Subgroups — D₄×#41 #3468

Order 128.

824 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41	C₂	1	2	yes	maximal, complemented
2	#550	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#548	C₂	1	2	yes	maximal, complemented
5	#547	C₂	1	3	yes	maximal, complemented
6	#546	C₂	1	6	yes	maximal, complemented
7	D₄²	C₂	1	3	yes	maximal, complemented
8	#532	C₂	1	3	yes	maximal, complemented
9	#525	C₂	1	6	yes	maximal, complemented
10	C₄×#41	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×D₄	C₂	1	3	yes	maximal, complemented
12	C₂×#41	—	2	16	no	—
13	C₂×#41	C₂²	1	1	yes	characteristic
14	C₂×#41	C₂²	1	6	yes	complemented
15	C₂×#41	C₂²	1	6	yes	complemented
16	C₂×#41	C₂²	1	2	yes	complemented
17	C₂²×Q₈	C₂²	1	2	yes	complemented
18	C₂²×D₄	C₂²	1	3	yes	—
19	C₂²×D₄	C₂²	1	6	yes	complemented
20	C₂³×C₄	C₂²	1	6	yes	complemented
21	C₄⋊Q₈	C₂²	1	3	yes	complemented
22	#127	C₂²	1	3	yes	complemented
23	#124	C₂²	1	6	yes	complemented
24	#123	C₂²	1	12	yes	complemented
25	C₂²⋊Q₈	C₂²	1	6	yes	complemented
26	C₂²⋊Q₈	C₂²	1	6	yes	complemented
27	#121	C₂²	1	6	yes	complemented
28	#121	C₂²	1	6	yes	complemented
29	#121	C₂²	1	6	yes	complemented
30	#121	C₂²	1	6	yes	complemented
31	#120	C₂²	1	12	yes	complemented
32	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
33	C₄×Q₈	C₂²	1	1	yes	characteristic, complemented
34	C₄×D₄	C₂²	1	1	yes	characteristic
35	C₄×D₄	C₂²	1	3	yes	—
36	C₄×D₄	C₂²	1	6	yes	complemented
37	C₄×D₄	C₂²	1	3	yes	complemented
38	C₄×D₄	C₂²	1	6	yes	complemented
39	C₄×D₄	C₂²	1	3	yes	complemented
40	C₄×D₄	C₂²	1	3	yes	complemented
41	C₄×D₄	C₂²	1	3	yes	complemented
42	C₄×D₄	C₂²	1	6	yes	complemented
43	#117	C₂²	1	3	yes	complemented
44	C₂×C₄⋊C₄	C₂²	1	3	yes	complemented
45	C₂×C₂²⋊C₄	C₂²	1	6	yes	complemented
46	C₂×C₄²	C₂²	1	3	yes	complemented
47	C₂⁴	C₂³	1	6	yes	—
48	#41	—	2	8	no	—
49	#41	—	2	24	no	—
50	#41	—	2	24	no	—
51	#41	D₄	1	8	yes	direct product factor
52	C₂×Q₈	—	2	8	no	—
53	C₂×Q₈	C₂³	1	6	yes	—
54	C₂×Q₈	C₂³	1	1	yes	characteristic
55	C₂×D₄	—	2	24	no	—
56	C₂×D₄	—	2	12	no	—
57	C₂×D₄	C₂³	1	6	yes	—
58	C₂×D₄	C₂³	1	3	yes	—
59	C₂×D₄	C₂³	1	12	yes	—
60	C₂×D₄	C₂³	1	6	yes	—
61	C₂×D₄	C₂³	1	6	yes	—
62	C₂×D₄	C₂³	1	3	yes	—
63	C₂×D₄	C₂³	1	1	yes	characteristic
64	C₂²×C₄	—	2	24	no	—
65	C₂²×C₄	C₂³	1	3	yes	—
66	C₂²×C₄	C₂³	1	6	yes	—
67	C₂²×C₄	C₂³	1	6	yes	—
68	C₂²×C₄	C₂³	1	6	yes	—
69	C₂²×C₄	C₂³	1	2	yes	—
70	C₂²×C₄	C₂³	1	6	yes	—
71	C₂²×C₄	C₂³	1	3	yes	—
72	C₂²×C₄	C₂³	1	3	yes	—
73	C₄⋊C₄	C₂³	1	1	yes	characteristic
74	C₄⋊C₄	C₂³	1	3	yes	—
75	C₄⋊C₄	C₂³	1	6	yes	—
76	C₄⋊C₄	C₂³	1	3	yes	—
77	C₄⋊C₄	C₂³	1	3	yes	—
78	C₄⋊C₄	C₂³	1	6	yes	—
79	C₂²⋊C₄	C₂³	1	2	yes	—
80	C₂²⋊C₄	C₂³	1	6	yes	—
81	C₂²⋊C₄	C₂³	1	6	yes	—
82	C₂²⋊C₄	C₂³	1	6	yes	—
83	C₂²⋊C₄	C₂³	1	6	yes	—
84	C₂²⋊C₄	C₂³	1	12	yes	—
85	C₄²	C₂³	1	3	yes	—
86	C₄²	C₂³	1	3	yes	—
87	C₄²	C₂³	1	1	yes	characteristic
88	C₄²	C₂³	1	3	yes	—
89	C₂³	—	2	12	no	—
90	C₂³	—	2	12	no	—
