	direct (N × H)
2. C₂×Q₈²	C₂	semidirect
3. C₂×#557	C₂	semidirect
4. C₂×#557	C₂	semidirect
10. Q₈²	C₂²	semidirect
11. #557	C₂²	semidirect
12. #557	C₂²	semidirect
33. C₂²×Q₈	Q₈	semidirect
72. C₂×Q₈	C₂×Q₈	semidirect
137. C₂	#3502	direct (N × H)

Maximal subgroups

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

subgroup	index
#3502	2
C₂×Q₈²	2
C₂×#557	2
C₂×#544	2
C₂×#533	2
C₂×#519	2
C₂²×C₄×Q₈	2

Subgroups — C2×#3502 #62784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3502 #62784