Subgroups — D₄×2₋¹⁺⁴ #62996

Order 256.

5525 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	#3499	C₂	1	10	yes	maximal, complemented
3	#3497	C₂	1	5	yes	maximal, complemented
4	#3489	C₂	1	20	yes	maximal, complemented
5	#3486	C₂	1	10	yes	maximal, complemented
6	D₄×#41	C₂	1	10	yes	maximal, complemented
7	C₂×D₄×Q₈	C₂	1	5	yes	maximal, complemented
8	C₄×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴	—	2	32	no	—
10	C₂×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
11	C₂×2₋¹⁺⁴	C₂²	1	10	yes	complemented
12	C₂×2₋¹⁺⁴	C₂²	1	20	yes	complemented
13	C₂²×#41	C₂²	1	20	yes	complemented
14	C₂³×Q₈	C₂²	1	10	yes	complemented
15	#550	C₂²	1	10	yes	complemented
16	D₄×Q₈	C₂²	1	10	yes	—
17	D₄×Q₈	C₂²	1	40	yes	complemented
18	D₄×Q₈	C₂²	1	20	yes	complemented
19	D₄×Q₈	C₂²	1	20	yes	complemented
20	#548	C₂²	1	20	yes	complemented
21	#548	C₂²	1	40	yes	complemented
22	#547	C₂²	1	30	yes	complemented
23	#547	C₂²	1	60	yes	complemented
24	#546	C₂²	1	60	yes	complemented
25	D₄²	C₂²	1	10	yes	—
26	#537	C₂²	1	30	yes	complemented
27	#536	C₂²	1	60	yes	complemented
28	#532	C₂²	1	30	yes	complemented
29	C₂×C₄⋊Q₈	C₂²	1	15	yes	complemented
30	#525	C₂²	1	60	yes	complemented
31	C₂×C₂²⋊Q₈	C₂²	1	30	yes	complemented
32	#520	C₂²	1	10	yes	complemented
33	#519	C₂²	1	5	yes	complemented
34	C₄×#41	C₂²	1	10	yes	complemented
35	C₂×C₄×Q₈	C₂²	1	5	yes	complemented
36	C₂×C₄×D₄	C₂²	1	15	yes	—
37	2₋¹⁺⁴	—	2	160	no	—
38	2₋¹⁺⁴	—	2	80	no	—
39	2₋¹⁺⁴	D₄	1	16	yes	direct product factor
40	C₂×#41	—	2	160	no	—
41	C₂×#41	C₂³	1	60	yes	—
42	C₂×#41	C₂³	1	60	yes	—
43	C₂×#41	C₂³	1	20	yes	—
44	C₂×#41	C₂³	1	10	yes	—
45	C₂²×Q₈	—	2	80	no	—
46	C₂²×Q₈	C₂³	1	30	yes	—
47	C₂²×Q₈	C₂³	1	20	yes	—
48	C₂²×Q₈	C₂³	1	40	yes	—
49	C₂²×Q₈	C₂³	1	5	yes	—
50	C₂²×D₄	C₂³	1	5	yes	—
51	C₂²×D₄	C₂³	1	20	yes	—
52	C₂³×C₄	C₂³	1	30	yes	—
53	C₄⋊Q₈	C₂³	1	30	yes	—
54	C₄⋊Q₈	C₂³	1	60	yes	—
55	#127	C₂³	1	10	yes	—
56	#124	C₂³	1	60	yes	—
57	#123	C₂³	1	120	yes	—
58	C₂²⋊Q₈	C₂³	1	60	yes	—
59	C₂²⋊Q₈	C₂³	1	120	yes	—
