Subgroups — C₂²×#4682 #11664785

Order 768.

2398 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#869	C₂	1	2	yes	maximal
2	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×#818	C₂	1	4	yes	maximal
4	C₂×#4682	C₂	1	24	yes	maximal, direct product factor
5	C₂²×#500	—	3	1	no	maximal, Sylow
6	C₂³×Dic₆	—	2	2	no	—
7	C₂³×C₃×C₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₃	—	2	2	no	—
10	C₂²×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₂×#869	—	2	12	no	—
12	C₂×#869	C₂²	1	24	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	12	yes	—
14	C₂×#818	C₂²	1	48	yes	—
15	#4682	C₂²	1	64	yes	direct product factor
16	C₂²×C₄⋊Q₈	—	3	2	no	—
17	C₂×#500	—	3	24	no	—
18	C₂²×#107	—	3	4	no	—
19	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
20	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
21	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
22	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂³×Dic₃	—	2	4	no	—
24	C₂²×Dic₆	—	2	24	no	—
25	C₂²×Dic₆	—	4	16	no	—
26	C₂²×C₃×C₈	Q₈	1	16	yes	complemented
27	C₂²×C₃×C₈	C₂³	1	12	yes	—
28	C₂×C₃×C₄²	D₄	1	6	yes	—
29	C₂×C₃×C₄²	C₂³	1	6	yes	—
30	C₂×C₄⋊Dic₃	—	2	12	no	—
31	C₂×C₄⋊Dic₃	—	2	24	no	—
32	C₂×C₄⋊Dic₃	C₂³	1	24	yes	—
33	#869	—	2	48	no	—
34	#869	C₂³	1	32	yes	—
35	C₃×C₄×C₈	C₂³	1	16	yes	—
36	#818	C₂³	1	64	yes	—
37	C₂³×Q₈	—	6	2	no	—
38	C₂³×C₈	C₂×D₃	1	2	yes	—
39	C₂×C₄⋊Q₈	—	3	24	no	—
40	C₂×C₄⋊Q₈	—	6	12	no	—
41	C₂²×C₄⋊C₄	—	3	4	no	—
42	C₂²×C₄⋊C₄	—	6	2	no	—
43	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
44	#500	—	3	64	no	—
45	C₂×#107	—	3	48	no	—
46	C₂×C₄×C₈	C₂×D₃	1	12	yes	—
47	C₂⁴×C₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₃×C₄	D₈	1	8	yes	—
49	C₂²×C₃×C₄	Q₁₆	1	8	yes	—
50	C₂²×C₃×C₄	C₂×D₄	1	3	yes	—
51	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
52	C₂²×C₃×C₄	C₂×D₄	1	6	yes	—
53	C₂²×C₃×C₄	C₂×Q₈	1	8	yes	—
54	C₂²×C₃×C₄	C₂⁴	1	3	yes	—
55	C₂²×Dic₃	—	2	24	no	—
56	C₂²×Dic₃	—	4	16	no	—
57	C₂×Dic₆	—	2	32	no	—
58	C₂×Dic₆	—	4	96	no	—
59	C₂×C₃×C₈	C₂×Q₈	1	48	yes	complemented
60	C₂×C₃×C₈	C₂⁴	1	8	yes	—
61	C₃×C₄²	C₂×D₄	1	12	yes	—
62	C₃×C₄²	C₂⁴	1	4	yes	—
63	C₄⋊Dic₃	—	2	24	no	—
64	C₄⋊Dic₃	—	2	32	no	—
65	C₄⋊Dic₃	C₂⁴	1	16	yes	—
66	C₂²×Q₈	—	6	24	no	—
67	C₂²×Q₈	—	12	16	no	—
68	C₂³×C₄	—	6	4	no	—
69	C₂³×C₄	D₁₂	1	1	yes	characteristic
70	C₂³×C₄	D₁₂	1	1	yes	characteristic
71	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₂²×C₈	Dic₆	1	16	yes	complemented
73	C₂²×C₈	C₂²×D₃	1	12	yes	—
74	C₄⋊Q₈	—	3	32	no	—
75	C₄⋊Q₈	—	6	48	no	—
76	C₂×C₄⋊C₄	—	3	24	no	—
77	C₂×C₄⋊C₄	—	6	12	no	—
78	C₂×C₄⋊C₄	—	6	24	no	—
79	C₂×C₄²	D₁₂	1	6	yes	—
80	C₂×C₄²	C₂²×D₃	1	6	yes	—
81	#107	—	3	64	no	—
82	C₄×C₈	C₂²×D₃	1	16	yes	—
83	C₂³×C₃	C₄⋊Q₈	1	4	yes	—
84	C₂³×C₃	C₂×D₈	1	4	yes	—
85	C₂³×C₃	C₂×Q₁₆	1	4	yes	—
86	C₂³×C₃	C₂²×D₄	1	3	yes	—
87	C₂×C₃×C₄	C₄⋊Q₈	1	12	yes	—
88	C₂×C₃×C₄	C₂×D₈	1	24	yes	—
89	C₂×C₃×C₄	C₂×Q₁₆	1	24	yes	—
90	C₂×C₃×C₄	C₂²×D₄	1	3	yes	—
91	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
92	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