91	C₂³	—	4	6	no	—
92	C₂³	C₂⁴	1	6	yes	—
93	C₂³	C₂⁴	1	3	yes	—
94	C₂³	C₂⁴	1	2	yes	—
95	Q₈	—	2	12	no	—
96	Q₈	C₂×D₄	1	4	yes	complemented
97	D₄	—	2	24	no	—
98	D₄	—	2	12	no	—
99	D₄	—	2	12	no	—
100	D₄	—	2	6	no	—
101	D₄	—	4	6	no	—
102	D₄	C₂×D₄	1	12	yes	complemented
103	D₄	#41	1	4	yes	direct product factor
104	C₂×C₄	—	2	12	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	12	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	2	12	no	—
109	C₂×C₄	—	2	6	no	—
110	C₂×C₄	—	4	3	no	—
111	C₂×C₄	C₂×D₄	1	12	yes	complemented
112	C₂×C₄	C₂⁴	1	1	yes	characteristic
113	C₂×C₄	C₂⁴	1	3	yes	—
114	C₂×C₄	C₂⁴	1	3	yes	—
115	C₂×C₄	C₂⁴	1	2	yes	—
116	C₂×C₄	C₂⁴	1	6	yes	—
117	C₂×C₄	C₂⁴	1	3	yes	—
118	C₂×C₄	C₂⁴	1	1	yes	characteristic
119	C₂×C₄	C₂⁴	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	—	2	3	no	—
122	C₂²	—	2	3	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	6	no	—
125	C₂²	—	2	6	no	—
126	C₂²	—	2	6	no	—
127	C₂²	—	4	24	no	—
128	C₂²	C₂²×D₄	1	6	yes	—
129	C₂²	C₂×#41	1	4	yes	complemented
130	C₂²	C₂⁵	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	6	no	—
133	C₄	—	2	3	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	3	no	—
136	C₄	C₂²×D₄	1	2	yes	—
137	C₄	C₂²×D₄	1	6	yes	—
138	C₄	C₂×#41	1	2	yes	complemented
139	C₂	—	2	4	no	—
140	C₂	—	2	6	no	—
141	C₂	—	4	6	no	—
142	C₂	C₂³×D₄	1	1	yes	characteristic
143	C₂	C₂²×#41	1	1	yes	characteristic
144	C₂	#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#41	C₂	semidirect
2. #550	C₂	semidirect
3. D₄×Q₈	C₂	semidirect
4. #548	C₂	semidirect
5. #547	C₂	semidirect
6. #546	C₂	semidirect
7. D₄²	C₂	semidirect
8. #532	C₂	semidirect
9. #525	C₂	semidirect
10. C₄×#41	C₂	semidirect
11. C₂×C₄×D₄	C₂	semidirect
14. C₂×#41	C₂²	semidirect
15. C₂×#41	C₂²	semidirect
16. C₂×#41	C₂²	semidirect
17. C₂²×Q₈	C₂²	semidirect
19. C₂²×D₄	C₂²	semidirect
20. C₂³×C₄	C₂²	semidirect
21. C₄⋊Q₈	C₂²	semidirect
22. #127	C₂²	semidirect
23. #124	C₂²	semidirect
24. #123	C₂²	semidirect
25. C₂²⋊Q₈	C₂²	semidirect
26. C₂²⋊Q₈	C₂²	semidirect
27. #121	C₂²	semidirect
28. #121	C₂²	semidirect
29. #121	C₂²	semidirect
30. #121	C₂²	semidirect
31. #120	C₂²	semidirect
32. C₄×Q₈	C₂²	semidirect
33. C₄×Q₈	C₂²	semidirect
36. C₄×D₄	C₂²	semidirect
37. C₄×D₄	C₂²	semidirect
38. C₄×D₄	C₂²	semidirect
39. C₄×D₄	C₂²	semidirect
40. C₄×D₄	C₂²	semidirect
41. C₄×D₄	C₂²	semidirect
42. C₄×D₄	C₂²	semidirect
43. #117	C₂²	semidirect
44. C₂×C₄⋊C₄	C₂²	semidirect
45. C₂×C₂²⋊C₄	C₂²	semidirect
46. C₂×C₄²	C₂²	semidirect
51. #41	D₄	direct (N × H)
96. Q₈	C₂×D₄	semidirect
102. D₄	C₂×D₄	semidirect
103. D₄	#41	direct (N × H)
111. C₂×C₄	C₂×D₄	semidirect
129. C₂²	C₂×#41	semidirect
138. C₄	C₂×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#41	2
#550	2
D₄×Q₈	2
#548	2
#547	2
#546	2
D₄²	2
#532	2
#525	2
C₄×#41	2
C₂×C₄×D₄	2

Subgroups — D4×#41 #3468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#41 #3468