60	C₂²⋊Q₈	C₂³	1	60	yes	—
61	#121	C₂³	1	20	yes	—
62	#121	C₂³	1	60	yes	—
63	#121	C₂³	1	20	yes	—
64	#121	C₂³	1	60	yes	—
65	#120	C₂³	1	40	yes	—
66	C₄×Q₈	C₂³	1	10	yes	—
67	C₄×Q₈	C₂³	1	20	yes	—
68	C₄×Q₈	C₂³	1	10	yes	—
69	C₄×D₄	C₂³	1	10	yes	—
70	C₄×D₄	C₂³	1	60	yes	—
71	C₄×D₄	C₂³	1	30	yes	—
72	C₄×D₄	C₂³	1	10	yes	—
73	C₄×D₄	C₂³	1	30	yes	—
74	C₄×D₄	C₂³	1	60	yes	—
75	#117	C₂³	1	30	yes	—
76	C₂×C₄⋊C₄	C₂³	1	15	yes	—
77	C₂×C₄⋊C₄	C₂³	1	15	yes	—
78	C₂×C₄⋊C₄	C₂³	1	30	yes	—
79	C₂×C₂²⋊C₄	C₂³	1	30	yes	—
80	C₂×C₄²	C₂³	1	15	yes	—
81	C₂⁴	C₂⁴	1	10	yes	—
82	#41	—	2	80	no	—
83	#41	—	2	240	no	—
84	#41	—	2	240	no	—
85	#41	C₂×D₄	1	80	yes	complemented
86	C₂×Q₈	—	2	80	no	—
87	C₂×Q₈	—	2	160	no	—
88	C₂×Q₈	—	2	120	no	—
89	C₂×Q₈	C₂×D₄	1	40	yes	complemented
90	C₂×Q₈	C₂⁴	1	60	yes	—
91	C₂×Q₈	C₂⁴	1	10	yes	—
92	C₂×D₄	—	2	80	no	—
93	C₂×D₄	—	2	20	no	—
94	C₂×D₄	C₂⁴	1	10	yes	—
95	C₂×D₄	C₂⁴	1	5	yes	—
96	C₂×D₄	C₂⁴	1	40	yes	—
97	C₂×D₄	C₂⁴	1	20	yes	—
98	C₂×D₄	C₂⁴	1	20	yes	—
99	C₂×D₄	C₂⁴	1	10	yes	—
100	C₂×D₄	C₂⁴	1	1	yes	characteristic
101	C₂²×C₄	—	2	120	no	—
102	C₂²×C₄	C₂⁴	1	5	yes	—
103	C₂²×C₄	C₂⁴	1	30	yes	—
104	C₂²×C₄	C₂⁴	1	20	yes	—
105	C₂²×C₄	C₂⁴	1	60	yes	—
106	C₂²×C₄	C₂⁴	1	15	yes	—
107	C₂²×C₄	C₂⁴	1	15	yes	—
108	C₄⋊C₄	C₂⁴	1	10	yes	—
109	C₄⋊C₄	C₂⁴	1	30	yes	—
110	C₄⋊C₄	C₂⁴	1	60	yes	—
111	C₄⋊C₄	C₂⁴	1	30	yes	—
112	C₄⋊C₄	C₂⁴	1	10	yes	—
113	C₂²⋊C₄	C₂⁴	1	20	yes	—
114	C₂²⋊C₄	C₂⁴	1	60	yes	—
115	C₂²⋊C₄	C₂⁴	1	20	yes	—
116	C₂²⋊C₄	C₂⁴	1	40	yes	—
117	C₄²	C₂⁴	1	10	yes	—
118	C₄²	C₂⁴	1	30	yes	—
119	C₂³	—	2	20	no	—
120	C₂³	—	2	20	no	—
121	C₂³	—	4	10	no	—
122	C₂³	C₂⁵	1	10	yes	—
123	C₂³	C₂⁵	1	5	yes	—
124	C₂³	C₂⁵	1	2	yes	—
125	Q₈	—	2	120	no	—
126	Q₈	C₂²×D₄	1	40	yes	—
127	D₄	—	2	80	no	—
128	D₄	—	2	20	no	—
129	D₄	—	2	40	no	—
130	D₄	—	2	10	no	—
131	D₄	—	4	20	no	—
132	D₄	C₂²×D₄	1	40	yes	—
133	D₄	2₋¹⁺⁴	1	4	yes	direct product factor
134	C₂×C₄	—	2	60	no	—
135	C₂×C₄	—	2	40	no	—
136	C₂×C₄	—	2	120	no	—
137	C₂×C₄	—	2	10	no	—
138	C₂×C₄	—	4	15	no	—
139	C₂×C₄	C₂²×D₄	1	60	yes	—
