Subgroups — C₂×#72216682 #72337612

Order 4096.

At least 13227 conjugacy classes of subgroups; only the 339 normal subgroups are shown.

The subgroup lattice has 166 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#72216484	C₂	1	2	yes	maximal, complemented
2	C₂×#72216329	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#72216449	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#72210547	C₂	1	2	yes	maximal, complemented
5	C₂×#134033	C₂²	1	2	yes	—
6	C₂×#133822	C₂²	1	2	yes	—
7	C₂×#72209716	C₂	1	1	yes	maximal, characteristic
8	C₂×#115690	C₄	1	2	yes	complemented
9	C₂×#115690	C₂²	1	1	yes	characteristic
10	C₂×#6499867	C₂²	1	1	yes	characteristic
11	#72216682	C₂	1	8	yes	maximal, direct product factor
12	#72216484	C₂²	1	8	yes	complemented
13	#72216449	C₂²	1	4	yes	complemented
14	#72216329	C₂²	1	4	yes	complemented
15	#72210547	C₂²	1	8	yes	complemented
16	C₂×#6342371	C₄	1	2	yes	complemented
17	C₂×#6342371	C₂²	1	1	yes	characteristic
18	#72209716	C₄	1	4	yes	complemented
19	#72209716	C₂²	1	4	yes	—
20	C₂⁹	C₂×C₄	1	1	yes	characteristic, complemented
21	C₂×#62817	D₄	1	1	yes	characteristic
22	C₂²×#2846	C₂×C₄	1	2	yes	—
23	C₂²×#2846	C₂³	1	1	yes	characteristic
24	#6499867	C₂×C₄	1	2	yes	—
25	#6499867	D₄	1	2	yes	—
26	#6499867	D₄	1	2	yes	—
27	#6499867	C₂³	1	2	yes	—
28	C₂×#21746	D₄	1	2	yes	—
29	C₂×#21746	D₄	1	1	yes	characteristic
30	#6342371	C₂×C₄	1	2	yes	complemented
31	#6342371	C₂×C₄	1	8	yes	complemented
32	#6342371	C₂³	1	2	yes	—
33	C₂×#14679	C₂×C₄	1	1	yes	characteristic, complemented
34	C₂×#14679	C₂×C₄	1	2	yes	complemented
35	C₂×#8547	D₄	1	2	yes	complemented
36	C₂×#8541	D₄	1	2	yes	complemented
37	#134033	C₂³	1	4	yes	—
38	#133822	D₄	1	4	yes	complemented
39	#133822	C₂³	1	4	yes	—
40	#115690	C₂×C₄	1	2	yes	complemented
41	#115690	C₂×C₄	1	8	yes	complemented
42	#115690	C₂³	1	2	yes	—
43	C₂⁸	C₂²×C₄	1	1	yes	characteristic
44	C₂⁸	C₂²×C₄	1	2	yes	complemented
45	#62817	C₂²⋊C₄	1	2	yes	—
46	#62817	C₂×D₄	1	2	yes	—
47	C₂³×#120	C₂²⋊C₄	1	2	yes	—
48	C₂³×#120	C₂×D₄	1	1	yes	characteristic
49	C₂³×#120	C₂×D₄	1	2	yes	—
50	C₂×#2846	C₂²⋊C₄	1	4	yes	—
51	C₂×#2846	C₂²×C₄	1	4	yes	—
52	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
53	C₂×#2846	C₂×D₄	1	1	yes	characteristic
54	C₂×#2846	C₂×D₄	1	1	yes	characteristic
55	C₂×#2846	C₂×D₄	1	2	yes	—
56	C₂×#2846	#41	1	2	yes	—
57	C₂×#2846	C₂⁴	1	1	yes	characteristic
58	C₂²×#379	C₂²⋊C₄	1	2	yes	—
59	C₂²×#379	C₂×D₄	1	1	yes	characteristic
60	C₂²×#379	#41	1	2	yes	—
61	#21746	C₂²⋊C₄	1	4	yes	—
62	#21746	C₂²⋊C₄	1	2	yes	—
63	#21746	C₂×D₄	1	4	yes	—
64	#21746	C₂×D₄	1	2	yes	—
65	#14679	C₂²×C₄	1	2	yes	complemented
66	#14679	C₂²×C₄	1	4	yes	complemented
