Subgroups — #3543

Order 128.

630 conjugacy classes of subgroups.

The subgroup lattice has 151 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#560	C₂	1	4	yes	maximal, complemented
2	#559	C₂	1	1	yes	maximal, characteristic, complemented
3	#550	C₂	1	2	yes	maximal, complemented
4	#546	C₂	1	4	yes	maximal, complemented
5	D₄²	C₂	1	4	yes	maximal, complemented
6	D₄²	C₂	1	1	yes	maximal, characteristic, complemented
7	#540	C₂	1	4	yes	maximal, complemented
8	#535	C₂	1	4	yes	maximal, complemented
9	#534	C₂	1	4	yes	maximal, complemented
10	C₂×#127	C₂	1	1	yes	maximal, characteristic, complemented
11	#529	C₂	1	2	yes	maximal, complemented
12	C₂×#41	C₂²	1	4	yes	complemented
13	C₂²×D₄	C₂²	1	2	yes	complemented
14	C₂²×D₄	C₂²	1	4	yes	complemented
15	C₂²×D₄	C₂²	1	4	yes	complemented
16	C₂²×D₄	C₂²	1	4	yes	complemented
17	#127	C₂²	1	1	yes	characteristic
18	#127	C₂²	1	4	yes	complemented
19	#127	C₂²	1	1	yes	characteristic, complemented
20	#127	C₂²	1	2	yes	complemented
21	#127	C₂²	1	4	yes	complemented
22	#126	C₂²	1	4	yes	complemented
23	#125	C₂²	1	2	yes	complemented
24	#124	C₂²	1	2	yes	complemented
25	#124	C₂²	1	4	yes	complemented
26	#123	C₂²	1	4	yes	complemented
27	#123	C₂²	1	4	yes	complemented
28	#123	C₂²	1	4	yes	complemented
29	#123	C₂²	1	8	yes	complemented
30	C₂²⋊Q₈	C₂²	1	2	yes	complemented
31	#121	C₂²	1	2	yes	—
32	#121	C₂²	1	8	yes	complemented
33	#121	C₂²	1	8	yes	complemented
34	#121	C₂²	1	8	yes	complemented
35	#121	C₂²	1	4	yes	complemented
36	#121	C₂²	1	4	yes	complemented
37	#121	C₂²	1	8	yes	complemented
38	#120	C₂²	1	4	yes	complemented
39	#120	C₂²	1	8	yes	complemented
40	#120	C₂²	1	4	yes	complemented
41	#120	C₂²	1	8	yes	complemented
42	C₄×Q₈	C₂²	1	2	yes	complemented
43	C₄×D₄	C₂²	1	4	yes	complemented
44	C₄×D₄	C₂²	1	2	yes	complemented
45	C₄×D₄	C₂²	1	4	yes	complemented
46	C₄×D₄	C₂²	1	4	yes	complemented
47	#117	C₂²	1	4	yes	complemented
48	C₂×C₂²⋊C₄	C₂²	1	4	yes	complemented
49	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴	C₂³	1	4	yes	—
51	C₂⁴	C₂³	1	4	yes	—
52	#41	—	4	8	no	—
53	C₂×Q₈	C₂³	1	2	yes	—
54	C₂×D₄	—	4	8	no	—
55	C₂×D₄	—	4	8	no	—
56	C₂×D₄	—	4	4	no	—
57	C₂×D₄	—	4	8	no	—
58	C₂×D₄	C₂³	1	4	yes	—
59	C₂×D₄	C₂³	1	2	yes	—
60	C₂×D₄	C₂³	1	4	yes	—
61	C₂×D₄	C₂³	1	4	yes	—
62	C₂×D₄	C₂³	1	8	yes	—
63	C₂×D₄	C₂³	1	8	yes	—
64	C₂×D₄	C₂³	1	2	yes	—
65	C₂×D₄	C₂³	1	8	yes	—
66	C₂×D₄	C₂³	1	4	yes	—
67	C₂×D₄	C₂³	1	8	yes	—
68	C₂×D₄	C₂³	1	2	yes	—
69	C₂²×C₄	C₂³	1	2	yes	—
70	C₂²×C₄	C₂³	1	1	yes	characteristic
71	C₂²×C₄	C₂³	1	4	yes	—
72	C₂²×C₄	C₂³	1	8	yes	—
73	C₂²×C₄	C₂³	1	4	yes	—
74	C₄⋊C₄	C₂³	1	4	yes	—
75	C₄⋊C₄	C₂³	1	4	yes	—
76	C₄⋊C₄	C₂³	1	2	yes	—
77	C₄⋊C₄	C₂³	1	2	yes	—
78	C₄⋊C₄	C₂³	1	4	yes	—
79	C₄⋊C₄	C₂³	1	4	yes	—
80	C₂²⋊C₄	C₂³	1	4	yes	—
81	C₂²⋊C₄	C₂³	1	4	yes	—
82	C₂²⋊C₄	C₂³	1	4	yes	—
83	C₂²⋊C₄	C₂³	1	4	yes	—
84	C₂²⋊C₄	C₂³	1	4	yes	—