140	C₂×C₄	C₂⁵	1	5	yes	—
141	C₂×C₄	C₂⁵	1	10	yes	—
142	C₂×C₄	C₂⁵	1	20	yes	—
143	C₂×C₄	C₂⁵	1	10	yes	—
144	C₂×C₄	C₂⁵	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	—	2	5	no	—
147	C₂²	—	2	5	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	10	no	—
150	C₂²	—	2	10	no	—
151	C₂²	—	2	10	no	—
152	C₂²	—	4	40	no	—
153	C₂²	C₂³×D₄	1	10	yes	—
154	C₂²	C₂×2₋¹⁺⁴	1	4	yes	complemented
155	C₂²	C₂⁶	1	1	yes	characteristic
156	C₄	—	2	20	no	—
157	C₄	—	2	5	no	—
158	C₄	—	2	10	no	—
159	C₄	C₂³×D₄	1	20	yes	—
160	C₄	C₂×2₋¹⁺⁴	1	2	yes	complemented
161	C₂	—	2	4	no	—
162	C₂	—	2	10	no	—
163	C₂	—	4	10	no	—
164	C₂	C₂⁴×D₄	1	1	yes	characteristic
165	C₂	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
166	C₂	2₋¹⁺⁶	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
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. C₂²×2₋¹⁺⁴	C₂	semidirect
2. #3499	C₂	semidirect
3. #3497	C₂	semidirect
4. #3489	C₂	semidirect
5. #3486	C₂	semidirect
6. D₄×#41	C₂	semidirect
7. C₂×D₄×Q₈	C₂	semidirect
8. C₄×2₋¹⁺⁴	C₂	semidirect
11. C₂×2₋¹⁺⁴	C₂²	semidirect
12. C₂×2₋¹⁺⁴	C₂²	semidirect
13. C₂²×#41	C₂²	semidirect
14. C₂³×Q₈	C₂²	semidirect
15. #550	C₂²	semidirect
17. D₄×Q₈	C₂²	semidirect
18. D₄×Q₈	C₂²	semidirect
19. D₄×Q₈	C₂²	semidirect
20. #548	C₂²	semidirect
21. #548	C₂²	semidirect
22. #547	C₂²	semidirect
23. #547	C₂²	semidirect
24. #546	C₂²	semidirect
26. #537	C₂²	semidirect
27. #536	C₂²	semidirect
28. #532	C₂²	semidirect
29. C₂×C₄⋊Q₈	C₂²	semidirect
30. #525	C₂²	semidirect
31. C₂×C₂²⋊Q₈	C₂²	semidirect
32. #520	C₂²	semidirect
33. #519	C₂²	semidirect
34. C₄×#41	C₂²	semidirect
35. C₂×C₄×Q₈	C₂²	semidirect
39. 2₋¹⁺⁴	D₄	direct (N × H)
85. #41	C₂×D₄	semidirect
89. C₂×Q₈	C₂×D₄	semidirect
133. D₄	2₋¹⁺⁴	direct (N × H)
154. C₂²	C₂×2₋¹⁺⁴	semidirect
160. C₄	C₂×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂²×2₋¹⁺⁴	2
#3499	2
#3497	2
#3489	2
#3486	2
D₄×#41	2
C₂×D₄×Q₈	2
C₄×2₋¹⁺⁴	2

Subgroups — D4×2−1+4 #62996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×2₋¹⁺⁴ #62996