67	#8547	C₂×D₄	1	4	yes	complemented
68	#8541	C₂×D₄	1	4	yes	complemented
69	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂⁷	C₄×D₄	1	2	yes	—
72	C₂⁷	C₄×D₄	1	2	yes	—
73	C₂⁷	C₂³×C₄	1	1	yes	characteristic
74	C₂⁴×D₄	#99	1	2	yes	—
75	C₂⁴×D₄	#120	1	1	yes	characteristic
76	C₂²×#120	#99	1	4	yes	—
77	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
79	C₂²×#120	C₄×D₄	1	2	yes	—
80	C₂²×#120	#120	1	1	yes	characteristic
81	C₂²×#120	#120	1	2	yes	—
82	C₂²×#120	#121	1	2	yes	—
83	C₂²×#120	#123	1	1	yes	characteristic
84	C₂²×#120	C₂²×D₄	1	2	yes	—
85	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
86	C₂³×C₂²⋊C₄	#99	1	2	yes	—
87	C₂³×C₂²⋊C₄	#121	1	2	yes	—
88	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
89	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
90	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#379	C₄×D₄	1	2	yes	—
92	C₂×#379	#121	1	2	yes	—
93	C₂×#379	#121	1	2	yes	—
94	C₂×#379	#123	1	2	yes	—
95	C₂×#379	C₂²×D₄	1	1	yes	characteristic
96	C₂×#379	C₂×#41	1	2	yes	—
97	C₂⁶	AΣL(1,𝔽₁₆)	1	2	yes	complemented
98	C₂⁶	#386	1	1	yes	characteristic
99	C₂⁶	#386	1	2	yes	—
100	C₂⁶	#386	1	1	yes	characteristic
101	C₂⁶	C₂×#99	1	2	yes	—
102	C₂⁶	C₂×#99	1	2	yes	—
103	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
104	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
105	C₂⁶	C₂×C₄×D₄	1	2	yes	—
106	C₂³×D₄	AΣL(1,𝔽₁₆)	1	4	yes	complemented
107	C₂³×D₄	#386	1	1	yes	characteristic
108	C₂³×D₄	C₂×#99	1	4	yes	—
109	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
110	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
111	C₂³×D₄	C₂×#120	1	1	yes	characteristic
112	C₂⁴×C₄	AΣL(1,𝔽₁₆)	1	2	yes	complemented
113	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
114	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
115	C₂²×C₂²⋊C₄	#386	1	2	yes	—
116	C₂²×C₂²⋊C₄	C₂×#99	1	4	yes	—
117	C₂²×C₂²⋊C₄	C₂×#121	1	2	yes	—
118	C₂²×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
119	C₂⁵	#1889	1	1	yes	characteristic
120	C₂⁵	#1889	1	1	yes	characteristic
121	C₂⁵	#1889	1	1	yes	characteristic
122	C₂⁵	#1898	1	1	yes	characteristic
123	C₂⁵	#1898	1	1	yes	characteristic
124	C₂⁵	#1898	1	2	yes	—
125	C₂⁵	#1913	1	1	yes	characteristic
126	C₂⁵	#1913	1	1	yes	characteristic
127	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
128	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
129	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	4	yes	complemented
130	C₂⁵	#2121	1	2	yes	—
131	C₂⁵	#2121	1	2	yes	—
132	C₂⁵	C₂×#386	1	1	yes	characteristic
133	C₂⁵	C₂²×#99	1	2	yes	—
134	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
135	C₂³×C₄	#1889	1	1	yes	characteristic
