Subgroups — C₂×#3444 #62668

Order 256.

4259 conjugacy classes of subgroups.

The subgroup lattice has 154 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3444	C₂	1	32	yes	maximal, direct product factor
2	C₂×#533	C₂	1	3	yes	maximal, complemented
3	C₂×#532	C₂	1	3	yes	maximal, complemented
4	C₂×#529	C₂	1	12	yes	maximal, complemented
5	C₂×#525	C₂	1	6	yes	maximal, complemented
6	C₂×C₄×#41	C₂	1	6	yes	maximal, complemented
7	C₂³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41	C₂²	1	6	yes	—
9	C₂⁴×C₄	C₂²	1	3	yes	—
10	#533	C₂²	1	48	yes	complemented
11	#532	C₂²	1	48	yes	complemented
12	C₂×C₄⋊Q₈	C₂²	1	3	yes	complemented
13	C₂×#127	C₂²	1	1	yes	characteristic, complemented
14	#529	C₂²	1	192	yes	complemented
15	C₂×#126	C₂²	1	8	yes	complemented
16	C₂×#125	C₂²	1	6	yes	complemented
17	C₂×#124	C₂²	1	6	yes	complemented
18	#525	C₂²	1	96	yes	complemented
19	C₂×#123	C₂²	1	12	yes	—
20	C₂×C₂²⋊Q₈	C₂²	1	12	yes	—
21	C₂×#121	C₂²	1	12	yes	—
22	C₂×#120	C₂²	1	4	yes	—
23	C₄×#41	C₂²	1	96	yes	complemented
24	C₂×C₄×Q₈	C₂²	1	12	yes	complemented
25	C₂×C₄×D₄	C₂²	1	24	yes	—
26	C₂×C₄×D₄	C₂²	1	12	yes	complemented
27	C₂×#117	C₂²	1	6	yes	—
28	C₂×#117	C₂²	1	12	yes	complemented
29	C₂²×C₄²	C₂²	1	6	yes	—
30	C₂²×C₄²	C₂²	1	3	yes	—
31	C₂²×C₄²	C₂²	1	3	yes	—
32	C₂²×C₄²	C₂²	1	16	yes	complemented
33	C₂²×C₄²	C₂²	1	4	yes	complemented
34	C₂⁵	C₂³	1	1	yes	characteristic
35	C₂×#41	—	2	48	no	—
36	C₂×#41	C₂³	1	48	yes	—
37	C₂²×Q₈	C₂³	1	6	yes	—
38	C₂²×D₄	C₂³	1	12	yes	—
39	C₂²×D₄	C₂³	1	6	yes	—
40	C₂³×C₄	—	2	24	no	—
41	C₂³×C₄	C₂³	1	12	yes	—
42	C₂³×C₄	C₂³	1	6	yes	—
43	C₂³×C₄	C₂³	1	6	yes	—
44	C₂³×C₄	C₂³	1	3	yes	—
45	C₂³×C₄	C₂³	1	24	yes	—
46	C₂³×C₄	C₂³	1	3	yes	—
47	C₄⋊Q₈	C₂³	1	24	yes	complemented
48	#127	C₂³	1	8	yes	complemented
49	#126	C₂³	1	64	yes	complemented
50	#125	C₂³	1	48	yes	complemented
51	#124	C₂³	1	48	yes	complemented
52	#123	C₂³	1	96	yes	—
53	C₂²⋊Q₈	C₂³	1	96	yes	—
54	#121	C₂³	1	96	yes	—
55	#120	C₂³	1	32	yes	—
56	C₄×Q₈	C₂³	1	96	yes	complemented
57	C₄×D₄	C₂³	1	192	yes	—
58	C₄×D₄	C₂³	1	96	yes	complemented
59	#117	C₂³	1	48	yes	—
60	#117	C₂³	1	96	yes	complemented
61	C₂×C₄⋊C₄	C₂³	1	12	yes	—
62	C₂×C₄⋊C₄	C₂³	1	24	yes	—
63	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
64	C₂×C₂²⋊C₄	C₂³	1	24	yes	—
65	C₂×C₄²	C₂³	1	12	yes	—
66	C₂×C₄²	C₂³	1	3	yes	—
67	C₂×C₄²	C₂³	1	6	yes	—
68	C₂×C₄²	C₂³	1	24	yes	—
69	C₂×C₄²	C₂³	1	3	yes	—
70	C₂×C₄²	C₂³	1	2	yes	—
71	C₂×C₄²	C₂³	1	48	yes	—
72	C₂×C₄²	C₂³	1	1	yes	characteristic
73	C₂×C₄²	C₂³	1	24	yes	—
74	C₂×C₄²	C₂³	1	1	yes	characteristic
75	C₂×C₄²	C₂³	1	32	yes	complemented
76	C₂⁴	—	2	12	no	—
77	C₂⁴	C₂⁴	1	4	yes	—
78	C₂⁴	C₂⁴	1	4	yes	—
79	C₂⁴	C₂⁴	1	3	yes	—
80	#41	—	2	192	no	—
81	C₂×Q₈	—	2	24	no	—
82	C₂×Q₈	C₂⁴	1	24	yes	—
83	C₂×D₄	—	2	48	no	—
84	C₂×D₄	—	2	24	no	—
85	C₂×D₄	C₂⁴	1	48	yes	—
86	C₂×D₄	C₂⁴	1	24	yes	—
87	C₂²×C₄	—	2	12	no	—
88	C₂²×C₄	—	2	24	no	—
