Subgroups — C₂²×#4466 #10947841

Order 768.

782 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×C₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₂²×#812	C₂	1	2	yes	maximal
3	C₂×#4466	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#347	—	3	1	no	maximal, Sylow
5	C₂³×C₃×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₃×#97	C₄	1	6	yes	—
9	C₂×C₃×#97	C₂²	1	6	yes	—
10	C₂×#812	C₄	1	12	yes	complemented
11	C₂×#812	C₂²	1	12	yes	—
12	C₂²×C₃⋊C₁₆	C₄	1	4	yes	complemented
13	#4466	C₂²	1	16	yes	direct product factor
14	C₂²×M₃₂	—	3	2	no	—
15	C₂²×#97	D₃	1	1	yes	characteristic
16	C₂×#347	—	3	12	no	—
17	C₂³×C₃×C₄	—	2	1	no	—
18	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₈	—	2	3	no	—
20	C₂²×C₃×C₈	—	2	1	no	—
21	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₃×C₈	C₂×C₄	1	6	yes	—
23	C₂²×C₃×C₈	C₂×C₄	1	6	yes	—
24	C₂²×C₃×C₈	C₂×C₄	1	3	yes	—
25	C₂²×C₃×C₈	C₂³	1	3	yes	—
26	C₂×C₃×C₄²	—	2	3	no	—
27	C₂×C₃×C₄²	C₂×C₄	1	6	yes	—
28	C₃×#97	C₂×C₄	1	12	yes	—
29	C₃×#97	C₂³	1	4	yes	—
30	#812	C₂×C₄	1	24	yes	complemented
31	#812	C₂³	1	8	yes	—
32	C₂×C₃⋊C₁₆	C₂×C₄	1	24	yes	complemented
33	C₂³×C₈	Dic₃	1	1	yes	characteristic
34	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₄²	Dic₃	1	1	yes	characteristic
36	C₂×M₃₂	—	3	12	no	—
37	C₂×M₃₂	—	3	12	no	—
38	C₂²×C₁₆	—	3	4	no	—
39	C₂×#97	Dic₃	1	6	yes	—
40	C₂×#97	C₂×D₃	1	6	yes	—
41	#347	—	3	16	no	—
42	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
43	C₂²×C₃×C₄	—	2	6	no	—
44	C₂²×C₃×C₄	—	4	4	no	—
45	C₂²×C₃×C₄	C₄²	1	3	yes	—
46	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
47	C₂²×C₃×C₄	M₁₆	1	3	yes	—
48	C₂²×C₃×C₄	M₁₆	1	3	yes	—
49	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
50	C₂²×C₃×C₄	C₂²×C₄	1	3	yes	—
51	C₂×C₃×C₈	—	2	6	no	—
52	C₂×C₃×C₈	—	2	6	no	—
53	C₂×C₃×C₈	C₄²	1	6	yes	—
54	C₂×C₃×C₈	C₄²	1	6	yes	—
55	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
56	C₂×C₃×C₈	C₂²×C₄	1	6	yes	—
57	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
58	C₂×C₃×C₈	C₂²×C₄	1	3	yes	—
59	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
60	C₃×C₄²	—	2	6	no	—
61	C₃×C₄²	C₂²×C₄	1	4	yes	—
62	C₃⋊C₁₆	C₂²×C₄	1	16	yes	complemented
63	C₂³×C₄	—	2	1	no	—
64	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	2	3	no	—
67	C₂²×C₈	C₄×D₃	1	6	yes	—
68	C₂²×C₈	C₄×D₃	1	2	yes	—
69	C₂²×C₈	C₂×Dic₃	1	6	yes	—
70	C₂²×C₈	C₂×Dic₃	1	3	yes	—
71	C₂²×C₈	C₂²×D₃	1	3	yes	—
72	C₂×C₄²	—	2	3	no	—
73	C₂×C₄²	C₂×Dic₃	1	6	yes	—
74	M₃₂	—	3	24	no	—
75	M₃₂	—	3	8	no	—
76	C₂×C₁₆	—	3	24	no	—
77	#97	C₂×Dic₃	1	12	yes	—
78	#97	C₂²×D₃	1	4	yes	—
79	C₂³×C₃	—	2	4	no	—
80	C₂³×C₃	#97	1	3	yes	—
81	C₂³×C₃	#97	1	1	yes	characteristic
82	C₂³×C₃	C₂×M₁₆	1	3	yes	—
83	C₂×C₃×C₄	—	2	4	no	—
84	C₂×C₃×C₄	—	4	12	no	—
85	C₂×C₃×C₄	#97	1	6	yes	—
86	C₂×C₃×C₄	#97	1	3	yes	—
87	C₂×C₃×C₄	#97	1	3	yes	—
88	C₂×C₃×C₄	C₂×C₄²	1	3	yes	—
89	C₂×C₃×C₄	C₂×C₄²	1	3	yes	—
90	C₂×C₃×C₄	C₂×M₁₆	1	3	yes	—
91	C₂×C₃×C₄	C₂×M₁₆	1	3	yes	—
92	C₂×C₃×C₄	C₂×M₁₆	1	3	yes	—
93	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
94	C₃×C₈	—	2	4	no	—
95	C₃×C₈	C₂×C₄²	1	6	yes	—