136	C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	1	4	yes	complemented
137	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
138	C₂⁴	#9495	1	1	yes	characteristic
139	C₂⁴	#9495	1	1	yes	characteristic
140	C₂⁴	#9495	1	1	yes	characteristic
141	C₂⁴	#9495	1	1	yes	characteristic
142	C₂⁴	#13086	1	2	yes	—
143	C₂⁴	#13086	1	2	yes	—
144	C₂⁴	#13086	1	2	yes	—
145	C₂⁴	#13086	1	2	yes	—
146	C₂⁴	C₂×#1889	1	1	yes	characteristic
147	C₂⁴	C₂×#1898	1	1	yes	characteristic
148	C₂⁴	C₂×#1913	1	1	yes	characteristic
149	C₂⁴	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
150	C₂⁴	C₂×#2121	1	2	yes	—
151	C₂³	#139716	1	1	yes	characteristic
152	C₂³	#139716	1	2	yes	—
153	C₂³	#139716	1	1	yes	characteristic
154	C₂³	#139750	1	2	yes	—
155	C₂³	#139750	1	2	yes	—
156	C₂³	C₂×#9495	1	1	yes	characteristic
157	C₂³	C₂×#13086	1	2	yes	—
158	C₂²	#72216681	1	1	yes	characteristic
159	C₂²	#72216681	1	2	yes	—
160	C₂²	#72216681	1	1	yes	characteristic
161	C₂²	C₂×#139750	1	2	yes	—
162	C₂²	C₂×#139716	1	1	yes	characteristic
163	C₂	#72216682	1	2	yes	direct product factor
164	C₂	C₂×#72216681	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72216681
commutator	C₂⁷	C₂³×C₄
Frattini, cosocle	C₂×#2846	C₂⁴
prime radical	C₂×#72209716	C₂
LCS	G ▷ C₂⁷ ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁶ ◁ C₂²×#2846 ◁ G
LECS	G ▷ C₂×#2846 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×#72216484	C₂	semidirect
2. C₂×#72216329	C₂	semidirect
3. C₂×#72216449	C₂	semidirect
4. C₂×#72210547	C₂	semidirect
8. C₂×#115690	C₄	semidirect
11. #72216682	C₂	direct (N × H)
12. #72216484	C₂²	semidirect
13. #72216449	C₂²	semidirect
14. #72216329	C₂²	semidirect
15. #72210547	C₂²	semidirect
16. C₂×#6342371	C₄	semidirect
18. #72209716	C₄	semidirect
20. C₂⁹	C₂×C₄	semidirect
30. #6342371	C₂×C₄	semidirect
31. #6342371	C₂×C₄	semidirect
33. C₂×#14679	C₂×C₄	semidirect
34. C₂×#14679	C₂×C₄	semidirect
35. C₂×#8547	D₄	semidirect
36. C₂×#8541	D₄	semidirect
38. #133822	D₄	semidirect
40. #115690	C₂×C₄	semidirect
41. #115690	C₂×C₄	semidirect
44. C₂⁸	C₂²×C₄	semidirect
65. #14679	C₂²×C₄	semidirect
66. #14679	C₂²×C₄	semidirect
67. #8547	C₂×D₄	semidirect
68. #8541	C₂×D₄	semidirect
97. C₂⁶	AΣL(1,𝔽₁₆)	semidirect
106. C₂³×D₄	AΣL(1,𝔽₁₆)	semidirect
112. C₂⁴×C₄	AΣL(1,𝔽₁₆)	semidirect
129. C₂⁵	C₂×AΣL(1,𝔽₁₆)	semidirect
136. C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	semidirect
163. C₂	#72216682	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#72216484	2
C₂×#72216329	2
C₂×#72216449	2
C₂×#72210547	2
C₂×#72209716	2
#72216682	2

Subgroups — C2×#72216682 #72337612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#72216682 #72337612