Finite Group Databasebeta

← C₂×#538 #3450

Subgroups — C₂×#538 #3450

Order 128.

668 conjugacy classes of subgroups; only the 396 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 396 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#538	C₂	1	16	yes	maximal, direct product factor
2	C₂×#126	C₂	1	2	yes	maximal, complemented
3	C₂×#124	C₂	1	2	yes	maximal, complemented
4	C₂×#123	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#121	C₂	1	2	yes	maximal, complemented
7	C₂×#121	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#120	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₂²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁵	C₂²	1	1	yes	characteristic
12	C₂²×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄	C₂²	1	2	yes	—
14	C₂²×D₄	C₂²	1	2	yes	—
15	C₂²×D₄	C₂²	1	2	yes	—
16	C₂²×D₄	C₂²	1	1	yes	characteristic, complemented
17	C₂³×C₄	C₂²	1	1	yes	characteristic
18	C₂³×C₄	C₂²	1	1	yes	characteristic
19	C₂³×C₄	C₂²	1	2	yes	complemented
20	#126	C₂²	1	16	yes	complemented
21	#124	C₂²	1	16	yes	complemented
22	#123	C₂²	1	16	yes	complemented
23	C₂²⋊Q₈	C₂²	1	8	yes	complemented
24	#121	C₂²	1	16	yes	complemented
25	#121	C₂²	1	8	yes	complemented
26	#120	C₂²	1	16	yes	complemented
27	C₄×D₄	C₂²	1	16	yes	complemented
28	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
29	C₂×C₄⋊C₄	C₂²	1	2	yes	complemented
30	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
33	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₄	C₂²	1	2	yes	—
37	C₂×C₂²⋊C₄	C₂²	1	8	yes	complemented
38	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
39	C₂×C₂²⋊C₄	C₂²	1	2	yes	complemented
40	C₂×C₄²	C₂²	1	2	yes	complemented
41	C₂⁴	—	2	4	no	—
42	C₂⁴	—	4	2	no	—
43	C₂⁴	—	4	2	no	—
44	C₂⁴	C₂³	1	2	yes	—
45	C₂⁴	C₂³	1	2	yes	—
46	C₂⁴	C₂³	1	1	yes	characteristic
47	C₂⁴	C₂³	1	4	yes	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	C₂³	1	4	yes	complemented
50	C₂×D₄	—	2	8	no	—
51	C₂×D₄	—	4	4	no	—
52	C₂×D₄	—	4	4	no	—
53	C₂×D₄	—	4	2	no	—
54	C₂×D₄	C₂³	1	8	yes	—
55	C₂×D₄	C₂³	1	8	yes	—
56	C₂×D₄	C₂³	1	8	yes	—
57	C₂×D₄	C₂³	1	4	yes	complemented
58	C₂²×C₄	—	4	2	no	—
59	C₂²×C₄	—	4	2	no	—
60	C₂²×C₄	—	4	4	no	—
61	C₂²×C₄	C₂³	1	2	yes	—
62	C₂²×C₄	C₂³	1	2	yes	—
63	C₂²×C₄	C₂³	1	2	yes	—
64	C₂²×C₄	C₂³	1	1	yes	characteristic
65	C₂²×C₄	C₂³	1	1	yes	characteristic
66	C₂²×C₄	C₂³	1	2	yes	—
67	C₂²×C₄	C₂³	1	4	yes	—
68	C₂²×C₄	C₂³	1	4	yes	—
69	C₂²×C₄	C₂³	1	8	yes	complemented
70	C₄⋊C₄	C₂³	1	8	yes	complemented
71	C₄⋊C₄	C₂³	1	8	yes	complemented
72	C₄⋊C₄	C₂³	1	4	yes	complemented
73	C₄⋊C₄	C₂³	1	4	yes	complemented
74	C₂²⋊C₄	C₂³	1	8	yes	—
75	C₂²⋊C₄	C₂³	1	8	yes	—
76	C₂²⋊C₄	C₂³	1	8	yes	—
77	C₂²⋊C₄	C₂³	1	8	yes	—
78	C₂²⋊C₄	C₂³	1	8	yes	—
79	C₂²⋊C₄	C₂³	1	8	yes	complemented
