Subgroups — C₂×#542 #3454

Order 128.

566 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#542	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#126	C₂	1	2	yes	maximal, complemented
4	C₂×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124	C₂	1	2	yes	maximal, complemented
6	C₂×#123	C₂	1	2	yes	maximal, complemented
7	C₂×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
8	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#121	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#117	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₈	C₂²	1	1	yes	characteristic
14	C₂²×Q₈	C₂²	1	2	yes	—
15	C₂²×D₄	C₂²	1	1	yes	characteristic
16	C₂²×D₄	C₂²	1	2	yes	—
17	C₂³×C₄	C₂²	1	1	yes	characteristic
18	C₂³×C₄	C₂²	1	2	yes	—
19	C₄⋊Q₈	C₂²	1	8	yes	complemented
20	#126	C₂²	1	16	yes	complemented
21	#124	C₂²	1	8	yes	complemented
22	#124	C₂²	1	16	yes	complemented
23	#123	C₂²	1	16	yes	complemented
24	C₂²⋊Q₈	C₂²	1	16	yes	complemented
25	C₂²⋊Q₈	C₂²	1	8	yes	complemented
26	#121	C₂²	1	8	yes	—
27	C₄×Q₈	C₂²	1	8	yes	complemented
28	C₄×D₄	C₂²	1	8	yes	complemented
29	#117	C₂²	1	8	yes	complemented
30	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂×C₄⋊C₄	C₂²	1	2	yes	—
36	C₂×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₂²⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
40	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
43	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
44	C₂×C₄²	C₂²	1	1	yes	characteristic
45	C₂×C₄²	C₂²	1	1	yes	characteristic
46	C₂×C₄²	C₂²	1	2	yes	—
47	C₂⁴	C₂³	1	1	yes	characteristic
48	C₂⁴	C₂³	1	2	yes	—
49	C₂×Q₈	—	2	4	no	—
50	C₂×Q₈	—	4	4	no	—
51	C₂×Q₈	C₂³	1	4	yes	—
52	C₂×Q₈	C₂³	1	8	yes	—
53	C₂×D₄	—	2	4	no	—
54	C₂×D₄	—	4	4	no	—
55	C₂×D₄	C₂³	1	4	yes	—
56	C₂×D₄	C₂³	1	8	yes	—
57	C₂²×C₄	—	2	4	no	—
58	C₂²×C₄	—	4	4	no	—
59	C₂²×C₄	C₂³	1	4	yes	—
60	C₂²×C₄	C₂³	1	1	yes	characteristic
61	C₂²×C₄	C₂³	1	1	yes	characteristic
62	C₂²×C₄	C₂³	1	2	yes	—
63	C₂²×C₄	C₂³	1	1	yes	characteristic
64	C₂²×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₄	C₂³	1	2	yes	—
66	C₂²×C₄	C₂³	1	1	yes	characteristic
67	C₂²×C₄	C₂³	1	2	yes	—
68	C₂²×C₄	C₂³	1	8	yes	—
69	C₂²×C₄	C₂³	1	1	yes	characteristic
70	C₄⋊C₄	C₂³	1	4	yes	—
71	C₄⋊C₄	C₂³	1	4	yes	—
72	C₄⋊C₄	C₂³	1	4	yes	—
73	C₄⋊C₄	C₂³	1	4	yes	—
74	C₄⋊C₄	C₂³	1	8	yes	—
75	C₄⋊C₄	C₂³	1	8	yes	—
76	C₄⋊C₄	C₂³	1	8	yes	—
77	C₂²⋊C₄	C₂³	1	4	yes	—
78	C₂²⋊C₄	C₂³	1	4	yes	—
79	C₂²⋊C₄	C₂³	1	8	yes	—
80	C₂²⋊C₄	C₂³	1	8	yes	—
81	C₂²⋊C₄	C₂³	1	8	yes	—
82	C₂²⋊C₄	C₂³	1	8	yes	—
83	C₂²⋊C₄	C₂³	1	8	yes	—
84	C₄²	C₂³	1	4	yes	—
85	C₄²	C₂³	1	4	yes	—
86	C₄²	C₂³	1	8	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	2	2	no	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂⁴	1	2	yes	—
94	C₂³	C₂⁴	1	4	yes	—
95	C₂³	C₂⁴	1	1	yes	characteristic
96	Q₈	—	2	8	no	—
97	Q₈	—	4	8	no	—
98	D₄	—	2	8	no	—
99	D₄	—	4	8	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	8	no	—
109	C₂×C₄	—	4	8	no	—
110	C₂×C₄	#41	1	4	yes	—
111	C₂×C₄	C₂⁴	1	2	yes	—
112	C₂×C₄	C₂⁴	1	2	yes	—
113	C₂×C₄	C₂⁴	1	4	yes	—
114	C₂×C₄	C₂⁴	1	2	yes	—
115	C₂×C₄	C₂⁴	1	2	yes	—
116	C₂×C₄	C₂⁴	1	4	yes	—
117	C₂×C₄	C₂⁴	1	2	yes	—
118	C₂×C₄	C₂⁴	1	4	yes	—
119	C₂×C₄	C₂⁴	1	2	yes	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	4	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	4	8	no	—
128	C₂²	C₂×#41	1	2	yes	—
129	C₂²	H(5,𝔽₂)	1	2	yes	—
130	C₂²	2₋¹⁺⁴	1	2	yes	—
131	C₂²	C₂⁵	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	4	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	4	no	—
136	C₄	—	2	4	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	2	no	—
140	C₄	C₂×#41	1	4	yes	—
141	C₂	—	4	2	no	—
142	C₂	—	4	4	no	—
143	C₂	#542	1	4	yes	direct product factor
144	C₂	C₂²×#41	1	1	yes	characteristic
145	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂×#121	C₂
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. #542	C₂	direct (N × H)
2. C₂×C₄⋊Q₈	C₂	semidirect
3. C₂×#126	C₂	semidirect
4. C₂×#124	C₂	semidirect
5. C₂×#124	C₂	semidirect
6. C₂×#123	C₂	semidirect
7. C₂×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₂²⋊Q₈	C₂	semidirect
10. C₂×C₄×Q₈	C₂	semidirect
11. C₂×C₄×D₄	C₂	semidirect
12. C₂×#117	C₂	semidirect
19. C₄⋊Q₈	C₂²	semidirect
20. #126	C₂²	semidirect
21. #124	C₂²	semidirect
22. #124	C₂²	semidirect
23. #123	C₂²	semidirect
24. C₂²⋊Q₈	C₂²	semidirect
25. C₂²⋊Q₈	C₂²	semidirect
27. C₄×Q₈	C₂²	semidirect
28. C₄×D₄	C₂²	semidirect
29. #117	C₂²	semidirect
143. C₂	#542	direct (N × H)

Maximal subgroups

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

subgroup	index
#542	2
C₂×C₄⋊Q₈	2
C₂×#126	2
C₂×#124	2
C₂×#123	2
C₂×C₂²⋊Q₈	2
C₂×#121	2
C₂×C₄×Q₈	2
C₂×C₄×D₄	2
C₂×#117	2

Subgroups — C2×#542 #3454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#542 #3454