89	C₂²×C₄	—	2	24	no	—
90	C₂²×C₄	—	2	48	no	—
91	C₂²×C₄	—	2	96	no	—
92	C₂²×C₄	#41	1	24	yes	complemented
93	C₂²×C₄	C₂⁴	1	6	yes	—
94	C₂²×C₄	C₂⁴	1	24	yes	—
95	C₂²×C₄	C₂⁴	1	24	yes	—
96	C₂²×C₄	C₂⁴	1	12	yes	—
97	C₂²×C₄	C₂⁴	1	12	yes	—
98	C₂²×C₄	C₂⁴	1	48	yes	—
99	C₂²×C₄	C₂⁴	1	3	yes	—
100	C₂²×C₄	C₂⁴	1	3	yes	—
101	C₂²×C₄	C₂⁴	1	12	yes	—
102	C₄⋊C₄	C₂⁴	1	48	yes	—
103	C₄⋊C₄	C₂⁴	1	96	yes	—
104	C₂²⋊C₄	C₂⁴	1	48	yes	—
105	C₂²⋊C₄	C₂⁴	1	96	yes	—
106	C₄²	C₂⁴	1	12	yes	—
107	C₄²	C₂⁴	1	24	yes	—
108	C₄²	C₂⁴	1	12	yes	—
109	C₄²	C₂⁴	1	8	yes	—
110	C₄²	C₂⁴	1	4	yes	—
111	C₄²	C₂⁴	1	4	yes	—
112	C₄²	C₂⁴	1	48	yes	—
113	C₂³	—	2	32	no	—
114	C₂³	—	2	24	no	—
115	C₂³	—	2	12	no	—
116	C₂³	—	2	24	no	—
117	C₂³	C₂×#41	1	12	yes	—
118	C₂³	C₂⁵	1	8	yes	—
119	C₂³	C₂⁵	1	6	yes	—
120	C₂³	C₂⁵	1	1	yes	characteristic
121	Q₈	—	2	48	no	—
122	D₄	—	2	96	no	—
123	D₄	—	2	48	no	—
124	C₂×C₄	—	2	24	no	—
125	C₂×C₄	—	2	12	no	—
126	C₂×C₄	—	2	24	no	—
127	C₂×C₄	—	2	48	no	—
128	C₂×C₄	—	2	48	no	—
129	C₂×C₄	—	2	96	no	—
130	C₂×C₄	C₂×#41	1	12	yes	—
131	C₂×C₄	C₂×#41	1	12	yes	—
132	C₂×C₄	C₂×#41	1	48	yes	complemented
133	C₂×C₄	C₂⁵	1	12	yes	—
134	C₂×C₄	C₂⁵	1	24	yes	—
135	C₂×C₄	C₂⁵	1	6	yes	—
136	C₂×C₄	C₂⁵	1	6	yes	—
137	C₂²	—	2	32	no	—
138	C₂²	—	2	12	no	—
139	C₂²	—	2	24	no	—
140	C₂²	—	2	24	no	—
141	C₂²	—	4	16	no	—
142	C₂²	C₂²×#41	1	12	yes	—
143	C₂²	C₂²×#41	1	6	yes	—
144	C₂²	C₂⁶	1	1	yes	characteristic
145	C₄	—	2	12	no	—
146	C₄	—	2	24	no	—
147	C₄	C₂²×#41	1	12	yes	—
148	C₄	C₂²×#41	1	12	yes	—
149	C₂	—	2	12	no	—
150	C₂	—	4	8	no	—
151	C₂	#3444	1	4	yes	direct product factor
152	C₂	C₂³×#41	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂³	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3444	C₂	direct (N × H)
2. C₂×#533	C₂	semidirect
3. C₂×#532	C₂	semidirect
4. C₂×#529	C₂	semidirect
5. C₂×#525	C₂	semidirect
6. C₂×C₄×#41	C₂	semidirect
7. C₂³×C₄²	C₂	semidirect
10. #533	C₂²	semidirect
11. #532	C₂²	semidirect
12. C₂×C₄⋊Q₈	C₂²	semidirect
13. C₂×#127	C₂²	semidirect
14. #529	C₂²	semidirect
15. C₂×#126	C₂²	semidirect
16. C₂×#125	C₂²	semidirect
17. C₂×#124	C₂²	semidirect
18. #525	C₂²	semidirect
23. C₄×#41	C₂²	semidirect
24. C₂×C₄×Q₈	C₂²	semidirect
26. C₂×C₄×D₄	C₂²	semidirect
28. C₂×#117	C₂²	semidirect
32. C₂²×C₄²	C₂²	semidirect
33. C₂²×C₄²	C₂²	semidirect
47. C₄⋊Q₈	C₂³	semidirect
48. #127	C₂³	semidirect
49. #126	C₂³	semidirect
50. #125	C₂³	semidirect
51. #124	C₂³	semidirect
56. C₄×Q₈	C₂³	semidirect
58. C₄×D₄	C₂³	semidirect
60. #117	C₂³	semidirect
75. C₂×C₄²	C₂³	semidirect
92. C₂²×C₄	#41	semidirect
132. C₂×C₄	C₂×#41	semidirect
151. C₂	#3444	direct (N × H)

Maximal subgroups

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

subgroup	index
#3444	2
C₂×#533	2
C₂×#532	2
C₂×#529	2
C₂×#525	2
C₂×C₄×#41	2
C₂³×C₄²	2

Subgroups — C2×#3444 #62668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3444 #62668