85	C₂²⋊C₄	C₂³	1	4	yes	—
86	C₂²⋊C₄	C₂³	1	4	yes	—
87	C₂²⋊C₄	C₂³	1	8	yes	—
88	C₂²⋊C₄	C₂³	1	8	yes	—
89	C₄²	C₂³	1	2	yes	—
90	C₄²	C₂³	1	1	yes	characteristic
91	C₄²	C₂³	1	1	yes	characteristic
92	C₄²	C₂³	1	4	yes	—
93	C₂³	—	2	8	no	—
94	C₂³	—	4	8	no	—
95	C₂³	—	4	8	no	—
96	C₂³	—	4	4	no	—
97	C₂³	C₂⁴	1	4	yes	—
98	C₂³	C₂⁴	1	4	yes	—
99	C₂³	C₂⁴	1	4	yes	—
100	C₂³	C₂⁴	1	1	yes	characteristic
101	Q₈	—	2	4	no	—
102	D₄	—	2	8	no	—
103	D₄	—	2	4	no	—
104	D₄	—	2	8	no	—
105	D₄	—	2	8	no	—
106	D₄	—	4	8	no	—
107	D₄	—	4	8	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	4	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	8	no	—
112	D₄	—	4	8	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	8	no	—
118	C₂×C₄	C₂⁴	1	1	yes	characteristic
119	C₂×C₄	C₂⁴	1	2	yes	—
120	C₂×C₄	C₂⁴	1	1	yes	characteristic
121	C₂×C₄	C₂⁴	1	4	yes	—
122	C₂×C₄	C₂⁴	1	4	yes	—
123	C₂×C₄	C₂⁴	1	4	yes	—
124	C₂×C₄	C₂⁴	1	2	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	—	2	4	no	—
127	C₂²	—	2	8	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	4	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	8	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	8	8	no	—
134	C₂²	—	8	8	no	—
135	C₂²	H(5,𝔽₂)	1	2	yes	—
136	C₂²	C₂⁵	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	4	no	—
142	C₄	—	2	4	no	—
143	C₄	H(5,𝔽₂)	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	4	4	no	—
146	C₂	—	4	4	no	—
147	C₂	—	4	4	no	—
148	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
149	C₂	C₂×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #560	C₂	semidirect
2. #559	C₂	semidirect
3. #550	C₂	semidirect
4. #546	C₂	semidirect
5. D₄²	C₂	semidirect
6. D₄²	C₂	semidirect
7. #540	C₂	semidirect
8. #535	C₂	semidirect
9. #534	C₂	semidirect
10. C₂×#127	C₂	semidirect
11. #529	C₂	semidirect
12. C₂×#41	C₂²	semidirect
13. C₂²×D₄	C₂²	semidirect
14. C₂²×D₄	C₂²	semidirect
15. C₂²×D₄	C₂²	semidirect
16. C₂²×D₄	C₂²	semidirect
18. #127	C₂²	semidirect
19. #127	C₂²	semidirect
20. #127	C₂²	semidirect
21. #127	C₂²	semidirect
22. #126	C₂²	semidirect
23. #125	C₂²	semidirect
24. #124	C₂²	semidirect
25. #124	C₂²	semidirect
26. #123	C₂²	semidirect
27. #123	C₂²	semidirect
28. #123	C₂²	semidirect
29. #123	C₂²	semidirect
30. C₂²⋊Q₈	C₂²	semidirect
32. #121	C₂²	semidirect
33. #121	C₂²	semidirect
34. #121	C₂²	semidirect
35. #121	C₂²	semidirect
36. #121	C₂²	semidirect
37. #121	C₂²	semidirect
38. #120	C₂²	semidirect
39. #120	C₂²	semidirect
40. #120	C₂²	semidirect
41. #120	C₂²	semidirect
42. C₄×Q₈	C₂²	semidirect
43. C₄×D₄	C₂²	semidirect
44. C₄×D₄	C₂²	semidirect
45. C₄×D₄	C₂²	semidirect
46. C₄×D₄	C₂²	semidirect
47. #117	C₂²	semidirect
48. C₂×C₂²⋊C₄	C₂²	semidirect
49. C₂×C₄²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#560	2
#559	2
#550	2
#546	2
D₄²	2
#540	2
#535	2
#534	2
C₂×#127	2
#529	2