80	C₂²⋊C₄	C₂³	1	8	yes	complemented
81	C₄²	C₂³	1	8	yes	complemented
82	C₂³	—	2	2	no	—
83	C₂³	—	2	2	no	—
84	C₂³	—	2	4	no	—
85	C₂³	—	2	8	no	—
86	C₂³	—	2	4	no	—
87	C₂³	—	2	4	no	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	4	no	—
90	C₂³	—	2	4	no	—
91	C₂³	—	4	4	no	—
92	C₂³	—	4	4	no	—
93	C₂³	—	4	16	no	—
94	C₂³	#41	1	4	yes	—
95	C₂³	C₂⁴	1	4	yes	—
96	C₂³	C₂⁴	1	4	yes	—
97	C₂³	C₂⁴	1	2	yes	—
98	C₂³	C₂⁴	1	1	yes	characteristic
99	Q₈	—	4	4	no	—
100	D₄	—	2	16	no	—
101	D₄	—	4	8	no	—
102	D₄	—	4	8	no	—
103	D₄	—	4	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	8	no	—
113	C₂×C₄	C₂⁴	1	4	yes	—
114	C₂×C₄	C₂⁴	1	4	yes	—
115	C₂×C₄	C₂⁴	1	4	yes	—
116	C₂×C₄	C₂⁴	1	2	yes	—
117	C₂×C₄	C₂⁴	1	2	yes	—
118	C₂×C₄	C₂⁴	1	4	yes	—
119	C₂²	—	2	8	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	2	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	4	no	—
128	C₂²	—	4	16	no	—
129	C₂²	—	4	8	no	—
130	C₂²	—	4	8	no	—
131	C₂²	C₂×#41	1	4	yes	—
132	C₂²	C₂×#41	1	2	yes	—
133	C₂²	H(5,𝔽₂)	1	2	yes	—
134	C₂²	H(5,𝔽₂)	1	2	yes	—
135	C₂²	C₂⁵	1	1	yes	characteristic
136	C₄	—	2	4	no	—
137	C₄	—	2	4	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	4	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	4	no	—
142	C₂	—	2	4	no	—
143	C₂	—	4	4	no	—
144	C₂	—	4	4	no	—
145	C₂	#538	1	4	yes	direct product factor
146	C₂	C₂²×#41	1	1	yes	characteristic
147	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
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. #538	C₂	direct (N × H)
2. C₂×#126	C₂	semidirect
3. C₂×#124	C₂	semidirect
4. C₂×#123	C₂	semidirect
5. C₂×C₂²⋊Q₈	C₂	semidirect
6. C₂×#121	C₂	semidirect
7. C₂×#121	C₂	semidirect
8. C₂×#120	C₂	semidirect
9. C₂×C₄×D₄	C₂	semidirect
10. C₂²×C₂²⋊C₄	C₂	semidirect
12. C₂²×Q₈	C₂²	semidirect
16. C₂²×D₄	C₂²	semidirect
19. C₂³×C₄	C₂²	semidirect
20. #126	C₂²	semidirect
21. #124	C₂²	semidirect
22. #123	C₂²	semidirect
23. C₂²⋊Q₈	C₂²	semidirect
24. #121	C₂²	semidirect
25. #121	C₂²	semidirect
26. #120	C₂²	semidirect
27. C₄×D₄	C₂²	semidirect
28. C₂×C₄⋊C₄	C₂²	semidirect
29. C₂×C₄⋊C₄	C₂²	semidirect
30. C₂×C₄⋊C₄	C₂²	semidirect
31. C₂×C₄⋊C₄	C₂²	semidirect
37. C₂×C₂²⋊C₄	C₂²	semidirect
38. C₂×C₂²⋊C₄	C₂²	semidirect
39. C₂×C₂²⋊C₄	C₂²	semidirect
40. C₂×C₄²	C₂²	semidirect
49. C₂×Q₈	C₂³	semidirect
57. C₂×D₄	C₂³	semidirect
69. C₂²×C₄	C₂³	semidirect
70. C₄⋊C₄	C₂³	semidirect
71. C₄⋊C₄	C₂³	semidirect
72. C₄⋊C₄	C₂³	semidirect
73. C₄⋊C₄	C₂³	semidirect
79. C₂²⋊C₄	C₂³	semidirect
80. C₂²⋊C₄	C₂³	semidirect
81. C₄²	C₂³	semidirect
145. C₂	#538	direct (N × H)

Maximal subgroups

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

subgroup	index
#538	2
C₂×#126	2
C₂×#124	2
C₂×#123	2
C₂×C₂²⋊Q₈	2
C₂×#121	2
C₂×#120	2
C₂×C₄×D₄	2
C₂²×C₂²⋊C₄	2