93	C₂×C₃×C₄	C₂²×Q₈	1	12	yes	—
94	C₂×C₃×C₄	C₂⁵	1	1	yes	characteristic
95	C₂×Dic₃	—	2	16	no	—
96	C₂×Dic₃	—	4	48	no	—
97	Dic₆	—	4	64	no	—
98	C₃×C₈	C₂²×Q₈	1	16	yes	complemented
99	C₂⁴	C₂×D₁₂	1	1	yes	characteristic
100	C₂×Q₈	—	6	32	no	—
101	C₂×Q₈	—	12	96	no	—
102	C₂²×C₄	—	6	24	no	—
103	C₂²×C₄	—	12	16	no	—
104	C₂²×C₄	D₂₄	1	8	yes	—
105	C₂²×C₄	Dic₁₂	1	8	yes	—
106	C₂²×C₄	C₂×Dic₆	1	8	yes	—
107	C₂²×C₄	C₂×D₁₂	1	3	yes	—
108	C₂²×C₄	C₂×D₁₂	1	6	yes	—
109	C₂²×C₄	C₂×D₁₂	1	6	yes	—
110	C₂²×C₄	C₂³×D₃	1	3	yes	—
111	C₂×C₈	C₂×Dic₆	1	48	yes	complemented
112	C₂×C₈	C₂³×D₃	1	8	yes	—
113	C₄⋊C₄	—	3	16	no	—
114	C₄⋊C₄	—	6	24	no	—
115	C₄⋊C₄	—	6	32	no	—
116	C₄²	C₂×D₁₂	1	12	yes	—
117	C₄²	C₂³×D₃	1	4	yes	—
118	C₂²×C₃	#500	1	16	yes	complemented
119	C₂²×C₃	C₂×C₄⋊Q₈	1	6	yes	—
120	C₂²×C₃	C₂²×D₈	1	6	yes	—
121	C₂²×C₃	C₂²×Q₁₆	1	6	yes	—
122	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
123	C₃×C₄	C₂×C₄⋊Q₈	1	6	yes	—
124	C₃×C₄	C₂²×D₈	1	8	yes	—
125	C₃×C₄	C₂²×Q₁₆	1	8	yes	—
126	C₃×C₄	C₂³×Q₈	1	2	yes	—
127	Dic₃	—	4	16	no	—
128	C₂³	#869	1	4	yes	—
129	C₂³	C₂×D₂₄	1	4	yes	—
130	C₂³	C₂×Dic₁₂	1	4	yes	—
131	C₂³	C₂²×D₁₂	1	3	yes	—
132	Q₈	—	12	64	no	—
133	C₂×C₄	—	6	16	no	—
134	C₂×C₄	—	12	48	no	—
135	C₂×C₄	#869	1	12	yes	—
136	C₂×C₄	C₂×D₂₄	1	24	yes	—
137	C₂×C₄	C₂×Dic₁₂	1	24	yes	—
138	C₂×C₄	C₂²×Dic₆	1	12	yes	—
139	C₂×C₄	C₂²×D₁₂	1	3	yes	—
140	C₂×C₄	C₂²×D₁₂	1	4	yes	—
141	C₂×C₄	C₂²×D₁₂	1	4	yes	—
142	C₂×C₄	C₂⁴×D₃	1	1	yes	characteristic
143	C₈	C₂²×Dic₆	1	16	yes	complemented
144	C₂×C₃	C₂×#500	1	12	yes	complemented
145	C₂×C₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
146	C₂×C₃	C₂³×D₈	1	1	yes	characteristic
147	C₂×C₃	C₂³×Q₁₆	1	1	yes	characteristic
148	C₂²	#4682	1	16	yes	direct product factor
149	C₂²	C₂×#869	1	6	yes	—
150	C₂²	C₂²×D₂₄	1	6	yes	—
151	C₂²	C₂²×Dic₁₂	1	6	yes	—
152	C₂²	C₂³×D₁₂	1	1	yes	characteristic
153	C₄	—	12	16	no	—
154	C₄	C₂×#869	1	6	yes	—
155	C₄	C₂²×D₂₄	1	8	yes	—
156	C₄	C₂²×Dic₁₂	1	8	yes	—
157	C₄	C₂³×Dic₆	1	2	yes	—
158	C₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
159	C₂	C₂×#4682	1	12	yes	direct product factor
160	C₂	C₂²×#869	1	1	yes	characteristic
161	C₂	C₂³×D₂₄	1	1	yes	characteristic
162	C₂	C₂³×Dic₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle, prime radical	C₂⁴×C₃	C₂×D₄
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₂²×#500
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₂×#4682	C₂	direct (N × H)
15. #4682	C₂²	direct (N × H)
26. C₂²×C₃×C₈	Q₈	semidirect
59. C₂×C₃×C₈	C₂×Q₈	semidirect
72. C₂²×C₈	Dic₆	semidirect
98. C₃×C₈	C₂²×Q₈	semidirect
111. C₂×C₈	C₂×Dic₆	semidirect
118. C₂²×C₃	#500	semidirect
143. C₈	C₂²×Dic₆	semidirect
144. C₂×C₃	C₂×#500	semidirect
148. C₂²	#4682	direct (N × H)
158. C₃	C₂²×#500	semidirect
159. C₂	C₂×#4682	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×#869	2
C₂²×C₃×C₄×C₈	2
C₂²×#818	2
C₂×#4682	2
C₂²×#500	3

Subgroups — C22×#4682 #11664785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#4682 #11664785