96	C₃×C₈	C₂³×C₄	1	2	yes	—
97	C₂⁴	#208	1	1	yes	characteristic
98	C₂²×C₄	—	2	6	no	—
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	#208	1	3	yes	—
101	C₂²×C₄	#208	1	1	yes	characteristic
102	C₂²×C₄	#208	1	3	yes	—
103	C₂²×C₄	C₄×Dic₃	1	3	yes	—
104	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²×Dic₃	1	3	yes	—
106	C₂×C₈	—	2	6	no	—
107	C₂×C₈	—	2	6	no	—
108	C₂×C₈	C₄×Dic₃	1	6	yes	—
109	C₂×C₈	C₄×Dic₃	1	6	yes	—
110	C₂×C₈	C₂×C₄×D₃	1	6	yes	—
111	C₂×C₈	C₂×C₄×D₃	1	6	yes	—
112	C₂×C₈	C₂²×Dic₃	1	4	yes	—
113	C₂×C₈	C₂²×Dic₃	1	3	yes	—
114	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
115	C₄²	—	2	6	no	—
116	C₄²	C₂²×Dic₃	1	4	yes	—
117	C₁₆	—	3	16	no	—
118	C₂²×C₃	—	2	6	no	—
119	C₂²×C₃	—	2	6	no	—
120	C₂²×C₃	#347	1	4	yes	complemented
121	C₂²×C₃	C₂×#97	1	3	yes	—
122	C₂²×C₃	C₂×#97	1	3	yes	—
123	C₂²×C₃	C₂²×M₁₆	1	1	yes	characteristic
124	C₃×C₄	—	4	4	no	—
125	C₃×C₄	C₂×#97	1	3	yes	—
126	C₃×C₄	C₂×#97	1	3	yes	—
127	C₃×C₄	C₂²×C₄²	1	1	yes	characteristic
128	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
129	C₂³	—	2	4	no	—
130	C₂³	#803	1	3	yes	—
131	C₂³	#803	1	1	yes	characteristic
132	C₂³	C₂×#208	1	3	yes	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	12	no	—
135	C₂×C₄	#803	1	6	yes	—
136	C₂×C₄	#803	1	3	yes	—
137	C₂×C₄	#803	1	3	yes	—
138	C₂×C₄	C₂×#208	1	3	yes	—
139	C₂×C₄	C₂×#208	1	3	yes	—
140	C₂×C₄	C₂×#208	1	3	yes	—
141	C₂×C₄	C₂×C₄×Dic₃	1	3	yes	—
142	C₂×C₄	C₂×C₄×Dic₃	1	3	yes	—
143	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	C₂×C₄×Dic₃	1	6	yes	—
146	C₈	C₂²×C₄×D₃	1	2	yes	—
147	C₂×C₃	—	2	1	no	—
148	C₂×C₃	—	2	3	no	—
149	C₂×C₃	C₂×#347	1	6	yes	complemented
150	C₂×C₃	C₂²×#97	1	1	yes	characteristic
151	C₂²	—	2	6	no	—
152	C₂²	—	2	6	no	—
153	C₂²	#4466	1	4	yes	direct product factor
154	C₂²	C₂×#803	1	3	yes	—
155	C₂²	C₂×#803	1	3	yes	—
156	C₂²	C₂²×#208	1	1	yes	characteristic
157	C₄	—	4	4	no	—
158	C₄	C₂×#803	1	3	yes	—
159	C₄	C₂×#803	1	3	yes	—
160	C₄	C₂²×#208	1	1	yes	characteristic
161	C₄	C₂²×C₄×Dic₃	1	1	yes	characteristic
162	C₃	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
163	C₂	—	2	1	no	—
164	C₂	—	2	3	no	—
165	C₂	C₂×#4466	1	6	yes	direct product factor
166	C₂	C₂²×#803	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₃
commutator	C₃×C₄	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂²×C₃×#97	C₂
socle	C₂³×C₃	#97
prime radical	C₂⁴×C₃	M₁₆
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂²×#97	D₃
hypocenter	C₃	C₂²×#347
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. C₂×#4466	C₂	direct (N × H)
10. C₂×#812	C₄	semidirect
12. C₂²×C₃⋊C₁₆	C₄	semidirect
13. #4466	C₂²	direct (N × H)
30. #812	C₂×C₄	semidirect
32. C₂×C₃⋊C₁₆	C₂×C₄	semidirect
62. C₃⋊C₁₆	C₂²×C₄	semidirect
120. C₂²×C₃	#347	semidirect
149. C₂×C₃	C₂×#347	semidirect
153. C₂²	#4466	direct (N × H)
162. C₃	C₂²×#347	semidirect
165. C₂	C₂×#4466	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×#97	2
C₂²×#812	2
C₂×#4466	2
C₂²×#347	3

Subgroups — C22×#4466 #10947841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4466 #10947841